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Multiplicative Operations on Cartesian Frames

3 ноября, 2020 - 22:27
Published on November 3, 2020 7:27 PM GMT

This is the seventh post in the Cartesian frames sequence.

Here, we introduce three new binary operations on Cartesian frames, and discuss their properties.

 

1. Tensor

Our first multiplicative operation is the tensor product, ⊗.mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0} .MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0} .mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table} .mjx-full-width {text-align: center; display: table-cell!important; width: 10000em} .mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0} .mjx-math * {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left} .mjx-numerator {display: block; text-align: center} .mjx-denominator {display: block; 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One way we can visualize our additive operations from before, ⊕ and &, is to imagine two robots (say, a mining robot Agent(C) and a drilling robot Agent(D)) that have an override mode allowing an AI supervisor to take over that robot's decisions.

  • C⊕D represents the supervisor deciding which robot to take control of, then selecting that robot's action. (The other robot continues to run autonomously.)
  • C&D represents something in the supervisor's environment (e.g., its human operator) deciding which robot the supervisor will take control of. Then the supervisor selects that robot's action (while the other robot runs autonomously).

C⊗D represents an AI supervisor that controls both robots simultaneously. This lets Agent(C⊗D) direct Agent(C) and Agent(D) to work together as a team.

Definition: Let C=(A,E,⋅) and D=(B,F,⋆) be Cartesian frames over W. The tensor product of C and D, written C⊗D, is given by C⊗D=(A×B,hom(C,D∗),⋄), where hom(C,D∗) is the set of morphisms (g,h):C→D∗ (i.e., the set of all pairs (g:A→F,h:B→E) such that b⋆g(a)=a⋅h(b) for all a∈A, b∈B), and ⋄ is given by (a,b)⋄(g,h)=b⋆g(a)=a⋅h(b).

Let us meditate for a moment on why this definition represents two agents working together on a team. The following will be very informal.

Let Alice be an agent with Cartesian frame C=(A,E,⋅), and let Bob be an agent with Cartesian frame D=(B,F,⋆). The team consisting of Alice and Bob should have agent A×B, since the team's choices consist of deciding what Alice does and also deciding what Bob does.

The environment is a bit more complicated. Starting from Alice, to construct the team, we want to internalize Bob's choices: instead of just being choices in A's environment, Bob's choices will now be additional options for the team A×B.

To do this, we want to first see Bob as being embedded in Alice's environment. This embedding is given by a function h:B→E, which extends each b∈B to a full environment e∈E. We will view Alice's possible environments as being constructed by combining a choice by Bob (that is, a b∈B) with a function from Bob's choices to possible environments (h:B→E). Then, we will move the B part across the Cartesian boundary into the agent.

Now, the agent looks like A×B, while the environment looks like B→E. However, we must have been able to do this starting from Bob as well, so a possible environment can also be viewed as function g:A→F.

Since we should get the same world regardless of whether we think of the team as starting with Alice or with Bob, these functions g and h should agree with each other. This looks a bit like currying. The environment for an Alice-Bob team should be able to take in a Bob to create an environment for Alice, and it should also be able to take in an Alice to create an environment for Bob.

 

1.1. Example

We will illustrate this new operation using a simple formal example.

Jack, Kate, and Luke are simultaneously casting votes on whether to have a party. Each agent can vote for or against the party. The possible worlds are encoded as strings listing which people vote for the party, W={ε,J,K,L,JK,JL,KL,JKL}. Jack's perspective is given by the frame

CJ=(JJKJLJKLεKLKL),

Kate's perspective is given by the frame

CK=(KJKKLJKLεJLJL),

and Luke's perspective is given by the frame

CL=(LJLKLJKLεJKJK).

Since Luke's environment can be thought of as the team consisting of Jack and Kate, one might expect that CJ⊗CK≅C∗L. Indeed, we will show this is the case.

Let CJ=(A,E,⋅), and let CK=(B,F,⋆). We label the elements of A, E, B, and F as follows:

CJ=eεeKeLeKLaJaε(JJKJLJKLεKLKL), and CK=fεfJfLfJLbKbε(KJKKLJKLεJLJL).

We will first enumerate all of the morphisms from CJ to C∗K. A morphism (g,h):CJ→C∗K consists of a function g:A→F and a function h:B→E. There are 16 functions from A to F and 16 functions from B to E, but most of the 256 pairs do not form morphisms.

Let us break the possibilities into cases based on g(aJ). Observe that bK⋆g(aJ)=aJ⋅h(bK): the possible worlds where (from Kate's perspective) Kate votes for the party and Jake-interfacing-with-Kate's-perspective votes for the party too, are the same as the possible worlds where (from Jake's perspective) Jake votes for the party and Kate-interfacing-with-Jake's-perspective does too. These possible worlds must have a J in them, so g(aJ) must be either fJ or fJL.

If g(aJ)=fJ, then

aJ⋅h(bK)=bK⋆g(aJ)=JK,

so h(bK)=eK. Similarly,

aJ⋅h(bε)=bε⋆g(aJ)=J,

so h(bε)=eε, and

bK⋆g(aε)=aε⋅h(bK)=K,

so g(aε)=fε.

Similarly, if g(aJ)=fJL, then h(bK)=eKL, h(bε)=eL, and g(aε)=fL.

Thus, there are only two candidate morphisms:

  • The first, which we will call ϕε=(gε,hε), is given by gε(aε)=fε, gε(aJ)=fJ, hε(bε)=eε, and hε(bK)=eK.
  • The second, ϕL=(gL,hL), is given by gL(aε)=fL, gL(aJ)=fJL, hL(bε)=eL, and hL(bK)=eKL.

It is easy to see that these are both indeed morphisms, by checking the definition of morphism on all four pairs in A×B.

Thus, Env(CJ⊗CK)={ϕε,ϕL}, and Agent(CJ⊗CK)=A×B, and we can compute Eval(CJ⊗CK) from the definitions of the morphisms. The result is as follows:

CJ⊗CK=ϕεϕL(aJ,bK)(aJ,bε)(aε,bK)(aε,bε)⎛⎜ ⎜ ⎜⎝JKJKLJJLKKLεL⎞⎟ ⎟ ⎟⎠.

This is clearly C∗L, up to reordering and relabeling rows and columns.

 

2. Properties of Tensor

Tensor introduces a lot of categorical structure to Chu spaces, in fact giving us a star-autonomous category. This post and the ones to come will be ignoring connections to larger topics in category theory, but only because my time and my familiarity with category theory are limited, not because these connections are unimportant.

I encourage the interested reader to learn more about the structure of Chu spaces on the excellent category theory wiki nLab, beginning with their article on the Chu construction.

 

2.1. Commutativity, Associativity, and Identity

Claim: ⊗ is commutative and associative, and 1 is the identity of ⊗ (up to isomorphism).

Proof: Commutativity is clear from the definition of ⊗, once one unpacks the definition of hom(C,D∗).

To see that 1 is the identity of ⊗, let C=(A,E,⋅), let 1=({b},W,⋆), and let C⊗1=(A×{b},hom(C,1∗),⋄).

Consider the isomorphism (ι0,ι1):C→C⊗1 given by ι0(a)=(a,b) and ι1(g,h)=h(b). We need to show that (ι0,ι1) is a morphism, and that both ι0 and ι1 are bijective. To see that (ι0,ι1) is a morphism, observe that for all a∈A and (g,h):C→1∗,

ι0(a)⋄(g,h)=a⋅h(b)=a⋅ι1(g,h).

 Clearly, ι0 is a bijection, so all that remains to show is that ι1 is bijective.

To see that ι1 is injective, observe that if ι1(g0,h0)=ι1(g1,h1), then h0(b)=h1(b), so h0=h1, and

g0(a)=b⋆g0(a)=a⋅h0(b)=a⋅h1(b)=b⋆g1(a)=g1(a)

for all a∈A, so g0=g1.

To see that ι1 is surjective, observe that for every e∈E, there exists a morphism (ge,he):C→1∗, given by he(b)=e and ge(a)=a⋅e. This is clearly a morphism, since 

b⋆ge(a)=ge(a)=a⋅e=a⋅he(b),

and ι1(ge,he)=e.

Next, we need to show that ⊗ is associative, which will be much more tedious. Let Ci=(Ai,Ei,⋅). Since we have already established commutativity, it suffices to show that (C0⊗C1)⊗C2≅(C0⊗C2)⊗C1.

Let D=(A0×A1×A2,F,⋆), where F is the set of all triples of functions (g0:A1×A2→E0,g1:A0×A2→E1,g2:A0×A1→E2), such that for all ai∈Ai, we have

a0⋅0g0(a1,a2)=a1⋅1g1(a0,a2)=a2⋅2g2(a0,a1),

 and ⋆ is given by

(a0,a1,a2)⋆(g0,g1,g2)=a0⋅0g0(a1,a2)=a1⋅1g1(a0,a2)=a2⋅2g2(a0,a1).

We will show that (C0⊗C1)⊗C2≅D, and since the definition of D is symmetric in swapping C1 and C2, it will follow that (C0⊗C1)⊗C2≅D, so (C0⊗C1)⊗C2≅(C0⊗C2)⊗C1.

We construct a morphism (ι0,ι1) from (C0⊗C1)⊗C2 to D as follows. ι0 is just the identity on A0×A1×A2. We will let ι1(g0,g1,g2) be the morphism (g2,h):C0⊗C1→C∗2, where h:A2→hom(C0,C∗1) is given by h(a2)=(ha20,ha21):C0→C∗1, where ha20(a0)=g1(a0,a2), and ha21(a1)=g0(a1,a2).

First, we need to show that ι1 is well-defined, by showing that h(a2) is a morphism from C0 to C∗1, and that (g2,h) is a morphism from C0⊗C1→C∗2. To see that h(a2)=(ha20,ha21) is a morphism, observe that for a0∈A0 and a1∈A1,

a1⋅1ha20(a0)=a1⋅1g1(a0,a2)=a0⋅0g0(a1,a2)=a0⋅0ha21(a1).

To see that (g2,h) is a morphism, observe for all (a0,a1)∈A0×A1 and all a2∈A2,

a2⋅2g2(a0,a1)=a0⋅0g0(a1,a2)=a0⋅0ha21(a1)=(a0,a1)⋄(ha20,ha21)=(a0,a1)⋄h(a2),

where ⋄=Eval(C0⊗C1).

Now that we know ι1 is well-defined, we need to show that (ι0,ι1) is a morphism. Indeed, for all (a0,a1,a2)∈A0,A1,A2, and for all (g0,g1,g2)∈F, we have

ι0(a0,a1,a2)⋆(g0,g1,g2)=a2⋅2g2(a0,a1)=(a0,a1,a2)∙(g2,h)=(a0,a1,a2)∙ι1(g0,g1,g2),

where ∙=Eval((C0⊗C1)⊗C2).

Finally, to show that (ι0,ι1) is an isomorphism, we need to show that ι0 and ι1 are bijective. ι0 is trivial, since it is the identity, so it suffices to show that ι1 is bijective.

To see that ι1 is surjective, let (g,h) be a morphism from C0⊗C1 to C∗2, so g:A0×A1→E2, and h:A2→hom(C0,C∗1). Again, let h(a2)=(ha20,ha21). We will define (g0,g1,g2) by g2=g, g1(a0,a2)=ha20(a0), and g0(a1,a2)=ha21(a1).

We need to show that (g0,g1,g2)∈F, by showing that for all (a0,a1,a2)∈A0×A1×A2, we have

a0⋅0g0(a1,a2)=a1⋅1g1(a0,a2)=a2⋅2g2(a0,a1).

Observe that since (g,h) is a morphism,

a2⋅2g2(a0,a1)=a2⋅2g(a0,a1)=(a0,a1)⋆h(a2)=(a0,a1)⋆(ha20,ha21),

where ⋆=Eval(C0⊗C1). Also, by the definition of C0⊗C1, we have that

(a0,a1)⋆(ha20,ha21)=a0⋅0ha21(a1)=a0⋅0g0(a1,a2),

and similarly

(a0,a1)⋆(ha20,ha21)=a1⋅1ha20(a1)=a1⋅1g1(a0,a2).

Thus,

a0⋅0g0(a1,a2)=a1⋅1g1(a0,a2)=a2⋅2g2(a0,a1),

so (g0,g1,g2)∈F. Finally, observe that ι1(g0,g1,g2) is in fact (g,h).

To show that ι1 is injective, assume ι1(g0,g1,g2)=ι1(g′0,g′1,g′2)=(g,h), and given an a2∈A2, let h(a2)=(ha20,ha21). Clearly, this means g2=g=g′2. Further, for all a0∈A0, a1∈A1, and a2∈A2,

g0(a1,a2)=ha21(a1)=g′0(a1,a2)

and

g1(a0,a2)=ha20(a0)=g′1(a0,a2).

Thus (g0,g1,g2)=(g′0,g′1,g′2). Thus, ι1 is bijective, so (ι0,ι1) is an isomorphism, so (C0⊗C1)⊗C2≅D≅(C0⊗C2)⊗C1. □

 

2.2. Biextensional Equivalence

Since many of our intuitions about Cartesian frames are up to biextensional equivalence, we should verify that tensor is well-defined up to biextensional equivalence.

Claim: If C0≃C1 and D0≃D1, then C0⊗D0≃C1⊗D1.

Proof: It suffices to show that for all D, C0⊗D≃C1⊗D. Then, by commutativity of tensor, 

C0⊗D0≃C0⊗D1≅D1⊗C0≃D1⊗C1≡C1⊗D1.

Let Ci=(Ai,Ei,⋅i), and let D=(B,F,⋆). Since C0≃C1, there must exist morphisms (g0,h0):C0→C1 and (g1,h1):C1→C0 such that (g1∘g0,idE0):C0→C0 and (g0∘g1,idE1):C1→C1 are both morphisms.

Let Ci⊗D=(Ai×B,hom(Ci,D∗),⋄i). Consider the morphisms (g′i,h′i):Ci⊗D→C1−i⊗D, where g′i:Ai×B→A1−i×B is given by g′i(a,b)=(gi(a),b) and h′i:hom(C1−i,D∗)→hom(CI,D∗) is given by h′i(g,h)=(g,h)∘(gi,hi).

To see that these are morphisms, observe that for any (a,b)∈Ai×B and (g,h):C1−i→D∗, we have

g′i(a,b)⋄1−i(g,h)=(gi(a),b)⋄1−i(g,h)=b⋆g(gi(a))=b⋆(g∘gi)(a))=(a,b)⋄i(g∘gi,hi∘h)=(a,b)⋄ih′i(g,h).

Finally, we need to show that (g′0,h′0) and (g′1,h′1) compose to something homotopic to the identity in both orders. This is equivalent to saying that (g′0∘g′1,idhom(C1,D∗)) and (g′1∘g′0,idhom(C0,D∗)) are both morphisms. Indeed, for all (a,b)∈Ai×B and (g,h):Ci→D∗, since (g1−i∘gi,idEi) is a morphism, we have

g′1−i(g′i(a,b))⋄i(g,h)=(g1−i(gi(a)),b)⋄i(g,h)=g1−i(gi(a))⋅ih(b)=a⋅ih(b)=(a,b)⋄i(g,h).

 

2.3. Distributivity

Claim: ⊗ distributes over ⊕, so for all Cartesian frames C0, C1, and D,(C0⊕C1)⊗D≅(C0⊗D)⊕(C1⊗D).

Proof: Since ⊕ is the categorical coproduct, there exist morphisms ι0:C0→C0⊕C1 and ι1:C1→C0⊕C1 such that for any morphisms ϕ0:C0→D∗ and ϕ1:C1→D∗, there exists a unique morphism ϕ:C0⊗C1→D∗such that ϕi=ϕ∘ιi.

Let Ci=(Ai,Ei,⋅i), and let D=(B,F,⋆). Consider the isomorphism (g,h):(C0⊗D)⊕(C1⊗D)→(C0⊕C1)⊗D, where g:(A0×B)⊔(A1×B)→(A0⊔Ai)×B is the natural bijection that sends (a,b) to (a,b), and h:hom(C0⊕C1,D∗)→hom(C0,D∗)×hom(C1,D∗) is given by h(ϕ)=(ϕ∘ι0,ϕ∘ι1).

Clearly, g is an bijection. h is also a bijection, since it is inverse to the function that sends (ϕ0,ϕ1) to the unique ϕ as above. Thus, all that remains to show is that (g,h) is a morphism.

Let ⋄=Eval((C0⊗D)⊕(C1⊗D)) and let ∙=Eval((C0⊕C1)⊗D). Given (a,b)∈(A0×B)⊔(A1×B) and (g′,h′)∈hom(C0⊕C1,D∗), without loss of generality, assume that a∈A0. Let (g′0,h′0)=(g′,h′)∘ι0. Observe that since the function on agents in ι0 is the inclusion of A0 into A0⊔A1, we have that g′0 is g′ restricted to A0. Thus, we have

g(a,b)∙(g′,h′)=(a,b)∙(g′,h′)=b⋆g′(a)=b⋆g′0(a)=(a,b)⋄(g′0,h′0)=(a,b)⋄h(g′,h′).

 

2.4. Tensor is for Disjoint Agents

It doesn't really make sense to talk about C⊗D when C and D's agents are the same agent, or otherwise overlap. This is because C⊗D's agent can make choices for both C and D, and if C and D overlap, C⊗D's agent could make choices for the intersection in two contradictory ways.

If you try to take the tensor of two frames whose agents overlap, you get a frame with an agent but no possible worlds.

Claim: If Ensure(C)∩Prevent(D) is nonempty, then C⊗D≃⊤.

Proof: Let C=(A,E,⋅), and let D=(B,F,⋆). Consider some S∈Ensure(C)∩Prevent(D). There is some a∈A such that a⋅e∈S for all e∈E, and some b∈B such that b⋆f∉S for all f∈F. First, observe that Agent(C⊗D) is nonempty, since it contains (a,b). Next, observe that Env(C⊗D) is empty, since if there were a morphism (g,h):C→D∗, it would need to satisfy b⋆g(a)=a⋅h(b), which is impossible since the left hand side is not in S, while the right hand side is S. Thus, C⊗D has empty environment and nonempty agent, so C⊗D≃⊤. □

Tensoring an agent with itself lets you play "both" agents, which has the neat consequence that if the agent has any control, you can have the agent make two different choices that put you in two different possible worlds, which is a contradiction. The result is that the agent has no possible worlds.

Corollary: If Ctrl(C) is nonempty, then C⊗C≃⊤.

Proof: Trivial. □

 

3. Tensor is Relative to a Coarse World Model

Recall that for any function p:W→V, the functor p∘:Chu(W)→Chu(V) preserves sums and products, meaning that for any Cartesian frames C and D over W, p∘(C⊕D)=p∘(C)⊕p∘(D) and p∘(C&D)=p∘(C)&p∘(D). However, the same is not true for ⊗. To see this, let's go back to the voting example above.

Let's assume that Jack, Kate, and Luke have a party if and only if a majority vote in favor, and let V={Y,N} be the two-element world that only tracks whether or not they have a party. Let p:W→V be the function such that p(ε)=p(J)=p(K)=p(L)=N and p(JK)=p(JL)=p(KL)=p(JKL)=Y. Then, 

p∘(CJ)≅p∘(CK)≅(NYYYNNNY)≃(NYYNNY),

and

p∘(CJ⊗CK)≅p∘(C∗L)≅⎛⎜ ⎜ ⎜⎝YYNYNYNN⎞⎟ ⎟ ⎟⎠≃⎛⎜⎝YYNYNN⎞⎟⎠,

but

(NYYYNNNY)⊗(NYYYNNNY)/≄⎛⎜ ⎜ ⎜⎝YYNYNYNN⎞⎟ ⎟ ⎟⎠.

We can see that p∘(CJ⊗CK) is not equivalent to p∘(CJ)⊗p∘(CK) by observing that the latter has a constant N environment while the former doesn't.

Let p∘(CJ)≅p∘(CK)≅(A,E,⋅), and let eN∈E denote the environment such that a⋅eN=N for both a∈A. (In the matrix representation above, this is the first column.) Observe that there exists a morphism (g,h):(A,E,⋅)→(A,E,⋅)∗, where g and h are both the constant eN function. This is a morphism because for all a0,a1∈A, a0⋅h(a1)=a1⋅g(a0)=N. This gives an environment in  p∘(CJ)⊗p∘(CK), all of whose entries must be N. p∘(CJ⊗CK) has no such environment, so p∘(CJ⊗CK) cannot be isomorphic to p∘(CJ)⊗p∘(CK), or even biextensionally equivalent. Indeed:

p∘(CJ⊗CK)≃⎛⎜ ⎜ ⎜⎝NYYYYYNNNYYYNNYNYYNNNNNY⎞⎟ ⎟ ⎟⎠.

To see what is going on here, consider another example where Jack and Kate and Luke vote on whether to have a party, but whether or not the party happens is not just a function of the majority's vote. Instead, after the three people cast their votes, a coin is flipped:

  • If heads, the votes are tallied and majority wins as normal.
  • If tails, one of the three voters is selected at random to be dictator, and the party happens if and only if they voted in favor.

Let us work up to biextensional collapse. Let DJ be the Cartesian frame over V representing Jack's perspective. We have

DJ≃(NYYNNY),

where the top row represents voting for the party, and the bottom row represents voting against.

The first column represents environments where the party does not happen and Jack's vote didn't matter—either the coin came up heads and the others both voted against, or Kate or Luke became dictator and voted against. The third column similarly represents outcomes where the party happens regardless of how Jack votes. The second column represents all environments in which Jack's vote matters, so either he is dictator, or Kate and Luke's votes were split.

Similarly, let DK be the Cartesian frame over V representing Kate's perspective,

DK≃(NYYNNY).

Then,

DJ⊗DK≃⎛⎜ ⎜ ⎜⎝NYYYYYNNNYYYNNYNYYNNNNNY⎞⎟ ⎟ ⎟⎠.

The rows represent, in order: both voting in favor; Jack voting in favor but Kate voting against; Kate voting in favor but Jack voting against; and both voting against.

The columns represent, in order: Luke is dictator and votes against; majority rules and Luke votes against; Kate is dictator; Jack is dictator; majority rules and Luke votes in favor; and Luke is dictator and votes in favor.

Here, DJ⊗DK looks more like what we would expect Jack and Kate working together on a team to look like. However, up to biextensional equivalence, DJ and DK are the same as p∘(CJ) and p∘(CK).

When we forget the actual votes and only look at whether the party happens, then up to biextensional collapse, the Cartesian frame representing Jack's perspective no longer has any way to distinguish between the simple majority rule vote and the complicated voting system with coins and dictators.

In general, just looking at two Cartesian frames does not tell you all of the information about the relationships between the people we might be using the frames to model. The Cartesian frames over V representing Jack and Kate's perspectives do not have any information that distinguishes between the two vote counting schemes.

When taking a tensor, we automatically include all of the possible ways the two agents can embed in each other's environments, even if a given embedding doesn't make sense in a given interpretation.

 

4. Par

Our next multiplicative operation is ⅋ , which is pronounced "par."

Definition: Let C=(A,E,⋅) and D=(B,F,⋆) be Cartesian frames over W. C⅋ D=(hom(C∗,D),E×F,⋄), where (g,h)⋄(e,f)=g(e)⋆f=h(f)⋅e.

Claim: ⅋  is De Morgan dual to ⊗, so C⅋ D=(C∗⊗D∗)∗.

Proof: Trivial. □

⅋  has much less of an intuitive interpretation than ⊗. One reason for this is that in order to par two agents together, they have to be large enough that each other's environments embed within them. If C and D are not large enough, we will have that C⅋ D≃0. (I am being informal with the word "large" here.)

One way that C and D can fail to be large enough is if Ensure(C∗)∩Prevent(D∗) is nonempty, which is dual to the above result about tensor being for disjoint agents. It is actually pretty difficult for C and D to be large enough. If there is any fact about the world that is determined outside of both agents, C⅋ D will be trivial.

We had a dual restriction for ⊗, but it didn't get in the way nearly as often: simple intuitive examples tend to be about small agents interacting with a large environment, so it is easy to imagine two agents that are disjoint. It is much harder to imagine simple examples of two agents that cover, which (informally) is what you would have to have for ⅋  to be nontrivial.

I expect to not use ⅋  very often, but I am including it here for completeness.

Claim: ⅋  is commutative and associative, and ⊥ is the identity of ⅋  (up to isomorphism).

Proof: Trivial from the fact that ⅋  is De Morgan dual to ⊗ and 1∗≅⊥. □

Claim: If C0≃C1 and D0≃D1, then C0⅋ D0≃C1⅋ D1.

Proof: Trivial from the fact that ⅋  is De Morgan dual to ⊗, and ≃ is preserved by −∗. □

Claim: ⅋  distributes over &, so for all Cartesian frames C0, C1, and D, we have (C0&C1)⅋ D≅(C0⅋ D)&(C1⅋ D).

Proof: Trivial from the fact that ⅋  is De Morgan dual to ⊗, and & is De Morgan dual to ⊕. □

 

5. Lollipop

We have one more operation to introduce, ⊸ (pronounced "lollipop"), which is a Cartesian frame that can be thought of as representing the collection of morphisms between two Cartesian frames.

Definition: Given two Cartesian frames over W, C=(A,E,⋅) and D=(B,F,⋆), we let C⊸D denote the Cartesian frame C⊸D=(hom(C,D),A×F,⋄), where ⋄ is given by (g,h)⋄(a,f)=g(a)⋆f=a⋅h(f).

One way to interpret C⊸D is as "D with a C-shaped hole in it." Indeed, let us think about Agent(C⊸D). and Env(C⊸D) separately. 

Agent(C⊸D)=hom(C,D) is the collection of morphisms from C to D. Morphisms from C to D are exactly interfaces through which the agent of C can interact with the environment of D. We can also think of this as the collection of interfaces that allow the agent of C to fill the role of the agent of D. This makes sense. The collection of ways that a "D with a C-shaped hole in it" can be is exactly the collection of interfaces that allow us to get a possible agent of D from a possible agent of C.

Similarly, Env(C⊸D)=A×F makes sense as the environment of a "D with a C-shaped hole in it." The environment needs to supply an environment for D, and also fill in the hole with an agent for C.

Previously, C's agent might have been part of D's agent; in C⊸D, however, this part of D gets moved into the environment.

Imagine a football team D with one team member, C, removed—the team with a football-player-shaped hole in it. Its environment, naturally, is pairs of "the kind of environment you get for a football team" and "the removed teammate".

Lollipop can be easily constructed from our other operations.

Claim: C⊸D≅C∗⅋ D≅(C⊗D∗)∗.

Proof: Trivial. □

Lollipop is well-defined up to biextensional equivalence.

Claim: If C0≃C1 and D0≃D1, then C0⊸D0≃C1⊸D1.

Proof: Trivial. □

Lollipop also has some identity-like properties.

Claim: For all Cartesian Frames C, C≃1⊸C and C∗≃C⊸⊥.

Proof: 1⊸C≅(1⊗C∗)∗≅C∗∗≅C and C⊸⊥≅(C⊗1)∗≅C∗. □

This last result is especially interesting because we can actually think of C⊸⊥ as an alternative definition for C∗.

 

In "Tensor is Relative to a Coarse World Model" above, we noted that two agents working together might sometimes have strictly fewer possible environments than show up in the tensor. In the next post, we will introduce the concept of a sub-tensor, which allows us to represent teams that have fewer possible environments than the tensor. Similarly, sub-sum will be sum with spurious possible environments removed.

I'll be hosting online office hours this Sunday at 2-4pm PT for discussing Cartesian frames.



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Sleeping Julia: Empirical support for thirder argument in the Sleeping Beauty Problem

3 ноября, 2020 - 03:21
Published on November 3, 2020 12:21 AM GMT

I've created an emulation of the Sleeping Beauty Problem in the Julia programming language which supports the thirder solution.

For those unfamiliar with the problem, I recommend this explanation by Julia Galef: https://www.youtube.com/watch?v=zL52lG6aNIY

In this explanation, I'll briefly explain the current situation with regard to this problem's status in academia, how the emulation works, and how we can formalize the intuitions gleaned from this experiment. Let's start with the code.

Originally I wrote this in Julia (hence the name), and that code can be found on GitHub: https://github.com/seisvelas/SleepingJulia/blob/main/sleeping.jl.ipynb

Here I'll do the same thing, but in Python, as that language is likely grokked by a broader audience of LessWrong readers. First, I create a class to run the experiment and track the state of various sleeping beauty experiments:

import random class SleepingBeautyExperiment: def __init__(self): self.wakeups = 0 self.bets = { 'heads' : { 'win' : 0, 'loss' : 0}, 'tails' : { 'win' : 0, 'loss' : 0}, } def run(self, bet): coin = ('heads', 'tails') coin_toss = random.choice(coin) win_or_loss = 'win' if coin_toss == bet else 'loss' self.bets[bet][win_or_loss] += 1 # Tuesday, in case of tails if coin_toss == 'tails': self.bets[bet][win_or_loss] += 1 def repeat(self, bet, times): for i in range(times): self.run(bet) def reset(self): self.__init__()

I apologize for the lack of code highlighting. I tried to write code that self-documents as much as possible, but if I failed, just leave a comment and I'll clarify to the best of my ability. The key observation is that in the case of tails, we wake SB twice. Ie, for every 100 experiments, there will be 150 wakeups. We don't care how many whole experiments SB summarily wins (if we did, though, the halfer interpretation would be the correct one!).

Let's see the code in action:

>>> heads_wins = Sb.bets['heads']['win'] >>> heads_losses = Sb.bets['heads']['loss'] >>> heads_wins / (heads_wins + heads_losses) 0.33378682085242317 >>> # As percent rounded to two decimal places: >>> int(heads_wins / (heads_wins + heads_losses) * 10000) / 100 33.37

There ya go, a nice right dose of thirderism. In the lay rationality community the SB problem is often treated as an open debate, with larger and smaller menshevik and bolshevik factions, but this has not been the case for some time. I made these emulations originally to prove halferism before reading up on academic work from decision theorists such as Silvia Milano's wonderful Bayesian Beauty paper. In academia, SB problem is resoundingly considered solved.

For a lay level overview of the situation, we can do some Bayesian mini-algebra to simply summarize what halfers get wrong.

A = heads
B = SB awoken

Halfers believe these priors:

P(B | A) = 1
P(A ) = 1/2
P(B) = 1

Therefore, following Bayes' theorem:

P(A | B) = (P(B | A) * P(A)) / P(B) = (1 * 1/2) / 1 = 1/2

But for every tails flip, SB is awoken twice (once on Monday then again on Tuesday), so the probable number of wakeups per experiment is 1.5, therefore P(B) = 1.5. If we run the math again with this new prior:

P(A | B) = (P(B | A) * P(A)) / P(B) = (1 * 1/2) / 1.5 = .5 / 1.5 = 1/3
QED

Readers learning about this problem from the rationality community or Wikipedia are given an outdated sense of the problem's openness. Perhaps some enterprising spirit would like to review the academic literature and give Wikipedia's article a makeover. I'll do it one day, if no one else beats me to it.



Discuss

Subagents of Cartesian Frames

3 ноября, 2020 - 01:02
Published on November 2, 2020 10:02 PM GMT

Here, we introduce and discuss the concept of a subagent in the Cartesian Frames paradigm.

Note that in this post, as in much of the sequence, we are generally working up to biextensional equivalence. In the discussion, when we informally say that a frame has some property or is some object, what we'll generally mean is that this is true of its biextensional equivalence class.

 

1. Definitions of Subagent

1.1. Categorical Definition

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src: local('MathJax_Vector Bold'), local('MathJax_Vector-Bold')} @font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax_Vector'); font-weight: bold} @font-face {font-family: MJXc-TeX-vec-Bw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Vector-Bold.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Vector-Bold.otf') format('opentype')}  and D be Cartesian frames over W. We say that C's agent is a subagent of D's agent, written C◃D, if for every morphism ϕ:C→⊥ there exists a pair of morphisms ϕ0:C→D and ϕ1:D→⊥ such that ϕ=ϕ1∘ϕ0.

Colloquially, we say that every morphism from C to ⊥ factors through D. As a shorthand for "C's agent is a subagent of D's agent," we will just say "C is a subagent of D."

At a glance, it probably isn't clear what this definition has to do with subagents. We'll first talk philosophically about what we mean by "subagent", and then give an alternate definition that will make the connection more clear.

When I say "subagent," I am actually generalizing over two different relationships that may not immediately seem like they belong together.

First, there is the relationship between the component and the whole. One football player is a subagent of the entire football team.

Second, there is the relationship between an agent before and after making a precommitment or a choice. When I precommit not to take a certain action, I am effectively replacing myself with a weaker agent that has fewer options. The new agent with the commitment is a subagent of the original agent.

These are the two notions I am trying to capture with the word "subagent". I am making the philosophical claim that we should think of them primarily as one concept, and am partially backing up this claim by pointing to the simplicity of the above definition. In a future post, we will discuss the formal differences between these two kinds of subagent, but I think it is best to view them as two special cases of the one simple concept.

(My early drafts of the "Embedded Agency" sequence used the word "subagent" in the title for both the Subsystem Alignment and Robust Delegation sections.)

 

1.2. Currying Definition

Definition: Let C and D be Cartesian frames over W. We say that C◃D if there exists a Cartesian frame Z over Agent(D) such that C≃D∘(Z).

Assume for this discussion that we only care about frames up to biextensional equivalence. In effect, the above definition is saying that "C is a subagent of D" means "C's agent is playing a game, Z, where the stakes are to help decide what D's agent does." (And this game may or may not have multiple players, and may or may not fully cover all the options of D's agent.)

Letting C=(A,E,⋅) and D=(B,F,⋆), it turns out (as we will see later) that we can explicitly construct Z. Z=(A,X,⋄), where X is the set of all morphisms from C to D, and ⋄:A×X→B is given by a⋄(g,h)=g(a).

We will later prove the categorical and currying definitions equivalent, but let's first interpret this definition using examples.

Z is a Cartesian frame whose agent is the agent of C and whose world is the agent of D. This seems like the kind thing we would have when C is a subagent of D.

Thinking about the football example: We have the football player A as the agent in a Cartesian frame C over the world W. We also have the football team B as the agent in a Cartesian frame D over the same world W.

Z is a Cartesian frame over the football team; and the agent of this frame is again the football player A. X, the environment of Z, represents the rest of the football team: the player's effect on the team as a whole (here treated as the player's world) is a function of what the player chooses and what the rest of the team chooses. We can think of Z as representing a  "zoomed-in" picture of A interacting with its local environment (the team), while C represents a "zoomed-out" picture of A interacting with its teammates and the larger world (rival teams, referees, etc.).

D∘(Z)=(A,X×F,∙), so E is equivalent to X×F, which is saying that the environment for the football player in its original frame (C) is equivalent to the Cartesian product of the rest of the team X with the team's environment F.

Thinking about the precommitment example: C has made a precommitment, so there is an inclusion morphism ι:A→B, which shows that C's agent's options are a subset of D's agent's options. Z is just CF∗({ι}), so X={ι} is a singleton. D∘(Z)=(A,X×F,∙), so E is equivalent to X×F=F, so here A is a subset of B and E is equivalent to F.

Although the word "precommitment" suggests a specific (temporal, deliberative) interpretation, formally, precommitment just looks like deleting rows from a matrix (up to biextensional equivalence), which can represent a variety of other situations.

A Cartesian frame Z=(A,X,⋄) over B is like a nondeterministic function from A to B, where X represents the the nondeterministic bits. When changing our frame from (B,F,⋆) to (A,E,⋅)≃(A,X×F,∙), we are identifying with A and externalizing the nondeterministic bits X into the environment.

 

1.3. Covering Definition

The categorical definition is optimized for elegance, while the currying definition is optimized to be easy to understand in terms of agency. We have a third definition, the covering definition, which is optimized for ease of use.

Definition. Let C=(A,E,⋅) and D=(B,F,⋆) be Cartesian frames over W. We say that C◃D if for all e∈E, there exists an f∈F and a (g,h):C→D such that e=h(f).

We call this the covering definition because the morphisms from C to D cover the set E.

 

2. Equivalence of Definitions

2.1. Equivalence of Categorical and Covering Definitions

The equivalence of the categorical and covering definitions follows directly from the fact that the morphisms from C to ⊥ are exactly the elements of Env(C).

Claim: The categorical and covering definitions of subagent are equivalent.

Proof: Let C=(A,E,⋅) and let D=(B,F,⋆). First, observe that the morphisms from C to ⊥ correspond exactly to the elements of E. For each e∈E, it is easy to see that (g,h):C→(W,{j},⋄), given by h(j)=e and g(a)=a⋅e, is a morphism, and every morphism is uniquely defined by h(j), so there are no other morphisms. Let ϕe denote the morphisms with h(j)=e.

Similarly, the morphisms from D to ⊥ correspond to the elements of F. Let ψf denote the morphisms corresponding to f∈F.

Thus, the categorical definition can be rewritten to say that for every morphism ϕe:C→⊥, there exist morphisms (g,h):C→D and ψf:D→⊥, such that ϕe=ψf∘(g,h). However, ψf∘(g,h):C→(W,{j},⋄) sends j to h(f), and so equals ϕe if and only if e=h(f). Thus the categorical definition is equivalent to the covering definition. □

 

2.2. Equivalence of Covering and Currying Definitions

Claim: The covering definition of subagent implies the currying definition of subagent.

Proof: Let C=(A,E,⋅) and D=(B,F,⋆) be Cartesian frames over W. Assume that C◃D according to the covering definition.

Let X be the set of all morphisms from C to D, and let Z=(A,X,⋄) be a Cartesian frame over B, with ⋄ given by a⋄(g,h)=g(a). We have that D∘(Z)=(A,X×F,∙), with

a∙((g,h),f)=(a⋄(g,h))⋆f=g(a)⋆f

for all a∈A, (g,h)∈X, and f∈F.

To show that C≃D∘(Z), we need to construct morphisms g0,h0:C→D∘(Z) and g1,h1:D∘(Z)→C which compose to something homotopic to the identity in both orders.

We will let g0 and g1 be the identity on A, and we let h0:X×F→E be given by h0((g,h),f)=h(f). Finally, we let h1(e)=((g,h),f) such that h(f)=e. We can always choose such a (g,h)∈X and f∈F by the covering definition of subagent.

We have that (g0,h0) is a morphism, since

g0(a)∙((g,h),f)=a∙((g,h),f)=g(a)⋆f=a⋅h(f)=a⋅h0((g,h),f).

Similarly, we have that (g1,h1) is a morphism since h1(e)=((g,h),f), where h(f)=e, so

g1(a)⋅e=a⋅e=a⋅h(f)=g(a)⋆f=a∙((g,h),f)=a∙h1(e).

It is clear that (g0,h0) and (g1,h1) compose to something homotopic to the identity in both orders, since g0 and g1 are the identity on A. Thus, C≃D∘(Z). □

Claim: The currying definition of subagent implies the covering definition of subagent.

Proof: Let C=(A,E,⋅) and D=(B,F,⋆) be Cartesian frames over W. Let Z=(Y,X,⋄) be a Cartesian frame over B, and let C≃D∘(Z). Our goal is to show that for every e∈E, there exists a (g,h):C→D and f∈F such that e=h(f). We will start with the special case where C=D∘(Z).

We have that D∘(Z)=(Y,X×F,∙), where y∙(x,f)=(y⋄x)⋆f. First, note that for every x∈X, there exists a morphism (gx,hx):D∘(Z)→D given by gx(y)=y⋄x, and gx(f)=(x,f). To see that this is a morphism, observe that

gx(y)⋆f=(y⋄x)⋆f=y∙(x,f)=f∙hx(f)

for all y∈Y and f∈F.

To show that D∘(Z)◃D according to the covering definition, we need that for all (x,f)∈X×F, there exists an f′∈F and a (g,h):D∘(Z)→D such that h(f′)=(x,f). Indeed we can take (g,h)=(gx,hx) and f′=f.

Now, we move to the case where C≃D∘(Z), but C≠D∘(Z). It suffices to show that under the covering definition of subagent, if C0◃D, and C1≃C0, then C1◃D.

Let Ci=(Ai,Ei,⋅i), and let (g0,h0):C0→C1 and (g1,h1):C1→C0 compose to something homotopic to the identity in both orders. Assume that C0◃D. To show that C1◃D, let the possible environment e∈E1 be arbitrary.

h0(e)∈E0, so there exists an f∈F and (g,h):C0→D such that h(f)=h0(e). Consider the morphism (g′,h′):C1→D, where g′=g∘g1, and h′(f)=e and h′(f′)=(h1∘h)(f′) on all f′≠f. To see that this is a morphism, observe that for all a∈A1, we have

g′(a)⋆f=g(g1(a))⋆f=a⋅1h1(h(f))=a⋅1h1(h0(e))=a⋅1e=a⋅1h′(f),

while for f′∈F, f′≠f, we have

g′(a)⋆f′=g(g1(a))⋆f′=a⋅1h1(h(f′))=a⋅1h′(f′).

Now, notice that for our arbitrary e∈E1, (g′,h′):C1→D and f∈F satisfy h′(f)=e, so C1◃D according to the to the covering definition.

Thus, whenever C≃D∘(Z), we have C◃D according to the covering definition, so the currying definition implies the covering definition of subagent. □

 

3. Mutual Subagents

The subagent relation is both transitive and reflexive. Surprisingly, this relation is not anti-symmetric, even up to biextensional equivalence.

Claim: ◃ is reflexive. Further, if C≃D, then C◃D.

Proof: Let C=(A,E,⋅) and D=(B,F,⋅) be Cartesian Frames over W, with C≃D. Consider the Cartesian frame Z over B given by Z=(B,{x},⋄), where b⋄x=b. Observe that D≅D∘(Z). Thus C≃D∘(Z), so C◃D, according to the currying definition. □

Claim: ◃ is transitive.

Proof: We will use the categorical definition. Let C0◃C1 and C1◃C2. Given a morphism, ϕ0:C0→⊥, since C0◃C1, we know that ϕ0=ϕ1∘ϕ2 with ϕ1:C1→⊥ and ϕ2:C0→C1. Further, since C1◃C2, we know that ϕ1=ϕ3∘ϕ4 with ϕ3:C2→⊥ and ϕ4:C1→C2. Thus,

ϕ0=(ϕ3∘ϕ4)∘ϕ2=ϕ3∘(ϕ4∘ϕ2),

with ϕ3:C2→⊥ and ϕ4∘ϕ2:C0→C2, so C0◃C2. □

As a corollary, we have that subagents are well-defined up to biextensional equivalence.

Corollary: If C0≃C1, D0≃D1, and C0◃D0, then C1◃D1.

Proof: C1◃C0◃D0◃D1. □

Sometimes, there are Cartesian frames C≄D with C◃D and D◃C. We can use this fact to define a third equivalence relation on Cartesian frames over W, weaker than both ≅ and ≃.

Definition: For Cartesian frames C and D over W, we say C⋈D if C◃D and D◃C.

Claim: ⋈ is an equivalence relation.

Proof: Reflexivity and transitivity follow from reflexivity and transitivity of ◃. Symmetry is trivial. □

This equivalence relation is less natural than ≅ and ≃, and is not as important. We discuss it mainly to emphasize that two frames can be mutual subagents without being biextensionally equivalent.

Claim: ⋈ is strictly weaker than ≃, which is strictly weaker than ≅.

Proof: We already know that ≃ is weaker than ≅. To see that ⋈ is weaker than ≃, observe that if C≃D, then C◃D and D◃C, so C⋈D.

To see that ≃ is strictly weaker than ≅, observe that ⊤⊕⊤≃⊤ (both have empty environment and nonempty agent), but ⊤⊕⊤≆⊤ (the agents have different size).

To see that ⋈ is strictly weaker than ≃, observe that ⊤⋈null (vacuous by covering definition), but ⊤≄null (there are no morphisms from null to ⊤). □

I do not have a simple description of exactly when C⋈D, but there are more cases than just the trivial ones like C≃D and vacuous cases like ⊤⋈null. As a quick example:

(xy)⋈⎛⎜⎝xxyyxy⎞⎟⎠.

To visualize this, imagine an agent that is given the choice between cake and pie. This agent can be viewed as a team consisting of two subagents, Alice and Bob, with Alice as the leader.

Alice has three choices. She can choose cake, she can choose pie, or she can delegate the decision to Bob. We represent this with a matrix where Bob is in Alice's environment, and the third row represents Alice letting the environment make the call:

⎛⎜⎝xxyyxy⎞⎟⎠.

If we instead treat Alice-and-Bob as a single superagent, then their interaction across the agent-environment boundary becomes agent-internal deliberation, and their functional relationship to possible worlds just becomes a matter of "What does the group decide?". Thus, Alice is a subagent of the Alice-and-Bob team:

⎛⎜⎝xxyyxy⎞⎟⎠◃(xy).

However, Alice also has the ability to commit to not delegating to Bob. This produces a future version of Alice that doesn't choose the third row. This new agent is a precommitment-style subagent of the original Alice, but using biextensional collapse, we can also see that this new agent is equivalent to the smaller matrix. Thus:

(xy)≃(xxyy)◃⎛⎜⎝xxyyxy⎞⎟⎠.

It is also easy to verify formally that these are mutual subagents using the covering definition of subagent.

I'm reminded here of the introduction and deletion of mixed strategies in game theory. The third row of Alice's frame is a mix of the first two rows, so we can think of Bob as being analogous to a random bit that the environment cannot see. I informally conjecture that for finite Cartesian frames, C⋈D if and only if you can pass between C and D by doing something akin to deleting and introducing mixed strategies for the agent.

However, this informal conjecture is not true for infinite Cartesian frames:

⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝zxzzz⋯zyzzz⋯zzxzz⋯zzyzz⋯zzzxz⋯zzzyz⋯zzzzx⋯zzzzy⋯⋮⋮⋮⋮⋮⋱⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠⋈⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝zyzzz⋯zzxzz⋯zzyzz⋯zzzxz⋯zzzyz⋯zzzzx⋯zzzzy⋯⋮⋮⋮⋮⋮⋱⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠.

We can see that these frames are mutual subagents by noting that one can transition back and forth by repeatedly committing not to take the top row.

I do not know of any examples of ⋈ that look qualitatively different from those discussed here, but I do not have a good understanding of exactly what the equivalence classes look like.

 

4. Universal Subagents and Superagents

We can view ⊤ as a universal subagent and ⊥ as a universal superagent.

Claim: ⊤◃C◃⊥ for all Cartesian frames C.

Proof: We use the categorical definition. That ⊤◃C is vacuous, since there is no morphism from ⊤ to ⊥. That C◃⊥ is also trivial, since any ϕ:C◃⊥ is equal to ϕ∘id⊥. □

Since null⋈⊤, we also have null◃C for all C.

We also have a that ⊥S is a superagent of all Cartesian frames with image in S.

Claim: C◃⊥S if and only if Image(C)⊆S.

Proof: Let C=(A,E,⋅), and let ⊥S=(S,{f},⋆), with s⋆f=s.

First, assume Image(C)⊆S. We will use the covering definition. Given an e∈E, let (g,h):C→⊥S be given by g(a)=a⋅e and h(f)=e. We have that g is well-defined because Image(C)⊆S, and (g,h) is a morphism because for all a∈A, 

a⋅h(f)=a⋅e=g(a)=g(a)⋆f.

Thus, there is a morphism (g,h):C→⊥S and an element f∈{f} such that h(f)=e for an arbitrary e∈E, so C◃⊥S.

Conversely, assume Image(C)⊈S, so let a∈A and e∈E be such that a⋅e∉S. If we assume for contradiction that C◃⊥S, then by the covering definition, there must be a morphism (g,h):C→⊥S such that h(f)=e. But then we have that

a⋅e=a⋅h(f)=g(a)⋆f=g(a)

must be both inside and outside of S, a contradiction. □

Convention: We will usually write C◃⊥S instead of Image(C)⊆S, as it is shorter.

Corollary: S∈Obs(C) if and only if C≃C0&C1 for some C0◃⊥S and C1◃⊥W∖S.

Proof: This is just rewriting our definition of observables from "Controllables and Observables, Revisited." □

 

In the coming posts, we will introduce multiplicative operations on Cartesian frames, and use these to distinguish between additive and multiplicative subagents and superagents.

I'll be hosting online office hours this Sunday at 2-4pm for discussing Cartesian frames.



Discuss

Confucianism in AI Alignment

3 ноября, 2020 - 00:16
Published on November 2, 2020 9:16 PM GMT

I hear there’s a thing where people write a lot in November, so I’m going to try writing a blog post every day. Disclaimer: this post is less polished than my median. And my median post isn’t very polished to begin with.

Imagine a large corporation - we’ll call it BigCo. BigCo knows that quality management is high-value, so they have a special program to choose new managers. They run the candidates through a program involving lots of management exercises, simulations, and tests, and select those who perform best.

Of course, the exercises and simulations and tests are not a perfect proxy for the would-be managers’ real skills and habits. The rules can be gamed. Within a few years of starting the program, BigCo notices a drastic disconnect between performance in the program and performance in practice. The candidates who perform best in the program are those who game the rules, not those who manage well, so of course many candidates devote all their effort to gaming the rules.

How should this problem be solved?

Ancient Chinese scholars had a few competing schools of thought on this question, most notably the Confucianists and the Legalists. The (stylized) Confucianists’ answer was: the candidates should be virtuous and not abuse the rules. BigCo should demonstrate virtue and benevolence in general, and in return their workers should show loyalty and obedience. I’m not an expert, but as far as I can tell this is not a straw man - though stylized and adapted to a modern context, it accurately captures the spirit of Confucian thought.

The (stylized) Legalists instead took the position obvious to any student of modern economics: this is an incentive design problem, and BigCo leadership should design less abusable incentives.

If you have decent intuition for economics, it probably seems like the Legalist position is basically right and the Confucian position is Just Wrong. I don't want to discourage this intuition, but I expect that many people who have this intuition cannot fully spell out why the Confucian answer is Just Wrong, other than “it has no hope of working in practice”. After all, the whole thing is worded as a moral assertion - what people should do, how the problem should be solved. Surely the Confucian ideal of everyone working together in harmony is not wrong as an ideal? It may not be possible in practice, but that doesn’t mean we shouldn’t try to bring the world closer to the Confucian vision.

Now, there is room to argue with Confucianism on a purely moral front - everyone working together in harmony is not synonymous with everyone receiving what they deserve. Harmony does not imply justice. Also, there’s the issue of the system being vulnerable to small numbers of bad agents. These are fun arguments to have if you’re the sort of person who enjoys endless political/philosophical debates, but I bring it up to emphasize that they are NOT the arguments I’m going to talk about here.

The relevant argument here is not a moral claim, but a purely factual claim: the Confucian ideal would not actually solve the problem, even if it were fully implemented (i.e. zero bad actors). Even if BigCo senior management were virtuous and benevolent, and their workers were loyal and did not game the rules, the poor rules would still cause problems.

The key here is that the rules play more than one role. They act as:

  • Conscious incentives
  • Unconscious incentives
  • Selection rules

In the Confucian ideal, the workers all ignore the bad incentives provided by the rules, so conscious incentives are no longer an issue (as long as we’re pretending that the Confucian ideal is plausible in the first place). Unconscious incentives are harder to fight - when people are rewarded for X, they tend to do more X, regardless of whether they consciously intended to do so. But let’s assume a particularly strong form of Confucianism, where everyone fights hard against their unconscious biases.

That still leaves selection effects.

Even if everyone is ignoring the bad incentives, people are still different. Some people will naturally act in ways which play more to the loopholes and weaknesses in the rules, even if they don’t intend to do so. (And of course, if there’s even just a few bad actors, then they’ll definitely still abuse the rules.) And BigCo will disproportionately select those people as their new managers. It’s not necessarily maliciousness, it’s just Goodhart’s Law: make decisions based on a less-than-perfect proxy, and it will cease to be a good proxy.

Takeaway: even a particularly strong version of the Confucian ideal would not be sufficient to solve BigCo’s problem. Conversely, the Legalist answer - i.e. fixing the incentive structure - would be sufficient. Indeed, fixing the incentive structure seems not only sufficient but necessary; selection effects will perpetuate problems even if everyone is harmoniously working for the good of the collective.

Analogy to AI Alignment

The modern Ml paradigm: we have a system that we train offline. During that training, we select parameters which perform well in simulations/tests/etc. Alas, some parts of the parameter space may abuse loopholes in the parameter-selection rules. In extreme cases, we might even see malicious inner optimizers: subagents smart enough to intentionally abuse loopholes in the parameter-selection rules.

How should we solve this problem?

One intuitive approach: find some way to either remove or align the inner optimizers. I’ll call this the “generalized Confucianist” approach. It’s essentially the Confucianist answer from earlier, with most of the moralizing stripped out. Most importantly, it makes the same mistake: it ignores selection effects.

Even if we set up a training process so that it does not create any inner optimizers, we’ll still be selecting for the same bad behaviors which a malicious inner optimizer would utilize.

The basic problem is that “optimization” is an internal property, not a behavioral property. A malicious optimizer might do some learning and reasoning to figure out that behavior X exploits a weakness in the parameter selection goal/algorithm. But some other parameters could just happen to perform behavior X “by accident”, without any malicious intent at all. The parameter selection goal/algorithm will be just as weak to this “accidental” abuse as to the “intentional” abuse of an inner optimizer.

The equivalent of the Legalists’ solution to the problem would be to fix the parameter-selection rule: design a training goal and process which aren’t abusable, or at least aren’t abusable by anything in the parameter space. In alignment jargon: solve the outer alignment problem, and build a secure outer optimizer.

As with the Confucian solution to the BigCo problem, the Confucian solution is not sufficient for AI alignment. Even if we avoid creating misaligned inner optimizers, bad parameter-selection rules would still select for the same behavior that the inner optimizers would display. The only difference is that we’d select for rules which behave badly “by accident”.

Conversely, the Legalist solution would be sufficient to solve the problem, and seems necessary if we want to keep the general framework of optimization.

The main takeaway I want to emphasize here is that making our outer objective “secure” against abuse is part of the outer alignment problem. This means outer alignment is a lot harder than I think a lot of people imagine. If our proxy for human values has loopholes which a hypothetical inner optimizer could exploit, then it’s a bad proxy. If an inner optimizer could exploit some distribution shift between the training and deployment environments, then performance-in-training is a bad proxy for performance-in-deployment. In general, outer alignment contains an implicit “for all” quantifier: for all possible parameter values, our training objective should give a high value only if those parameters would actually perform well in practice.

The flip side is that, since we probably need to build the Legalist solution anyway, the Confucian solution isn’t really necessary. We don’t necessarily need to make any special effort to avoid inner optimizers, because our selection criteria need to be secure against whatever shenanigans the inner optimizers could attempt anyway.

That said, I do think there are some good reasons to work on inner optimizers. The biggest is imperfect optimization. In this context: our outer optimizer is not going to check every single point in the parameter space, so the basin of attraction of any misaligned behavior matters. If we expect that malicious inner optimizers will take up a larger chunk of the parameter space than “accidental” bad behavior, then it makes sense to worry more about “intentional” than “accidental” maligness. At this point, we don’t really know how to tell how much of a parameter space is taken up by malicious agents, or any sort of inner agents; one example of this kind of problem is Paul’s question about whether minimal circuits are daemon-free.

Taking the analogy back to the BigCo problem: if it’s very rare for workers to accidentally game the rules, and most rule-gaming is intentional, then the Confucian solution makes a lot more sense.

I also expect some people will argue that malicious inner optimizers would be more dangerous than accidental bad behavior. I don’t think this argument quite works - in sufficiently-rish parameter spaces, I’d expect that there are non-agenty parameter combinations which exhibit the same behavior as any agency combinations. Optimization is an internal property, not a behavioral property. But a slight modification of this argument seems plausible: more dangerous behaviors take up a larger fraction of the agenty chunks of parameter space than the non-agenty chunks. It’s not that misaligned inner optimizers are each individually more dangerous than their behaviorally-identical counterparts, it’s that misaligned optimizers are more dangerous on average. This would be a natural consequence to expect from instrumental convergence, for instance: a large chunk of agenty parameter space all converges to the same misbehavior. Again, this threat depends on imperfect optimization - if the optimizer is perfect, then "basin of attraction" doesn't matter.

Again taking the analogy back to the BigCo problem: if most accidental-rule-abusers only abuse the rules a little, but intentional-rule-abusers usually do it a lot, then the Confucian solution can help a lot.

Of course, even in the cases where the Confucian solution makes relatively more sense, it’s still just an imperfect patch; it still won’t fix “accidental” abuse of the rules. The Legalist approach is the full solution. The selection rules are the real problem here, and fixing the selection rules is the best possible solution.



Discuss

Location Discussion Takeaways

3 ноября, 2020 - 00:14
Published on November 2, 2020 9:14 PM GMT

It's been a month and a half since I posted The rationalist community's location problem, and there's been a ton of good discussion both in the comments and elsewhere. In this post I hope to summarize the discussion so far, provide additional data, and give my take on what I think we should do.

It's worth noting that there are two related discussions going on in parallel. The first is this one: rationalists as a whole have become less tied down geographically during the pandemic, and it seemed like a good time to reassess whether the Berkeley hub was the optimal setup. The second is internal to MIRI (though it's been posted about publicly a bit) – like everyone else, MIRI has had to experiment with new and nonstandard working setups in 2020, and that's also put them in a place of wanting to reassess whether being in Berkeley is optimal. 

Note: I work for MIRI, but all opinions expressed here are entirely my own. I've been thinking about this problem since long before I joined MIRI, and it should be obvious that I'm not just spouting MIRI's party line here given that I disagree with them about almost everything :) 

The discussion so far + data

People have shown the most serious interest in college towns (e.g. Austin, Ann Arbor, Oxford) and several European cities (e.g. Berlin, Tallinn, Prague, London). Options in New Zealand and Australia have been floated but appear to not be under very serious consideration. MIRI's main options under consideration are (1) the Toronto area, or (2) rural New Hampshire, within a couple hours' drive of Boston. 

After looking at all the comments I thought, "Wow, there sure are a lot of places people are interested in and a lot of different things they care about. This seems like a great time to use a spreadsheet!" So here it is. You will quickly notice that it is not all the way filled out; this is because it has nearly 1500 cells and I was doing it alone. It is publicly editable, so please contribute a bit of your time to fill out a column or two if you think it's valuable to have this data!

The spreadsheet is informative, but it doesn't make the decision for us. In order for it to be useful, we have to figure out what we value. 

My take

The rationalist community is many things, but fundamentally, it exists to carry out the project of rationality, broadly defined. As such, our goal in choosing a location should be to preserve (or improve) the ability of individuals in the community to do productive work on this project. 

I've come to believe that the two main cruxes for our ability to advance our goals are:

  1. Ensuring the existence of a thriving in-person community
  2. A sense of safety

Apart from those, I think people will have a huge range of idiosyncratic preferences leading to competing access needs. For example, some people really want low cost of living while others really want high salaries, but the two are generally anticorrelated. Or, some people strongly prefer winter to wildfires and some people strongly prefer wildfires to winter. These are considerations that every individual will need to weigh for themselves, since there's no objective right answer. On the other hand, I think the in-person community and safety considerations are more universally important and easier to evaluate objectively.

In-person community

Habryka suspects that the rationalist community, writ large, basically wouldn’t exist without the Berkeley hub. If we go with that assumption[1], we need to be really careful about how and whether we move away from Berkeley, because it's very possible that if there's no hub in Berkeley, the in-person community will essentially cease to exist. 

(Note: Pre-pandemic, I estimated that there were at least 400 rationalists living in the Bay community.[2] I've seen and participated in several just-for-fun efforts to draw out all the social and professional connections between people in the community, and it always ended up as a connected graph. Point being: That's a lot of people, and you can't neatly remove just a subset of them without disrupting the whole social graph.)

I think that MIRI has been underweighting this concern, given their interest in moving to places like rural NH and Toronto, where it would be hard for others to follow. MIRI may care far more about researcher productivity than about the community as a whole, but MIRI is also very linked to CFAR and LessWrong – two organizations very focused on building and nurturing the rationalist community – and is perhaps tacitly hoping for a MIRI/CFAR/LW move rather than just a MIRI move. Moving CFAR and LW away from the center of the community strikes me as an obviously wrong move. I also think that MIRI itself should want to be where the community is for reasons of recruiting, idea exchange, and social well-being.

Why is it important?

Why is having an in-person community important for the mission? So many reasons, according to me! Here are some bullet points:

  • Eliezer wrote in the Sequences about the motivating power of being physically near other people who share your goals and values.
  • High-context, face-to-face interactions are really important for exchange of ideas, working together on a team, and founding organizations (see Elizabeth's lit review on distributed teams).
  • Talent flow between organizations
    • Habryka and I have each worked at four different rationalist/EA orgs (not the same four), a level of mobility only made possible by the critical mass of rationalists in the area.
  • Serendipitous encounters
    • Habryka points out that it's very unlikely that he would have started LessWrong 2.0 if he hadn't known Vaniver socially. Both Habryka and Vaniver(?) initially moved to the Bay because that's where MIRI/CFAR was. I don't think this is the only such story but it's the only one I feel confident telling.
  • Finding close friends and life partners
    • Being happy and stable is good for your productivity and motivation, and you gotta have good social relationships in order to be a happy and stable person. (Arguably happiness is intrinsically good as well :P).

Takeaway

So what do I concretely think we should do? Basically, I think if we move away from the Bay, we need to optimize the new location for ease of following. When my housemates and I went through a list of all the rationalists we knew to see how many might follow MIRI to rural New Hampshire, we were only able to think of five, and that seems really quite bad.

Here are some things that I think go into ease of following:

  • Dating/friending/employee pool
    • If the area is isolated or even just if there isn't a very strong existing intellectual culture there, it will be hard to find friends or people to date. This is a dealbreaker for a lot of people. Those people for whom it's not a dealbreaker will likely be mostly older, married people who are looking to settle down, and I don't think selectively moving all those people away from everyone else would be good for anyone.
    • Anywhere with a reasonable pool of intellectual young people would likely meet these needs – i.e. an urban center or a college town.
    • Employees are also a big concern. This is a strong argument for being near a tech hub or a tech-focused university. e.g., it's easy to have a continuous influx of people into the community and the organizations in Berkeley, because smart and talented people are always coming to the area for tech jobs and to study at UC Berkeley.
  • Job availability – Even if people are willing to move to a particular new location, they need to be able to financially support themselves in order to actually live there.
    • Literal availability – Somewhere like Gowrie Park in Tasmania may have many points in its favor, but it probably doesn't have enough jobs for 200 extra people, let alone the kinds of jobs that most rationalists would be looking for.
    • Salaries – Many people do long-term financial planning and would not be willing to move somewhere if it meant a major pay cut (or a major tax increase).
    • Visas can be a major barrier that will prevent people from following to a non-US location, even if there are well-paying jobs there suited to their skillset. This makes me concerned about MIRI's Canada plan. (Sorry for the US-centric viewpoint; I know it's hard to immigrate here, it's just that so many of us are already here.)
  • Lines of retreat
    • If MIRI moved to a rural area, I would be quite worried about an 'all-in or all-out dynamic', wherein there would be nothing there for rationalists except MIRI. This means you'd need a high level of commitment to go there at all, and that it could be quite scary to consider leaving. I think this leads to a lot of bad incentives and dynamics.
    • A well-put point from Adam Scholl: "The Bay has lots of high-paying jobs (especially for programmers) and hence provides many folks more lines of retreat, I think, than one is likely to find elsewhere. (In the sense of e.g. being able to get a new job without also having to move/get a new social network)."
    • This is another point in favor of urban areas.
  • Ease of travel to the location
    • Most people like being able to see their family and friends, at least once in a while. Many people just like traveling. Sometimes you (either as an organization or as an individual) want people to be able to travel to you. So it would be pretty hard to establish a hub many hours away from an international airport, since traveling from arbitrary other locations would be such a huge hassle. (This is the case for some areas under consideration, though not many). This narrows the search space to Europe and North America, within ~2 hours' drive of an international airport.
    • Ideally a person would not need to make an extra special trip to the location of the community – there should at the very least be food and lodging nearby. Also, driving is garbage, and a location that's "within a 30-minute drive of" a place where people might actually want to be is still isolated. 
  • Modern conveniences, medical care
    • Being in an isolated area makes it generally more difficult to pay other people to do tasks for you (e.g. restaurant delivery, grocery delivery, childcare, housekeeping). You also don't want to be too far from good medical care. The latter is only a problem at extremes of isolation (e.g. rural Tasmania, but not e.g. an average US suburb). I think a lot of people value these types of modern conveniences highly, especially insofar as automating or hiring away many daily tasks allows them to be more productive. But I'm sure there are people who think the quiet peace of rural life is worth the trade-off, so this point isn't as strong as the others on this list.
Sense of safety

A sense of safety is fairly necessary for getting productive work done. If you don't feel physically, socially, and financially safe, you're more likely to be anxious and distracted. As an extreme example, you don't want to be a penniless refugee alone in a war zone with no family or friends. 

I think people are indeed treating this as a major consideration, tacitly or otherwise. So let's dig into it.

Physical safety

Factors that go into this are crime (particularly but not solely violent crime), laws, and the general feel of the place (maybe this just straightforwardly correlates with crime, I don't know). 

One example when it comes to laws is the fact that male sodomy is a crime in Singapore. Singapore has other good things going for it, but we don't want to move somewhere where the gay members of our community will have to worry about being arrested just for living their normal lives. (Note that there's some disagreement as to how much of a problem this would actually be, but in terms of feeling safe I think it matters quite a bit either way.)

As for general feel, I'd count things like street homelessness, trash and feces on the sidewalk, and busy roadways cutting through residential areas as negatives; and things like clean and well-maintained parks, lots of children running around, and people systematically smiling at you on the street as positives. 

I think some people probably are attracted to rural living because they like having control over their environment – in that they both have more property and are less restricted in what they can do with it. I can see this contributing to a feeling of physical safety. However, I think buying and building on rural land probably takes quite a lot of time and attention, and after really taking that into consideration I don't feel as excited about it as I initially expected to.

Political unrest

A lot of people's desire to move seems to be motivated by nebulous fear of the US political situation. I am not one of those people, and I'm frankly a little confused about their epistemic state. Despite inside view on the situation, widespread violent political unrest in the US continues to seem incredibly unlikely to me. 

It makes sense to me that we might want to move out of the US to a country with generally saner governance. That makes sense to me if you think that political polarization and general societal decline is going to proceed apace in the US, and want to establish yourself somewhere saner and stabler sooner rather than later.

However, that doesn't seem to be the goal – the goal seems to be just "flee the US." And I don't think we're at a point where this is a reasonable goal.

For an interesting example, let's look at Wei Dai's story about how his grandparents ended up trapped in communist China. Very importantly, his grandparents did not fail to see the warning signs of a hostile takeover. They knew things were bad, but they stayed because they hoped that they would be favored under the new system – and got burned when that turned out not to be the case. This just does not apply to our current situation, and it seems really unlikely that it will suddenly become impossible for us to leave, without significantly more warning signs beforehand than we're currently seeing.

Something that's even harder for me to understand is the "flee to a rural area of the US" plan. If we are worried about violent political unrest, it's not clear to me that rural areas are that much safer. Yes they're less densely populated, but they're often very politically polarized and have relatively high rates of gun ownership. Even if we grant that violent political unrest is more likely in cities, the historical likelihood of dying in this manner in the US is statistically zero; and if the situation looks bad, having a car and a passport seems good enough to be able to get you out in plenty of time.

The only situation where you really want to get out of all urban centers is if you're expecting a nuclear attack, which seems vastly less likely than political violence in general. (My models here aren't super explicit, but the fact that despite the entire Cold War the only time nuclear weapons have been deployed against enemies was in 1945, priors on nuclear attack are just incredibly low.)

For more on the subject of political unrest, see kdbscott's question post from last week.

Social safety

Social safety is a much tougher nut to crack. At a smaller scale, everything I said above about having a rationalist community with good lines of retreat seems important. But what about social safety at the broader societal level? The prominence of cancel culture and its effects on people such as Robin Hanson and Steve Hsu have put some rationalists on edge, especially those who write (or act) publicly in a way that's traceable to their real identity.

But, is there anywhere, physically, that one can go to escape cancel culture? My instinct is no, but I didn't do a ton of thinking about this. As a toy example of the type of thing that might matter, maybe European universities are less (or more) likely than their American counterparts to fire a professor for saying something that's not PC. Also, I guess cities are maybe worse on the cancel culture dimension because if you're hidden in the middle of nowhere it's harder for people to credibly threaten to physically attack you. But again, this is all very off-the-cuff and not grounded in strong models. I'd welcome other people's thoughts here.

Financial safety

This one's pretty straightforward. All else equal, you want high salaries, low cost of living, low taxes, and good job security and job availability. Obviously many of those trade off against each other, but bottom line, more money is better. 

My recommendations

From all this, I think it's pretty clear that we need to be in either a college town or a major metropolitan area (though not necessarily in the heart of one). 

Primary recommendation

My primary recommendation is that the community – and MIRI – should stay in the Bay.[3] This seems like the surest way to not destroy the large (and, pre-pandemic, thriving) in-person community we've built. It's pretty likely we could move MIRI and/or a sizable contingent of people to the South Bay without disrupting things too much, if we wanted to. If some MIRI researchers want a more secluded, less urban lifestyle, they can move to the suburbs (e.g. Moraga, if the bulk of the community stays in Berkeley), or even just North Berkeley. That way they can have a quieter environment but still be within easy commuting distance of everyone else.

Non-California recommendation

If we decide that we need to leave California but not the US – for example, because CA passes laws that make it unfriendly to tech and business, or because wildfire season appears to be getting significantly longer every year – we should move to the Boston/Cambridge area. It of course ticks the boxes of both being in a metropolitan area and being near a good college. MIT is the best technical university in the world by most measures, and the Boston area has plenty of tech jobs. There are trade-offs in terms of climate and costs (e.g. the Bay has higher rents but Boston has higher taxes), and the cultural milieu is certainly quite different, but overall I'm optimistic about the possibility of building a strong community in the Boston area.

Non-US recommendation

If we decide we need to leave the US, I think we should probably move to Oxford. Oxford is home to FHI and Nick Bostrom; many rationalists have spent time there and enjoyed it; and both these things contribute to the fact that it's had a stable x-risk/rationality culture for many years. It also ticks the box of being a university town, and it's an hour's train ride from London (commuting distance by Bay Area standards!), which is not only a huge metropolis but also home to 80,000 Hours

I have not actually looked into the practical aspects of living in England very much. For example, I don't know how hard it is for US or EU citizens to get work there, and I'm very ignorant of Britain's political situation (beyond having heard of Brexit and Dominic Cummings). Insight into these or any related questions would be much appreciated :)

Unconditional recommendation

No matter where MIRI ends up, I think it would be good in the abstract for the community as a whole to pay more attention to and nurture secondary hubs (such as Prague) and enclaves (such as the EA Hotel). I say "it would be good in the abstract", because I don't really have any concrete plans for this. At the very least we should acknowledge that each provides things the Bay can't – e.g. the quiet focus and financial support of the EA Hotel, or how much easier it is for someone in the EU to move to Prague than to the US. 

Some examples of beneficial exchange between hubs: CFAR has been running several workshops a year in Prague and training a whole cohort of Czech CFAR instructors. Buck has previously suggested 'EA residencies', where employees from Bay Area orgs go live near different local groups (e.g. Yale EA) for a couple months. Scott Alexander and six other Bay rationalists/EAs took a road trip last year to visit a bunch of East Coast EA groups and SlateStarCodex meetups. 

While the pandemic has eliminated most opportunities like these due to the drastic reduction in face-to-face interaction, it's also untethered many people from their physical locations, a change which if it persists will make inter-hub exchange much easier. I also expect that there's plenty of low-hanging fruit in this domain, since not much attention has been put on it up to this point.

The end

Thank you all for reading; please let me know your thoughts in the comments, and help fill in the location spreadsheet if you're so inclined!

[1] I'm not sure I agree with him there, since if there wasn't one large hub, I can imagine a world where local meetup groups became more important, as in the early days of OBNYC. Lots of places had and still have fairly successful local groups, and they might be even stronger without the "Berkeley brain drain" problem. 

[2] I counted ~30 rationalist group houses in Berkeley and ~8 in San Francisco; the calculation assumes that a group house has an average of 5 people and that about half of people in the community live in group houses. Also, this is probably a low estimate since we sold nearly 300 tickets for the 2019 Winter Solstice. Additional data: Open-invite SSC or LW meetups and parties usually get about 150 people, and it's definitely not the same people every time.

[3] I did not write this bottom line beforehand – I started this discussion to see what the correct answer was, and I just think this is probably it. I'm pretty sure I dislike living in the Bay more than the average Bay Area rationalist, and I've often personally dreamed of moving. If I could wave a magic wand and move the whole community to Oxford with no roadblocks I probably would. But in the end I value the continued existence of the in-person community more than my own personal feelings about the city.



Discuss

Kelly Bet or Update?

2 ноября, 2020 - 23:26
Published on November 2, 2020 8:26 PM GMT

I suggest augmenting the classic "bet or update" with "Kelly bet or update".

Epistemic status: feels like maybe going too far, but worth considering?

On the one hand, we have a line of thinking in rationalist discourse which says probability is willingness to bet. This line of thinking suggests that many bad thinking patterns and misapplications of probability theory can broadly be discouraged by a culture of betting.

On the other hand, there's the longstanding discussion of Aumann Agreement and modest epistemology. According to this line of thinking, beliefs should be contagious among honest and rational folk, so long as they believe each other to be honest and rational. One should never agree to disagree; where two disagree, at least one is wrong (irrationally wrong -- or dishonest). This line of thinking has been developed extensively by Robin Hanson and others, and criticized heavily in Eliezer's Inadequate Equilibria, as well as some earlier essays.

These two perspectives are reconciled in the creedo bet or update, which has been proposed as a norm for rationalists: in any significant disagreement, you should either come to an agreement (at least of the Aumann sort, where you may not understand each other's exact reasons, but assign enough outside-view credibility to each other that your final probabilities align), or, failing that, you should bet. The Hansons of the world can take the outside view and update to agree with each other, while the Yudkowskys of the world put their money where their mouth is.

But how much should you bet?

Risk Aversion

A common norm in the rationalist circles I've frequented is to bet small amounts. I think there are some arguments in favor of this:

  • This makes winning feel good and losing hurt, without putting too much on the line. We want opportunities to learn, we want some useful social pressure against overconfidence and other biases, but we're not out to bankrupt anyone.
  • Often, bets have to do with things we have control over, such as placing a bet about when you'll finish an important project. Bets which are small in relationship to the project's importance help guarantee that no one gains a perverse financial incentive to delay or sabotage an important project. (On the other hand, people usually avoid perverse bets anyway; and, sometimes it is useful to use bets in the opposite way -- setting up extra incentives in virtuous directions. So I'm not clear on how big this advantage is.)

However, small bets might also be the result of irrational risk aversion. This seems like the more likely causal explanation, at least.

I was recently reminded that I tend to act like I'm much more risk-averse than Kelly betting would advise, with no justification I can think of.

Most investors seem to be similar. Even though Kelly betting itself captures fairly extreme risk-aversion (equating an empty bank account with death), "fractional Kelly", where one bets some percentage of what a true Kelly better would, appears to be much more popular than true Kelly betting.

One interpretation of this is not that investors are more risk-averse than true Kelly, but that they are not so confident of their own probability assessments. This seems sensible in the abstract, but doesn't line up with the math of fractional Kelly.

Let's say I'm betting on horse races and I use a mathematical model which I feel 90% confident in (that is, I think there's a 10% chance I made a dumb mistake in my math; otherwise, I think the model is a good accounting of my subjective uncertainty). I don't know what the other 10% of my beliefs look like, so I do a worst-case analysis on all my bets, acting like my probabilities of winning are 90% of whatever the model tells me.

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src: local('MathJax_Size4'), local('MathJax_Size4-Regular')} @font-face {font-family: MJXc-TeX-size4-Rw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Size4-Regular.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Size4-Regular.otf') format('opentype')} @font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax_Vector'), local('MathJax_Vector-Regular')} @font-face {font-family: MJXc-TeX-vec-Rw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Vector-Regular.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Vector-Regular.otf') format('opentype')} @font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax_Vector Bold'), local('MathJax_Vector-Bold')} @font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax_Vector'); font-weight: bold} @font-face {font-family: MJXc-TeX-vec-Bw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Vector-Bold.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Vector-Bold.otf') format('opentype')}  of my bankroll, where p,q are my probabilities for winning and losing respectively, and a,b are the house's calculation of those probabilities. Adjusting for my 10% uncertainty in my math, this becomes .9p−q+.1pb/a. This can take me over the line from betting to not betting at all, for example if p=.85 and a=.80; fractional Kelly, on the other hand, never changes willingness to bet, only quantity.

The Strong Position

Mirroring the "probability is willingness to bet" argument I mentioned at the beginning, the purist argument for Kelly bet or update is that "probability is willingness to Kelly bet". Within a VNM expected value framework, this argument becomes: if you don't Kelly bet, then you have to either admit that your probability is not what you said it was, or you have to give a good reason why your utility function is not approximately logarithmic.

Now, there could be lots of reasons why your utility function is not approximately logarithmic. Ability to actually buy something you within a fixed time period basically creates lots of discrete step-functions that go into your actual utility. 

However, utility logarithmic in money does seem like a pretty good approximation of most people's values, and I doubt that more detailed analysis will reveal a justification of the extreme risk aversion which accompanies most bets. (If you think I'm wrong here, I'm very curious to hear the reason!)

A Weaker Case

OK, but all said and done, Kelly-betting with my savings still seems too risky. 

Here's an alternative proposal. Make a special account in which you put some amount of money, say, $100. Call this your betting fund. Kelly bet with this. You might have rules such as "I can take money out if I accumulate a lot from betting, but I can never add more" -- this means that if you lose the majority of your seed money, you have no choice but to crawl back up past $100 by betting well. (I'm not sure about this, just throwing it out there.)

The size of your betting account compared to its starting seed could be a mark of shame/honor among those who chose to engage in this kind of practice.

This is sort of like fractional Kelly, which I've already argued is irrational... but ah well.

Jacobian recently argued that we should Kelly bet more. One thing I noticed after reading that article is that I'm pretty bad at the Kelly formula. I barely ever approach Kelly-like calculations because I don't really know how; I have to look up the formula every time, and I'm always using different versions of it, and have to double-check the meaning of the variables.

So, it seems like a good exercise for me to at least do the math more often.



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What Belongs in my Glossary?

2 ноября, 2020 - 22:52
Published on November 2, 2020 7:52 PM GMT

I've been working on a Glossary that will give short descriptions of concepts I've introduced, so people don't have to seek out entire posts to grok what they mean in context and then decide later if they'd like to know more. So far I've got Easy Mode, Everything is Trying to Kill You, Easy Mode, Hard Mode, Perfect/Superperfect/Imperfect Competition, Immoral Maze, Maze Behaviors, Maze Levels, Slack, Out to Get You, Simulacra Levels, Sacrifice to the Gods.  

What other things should I make sure to include? 

(Also any cool elegant/short definitions for the above or other Zvi-concepts are welcome, including Devil's Dictionary style versions)



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Generalize Kelly to Account for # Iterations?

2 ноября, 2020 - 19:36
Published on November 2, 2020 4:36 PM GMT

On my post weird things about money, Bunthut made the following comment:

I think the interesting question is what to do when you expect many more, but only finitely many rounds. It seems like Kelly should somehow gradually transition, until it recommends normal utility maximization in the case of only a single round happening ever. Log utility doesn't do this. I'm not sure I have anything that does though, so maybe it's unfair to ask it from you, but still it seems like a core part of the idea, that the Kelly strategy comes from the compounding, is lost. 

Kelly betting seems somehow less justified when we're not doing it a bunch of times. If I were making bets left and right, I would feel more inclined to use Kelly; I could visualize the growth-maximizing behavior, and know that if I trusted my own probability assessments, I'd see that growth curve with high probability.

Several prediction markets have recently offered a bet at around 62¢ which superforecasters assign around 85% probability. This resulted in a rare temptation for me to Kelly bet. Calculating the Kelly formula, I found that I was supposed to put 60% of my bankroll on this.

Now, 60% of my savings seems like a lot. But am I really more risk-averse than Kelly? It struck me that if I were to do this sort of thing all the time, I would feel more strongly justified using Kelly.

Bunthut is suggesting a generalization of Kelly which accounts for the number of times we expect to iterate investment. In Bunthut's suggestion, we would get less risk-averse as number of iterations dropped, approaching expectation maximization. This would reflect the idea that the Kelly criterion arises because of long-term performance over many iterations, and normal expectation maximization is the right thing to do in single-shot scenarios.

But I sort of suspect this "origin of Kelly" story is wrong. So I'm also interested in number-iteration formulas which reach different conclusions.

The obvious route is to modulate Kelly by the probability that the result will be close to the median case. With arbitrarily many iterations, we are virtually certain that the fraction of bets which pay out approaches their probabilities of paying out, which is the classic argument in favor of Kelly. But with less iterations, we are less sure. So, how might one use that to modulate betting behavior?

I suggest explicitly stepping outside of an expected-utility framework here. The classic justification for Kelly is very divorced from expected utility, so I doubt you're going to find a really appealing generalization via an expected-utility route.



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Waffle Crepes

2 ноября, 2020 - 18:30
Published on November 2, 2020 3:30 PM GMT

For years now I've been making eggy crepes or waffles for the kids' breakfast as a way to get them eating more protein. One thing I discovered recently, is that if you cook the crepes on both sides at the same time it's faster and they come out slightly fluffy.

We have a combination waffle maker/griddle with reversible plates (an older model of this one) and when I made crepes on it I would normally open it fully flat. I found a closing it, however, offers another way to cook them:

It's still thin like a crepe:

Don't cook it too long, or it gets hard. Cooked just enough and it rolls nicely:

I find these are tastier than what I was making before, you don't have to flip them, and they cook really quickly.



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Non Polemic: How do you personally deal with "irrational" people?

2 ноября, 2020 - 16:44
Published on November 2, 2020 1:44 PM GMT

I'm finally managing to finish my "basic" training in rationality, which is to mean finish studying "Rationality: A-Z" (I had studied the first half years ago, but I foolishly stopped when I got to the part about reductionism, which was unbelievably stupid of me even with all the reasons the led to me doing so). I plan to continue studying even more material once I'm done with it, to train myself in instrumental rationality and everything else I can find to make myself as smart as I could possibly be. I'm very satisfied with my progresses, the first half of the sequences helped me improve tremendously years ago, and now I can see myself improving again.

 

But, even while I am still at what I think is just the beginning of my improvement, I'm noticing more and more a rather serious problem.

To put it politely, I hate how people think, now.

 

I know it's really unfair because I didn't know any better mere weeks ago, and years ago I was a good textbook example of an intelligent person who'd keep mainly using his intelligence to rationalise whatever questionable decisions he made, but I just can't help it.

 

I notice logic leaps, cognitive missteps and dumb conclusions of people who are considered smart, deep and expert on stuff while they talk on the radio or on other medias and I get angry.

I notice idiotic ideas, as well as practices of thoughts that are the cognitive equivalent of shooting yourself in both knees, spreading inside ideologies I deeply care about, because the evils they fight are very real and demonstrated by science, but now I can see how all the truth is hopelessly getting mixed up with stuff that's just stupid or wrong, and that the intelligent people that once introduced me to these ideologies are absolutely incapable of judging and criticising any bad idea that's coming from their own side, and I get livid.

Half the time I hear someone talking I have to choose between politely tearing apart the majority of what he said, growing more and more annoyed, or just shutting off my attention and think about something else while pretending to listen to them.

And all this is just when I have to deal with intelligent people. 

I can't comprehend how a stupid person thinks unless I just stop thinking of him as an actual human being, switch off my empathy completely and just model him as a badly designed computer program with a bunch of floating beliefs in his memory and no analytical or critical skill whatsoever. If I try doing it the intuitive way, using empathy and telling my brain to think like him, my brain just keeps running out of suspension of disbelief as I can't avoid thinking that, no matter how much I could believe that political party/religion/philosophy x is right, I'd still recognise that blatantly idiotic part of it as a very, very stupid idea the first time I'd heard it, since even before rationality I've never actually been stupid enough to believe something that even at surface level was just plain dumb, so I can't even understand why he's doing what he's doing, forget predicting it.

 

And all this is really starting to weight on me. I think my mood has changed for the worse in the last weeks. 

If you have read HPMOR, I think I'm starting to feel like professor Quirrel, and my brain has started to actually think the words "avada kedavra" when I hear something particularly stupid and hateful. I wouldn't do that even if I could get away with it, but, emotion-wise, I have to consciously remind myself reasons why to kill someone that stupid wouldn't just be a net positive gain for mankind and wouldn't just spare us a waste of oxygen. The me of several years ago would have just smirked and nodded at this kind of thoughts, but I want to be smarter than the old me, and smarter than professor Quirrel as well.

 

I'm sorry if that was longer and more emotional than what strictly necessary, I wanted to communicate exactly how I feel and really needed to say these things to someone. I'll try to go straight to the point now. 

I think that rationality is completely worth it, I don't regret at all studying it, I don't want anyone to think that I regret studying it or suggest not studying it, and I will continue to move forward and improve myself. But I also think that the smart thing to do is look for ways to cheat and avoid paying this "price" as well.

 

So, what I want to know is: 

  1. Did other people who already learned rationality went through this as well?
  2. If so, does it continue or eventually you just get used to other people being insane and you don't emotionally mind it that much anymore? I can't remember being this annoyed at people when I had read the first half of the sequences.
  3. Do you know of or have you tried any particular strategy to not being annoyed or feel... disinterested in other people? If so, did it worked? Could you suggest any material that explains it in more details?
  4. What do you currently do when you have to deal with the kind of problem I have described? (If your answer to this is similar to 3. you can just skip this)
  5. Can you suggest me any material or strategy to effectively model and predict stupid people's behaviour?

And, on a side note:

6. Can you recommend me any reading material or training you think it made you smarter or better at predicting the world or other people? I have checked some of the posts about it on this side but still thought it was worth asking. If you know of posts and lists about this, linking those would also be a huge help.

 

Thanks to everyone who will choose to answer this, I'll really appreciate any help and information I can get. 



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Surface Thoughts Suck

2 ноября, 2020 - 16:24
Published on November 2, 2020 1:24 PM GMT

I've been thinking about the differences between the ways in which people typically pursue wealth and the ways in which people typically pursue their altruistic goals. Charity is significantly signalling, and so on... But, there's something else I noticed.

Whenever you want wealth, you pursue a high paying career, open a business, and so on. Whenever you (a hypothetical average person unfamiliar with the rationalist or effective altruism community) want to accomplish some altruistic goal, you're going to consider personally volunteering yourself to perform the associated cheap labor. If you decide to donate instead, you probably see it as the less impactful option. It would be better (you would think) to go to third world villages yourself than to donate to charity.

The underlying reason behind this mistake is the notion of topicality- proximity in an unconsciously constructed web of connections. Traditionally, different paths to power (business, government, personal charitable work, etc.) are associated with different usages. Volunteering is for helping starving children, business is for making money, government work is for changing what the government does. This association is often valid- different types of power are better at accomplishing different goals. 

It's less valid when those types of power are highly interchangeable. Imagine money and political power have a standardized rate of exchange in the vein of one US dollar per DemocracyCoin. It's either going to be better to acquire money to buy political power, or it's going to be better to acquire political power to buy money- manipulating the government isn't easier if you're a politician, and buying gold jewelry isn't easier if you're a business owner. One profession or the other is better at both.

In the case of standard employment (the trade of your man-hours for the money of others) and charity (the trade of your money for the man-hours of others), there are clear exchange rates. For obvious reasons, average westerners have a comparative advantage in money-making relative to the average human. Thus, westerners (usually) shouldn't directly volunteer their time to altruistic organizations which work in e.g. the third world; They should perform the two-step resource conversion of My Time -> Money -> Someone Else's Time. 

More generally: In a big and complicated world where so many resources are directly convertible and generally useful, it's highly unlikely that the best human-level strategies for accomplishing medium to long term goals are composed out of steps which are in any way "topical"- superficially related in the way that starting a charity that plants trees is related to environmental concerns but (for example) being a computer science professor at a prestigious university isn't.

The mental mechanisms behind the unconscious associations we draw, and thus the borders of topicality, pale in comparison to our deliberate planning ability. Making your plans out of a handful of those unconscious associations which you've haphazardly cobbled together is a bad start- there is only so much genius in a single step.

Much of the intellectual progress of slightly younger me was in replacing bad low level ideas with better ones- changes analogous to deciding to work and donate to a charity which plants trees instead of planting trees myself. But, both are bad ways to save the environment. The strategy behind their generation (stringing together some select ideas that floated to the surface of my wandering mind) was bad, so all such strategies are cursed by methodological badness. This effect doesn't go away if you're really smart- these strategies are always going to be bad relative to what you can produce with better methods.

There's value in slowly refining low level ideas (working and donating > volunteering) by "just thinking" about the topic, but only so much. There's much more value in developing an idea of why you were previously mistaken, of which other legacy opinions are affected by the same methodological error and will need to be reevaluated, and more importantly in developing and using methods of reasoning which don't let individual cached thoughts represent single points of failure. 

You can't develop a heavier than air flying machine by "just thinking" about flight. Human brains are just not advanced enough to successfully perform those types of tasks without building huge sequences of successive layers. There's, as a zeroth order approximation, hundreds of layers of theories (the borders are fuzzy, of course) which are necessary to build a primitive heavier than air flying machine, starting from the peak science of the year 2000 BCE. 

Thinking about flight and taking what your surface thoughts give you- birds, feathers, flapping wings- will not cut it. As a start, you need something like classical mechanics, which will require a bunch of things including a complicated theory of mathematics and something like the scientific method, which will require blah blah blah.

Similar concepts follow if you want to do good altruistic work. You can't just make up an answer in the same way you can't just make up a design for a heavier than air flying machine. As a start, you need a theory of what you value and a theory of what the world looks like, which will in turn require such and so on. 

You realize there are on the order of 10^20 stars out there whose future usage you're partially determining, right? Well, that's a bit of an underestimate... That's the sort of thing which you need to include within your altruistic decision making. You can't expect to have a good strategy if you didn't think about that!

Philosophy, Politics, Altruism, and so on- some questions deserve correct answers. So, think deliberately. Divide questions into trees of subquestions whose answers can be repeatedly recombined to help answer the original question. You can't just have an answer, directly delivered from the surface of your idle mind- an answer built from thoughts which are not themselves the children of a highly relevant deeper theoretical understanding. It's going to be garbage if the question is complicated. You need to build a good answer.

Is this too obvious? For the general public, absolutely not... Somehow? For the reader of this post, I hope the extra clarity provided by reading this idea written down is useful.



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"Portland" SSC Meetup 11/15/20

2 ноября, 2020 - 08:39
Published on November 2, 2020 5:39 AM GMT

We are having an online Slate Star Codex Meetup on Sunday, November 15th, at 12:30 PM. Our meetups typically consist of people from Portland and the surrounding area, but all are welcome to this one. We tend toward open discussion; sometimes we play games or do other activities. If you're interested, email me or send me a message on LessWrong, and I will send you a Google Meet link when the meetup starts. 



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Should students be allowed to give good teachers a bonus?

2 ноября, 2020 - 03:19
Published on November 2, 2020 12:19 AM GMT

Alice is a STEM student taking general chemistry, linear algebra, and intro to computer programming. At the end of her term, the school emails her an online form with a link to her Student-Allocated Bonus.

For taking three classes, Alice gets 3 points. She can divide them up however she wants between her teachers from this quarter. Her favorite teacher was Prof. Bruin, her math teacher. Her least favorite teacher was Prof. Cameron, her computer programming teacher. Prof. Dorry, her chemistry teacher, was all right. She gives Prof. Bruin 2 points, Prof. Dorry 1 point, and Prof. Cameron 0 points.

Jack is her classmate, but he forgets to fill out the SAB form. The system divides his points equally, 1 per teacher.

Once the due date for submission has passed, the system totals up the points that students have allocated to each teacher. Each teacher gets $5 per point. It's a modest but not insignificant bonus. Prof. Bruin's 30 students award him 50 points, so he earns $250 extra. At the local college, teachers make about $3,000 per class, so that's an 8% bump in his earnings.

People were worried at first that this was basically the same thing as students bribing their teacher. What if Jack promises his math teacher that he'll give him all his points on the SAB in exchange for "going easy on his grading?"

The beauty of the anonymized points system is that it makes this impossible to verify. Which student gave which points to what teacher is anonymous. Since points can only be given in integers, students can't do any shenanigans like giving their teacher some weird decimal point number of points (2.6893 points) so that their teacher will see the decimal number and know who the money was from. And the SAB is assigned after all grades are due at the end of the quarter.

The SAB is a modest portion of the teacher's total salary. It's not a club that students can hold over a teacher's head. $5 is not a meaningful bribe. It's just one small but non-zero part of the incentive structure.

But it has some very attractive advantages.

Students are the consumers of the product schools are selling. They have at the very least an important role in evaluating its quality. Currently, they can only express their opinion before trying out a teacher's offering, by choosing whether or not to pay tuition for the class. Wouldn't it be better if at least some small fraction of their money was allocated after the class, when they have an informed opinion?

Students are also transient. If their only option for evaluating teachers is to grade them or give written feedback at the end of the quarter, it's on an administrator to decide whether and how to discipline teachers who get bad reviews and reward those who get good ones. They don't know exactly what the students meant in their feedback forms. Egos and relationships are on the line. Inertia sets in. This way, there's a little bit of a carrot that administrators can offer to good teachers, without having to take responsibility for deciding who the good teachers are.

Finally, this system gives students a small but meaningful sense of responsibility for their school. When they fill out the SAB, they're not going to alter their own outcomes. They already took the class. What they're doing is helping steer the school in a better direction for the next class of students. This promotes a sense of stewardship.

Teachers might be afraid that they're being given a pay cut that they have to "placate students" to get back. This fear could be alleviated by adding a $5 fee per class. Present teacher pay is unchanged. The SAB is truly a bonus - a pure increase in pay - for teachers who earn it.

I'd appreciate any feedback on ways this system could fail, and also ideas on the practical challenges administrators might face in implementing it. If similar systems already exist, please let me know examples.



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Zeno's Paradoxes - Dichotomy + Archilles and the Tortoise

2 ноября, 2020 - 02:39
Published on November 1, 2020 11:39 PM GMT

Dichotomy Paradox

Imagine the following situation:

A runner is trying to run 1km. Before they can run 1km, then must run 0.5km. Before they can run 0.5km, they must run 0.25km. Before they can run 0.25km, they must run 0.125km and so on. This would require completing an infinite number of tasks, which Zeno claimed to be impossible.

This naturally divides the space into intervals:

0.5-1, 0.25-0.5, 0.125-0.25...

Note that Zeno assumes that an infinite number of non-overlapping space intervals can fit within a finite space. But he seems to doubt that an infinite number of non-overlapping time intervals can fit within a finite time. Why the asymmetry?

Archilles and the Tortoise:

Suppose Achilles is in a foot race with a tortoise over 100m. By the time Archilles runs 100m, the tortoise has advanced a further 1m. By the time he has run the 1m, the tortoise has moved 1cm. By the time he runs the 1cm, the tortoise has moved 0.01cm, ect. Thus whenever Archilles has arrived where the tortoise was, he still has further distance to go

Let's simplify this and imagine that the tortoise moves half as fast as Archilles. To find when they intersect absent this paradox, let t be how far the tortoise has run and a be how far Archilles has run. They should intersect when

2t=a=t+100; that is t=100m, a=200m

We can then transform the argument about Archilles always being behind the tortoise into one about Archilles never quite reaching the 200m. This then becomes equivalent to the previous paradox.

Final Thoughts:

Hopefully you agree with me that these resolutions are more satisfying than just invoking geometric series. I'm still working on a write-up for the arrow paradox. I feel my current write-up is 80% there, but still has a few loose ends.



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Automated intelligence is not AI

2 ноября, 2020 - 02:32
Published on November 1, 2020 11:30 PM GMT

Crossposted from world spirit sock puppet.

Sometimes we think of ‘artificial intelligence’ as whatever technology ultimately automates human cognitive labor.

I question this equivalence, looking at past automation. In practice human cognitive labor is replaced by things that don’t seem at all cognitive, or like what we otherwise mean by AI.

Some examples:

  1. Early in the existence of bread, it might have been toasted by someone holding it close to a fire and repeatedly observing it and recognizing its level of doneness and adjusting. Now we have machines that hold the bread exactly the right distance away from a predictable heat source for a perfect amount of time. You could say that the shape of the object embodies a lot of intelligence, or that intelligence went into creating this ideal but non-intelligent tool.
  2. Self-cleaning ovens replace humans cleaning ovens. Humans clean ovens with a lot of thought—looking at and identifying different materials and forming and following plans to remove some of them. Ovens clean themselves by getting very hot.
  3. Carving a rabbit out of chocolate takes knowledge of a rabbit’s details, along with knowledge of how to move your hands to translate such details into chocolate with a knife. A rabbit mold automates this work, and while this route may still involve intelligence in the melting and pouring of the chocolate, all rabbit knowledge is now implicit in the shape of the tool, though I think nobody would call a rabbit-shaped tin ‘artificial intelligence’.
  4. Human pouring of orange juice into glasses involves various mental skills. For instance, classifying orange juice and glasses and judging how they relate to one another in space, and moving them while keeping an eye on this. Automatic orange juice pouring involves for instance a button that can only be pressed with a glass when the glass is in a narrow range of locations, which opens an orange juice faucet running into a spot common to all the possible glass-locations.

Some of this is that humans use intelligence where they can use some other resource, because it is cheap on the margin where the other resource is expensive. For instance, to get toast, you could just leave a lot of bread at different distances then eat the one that is good. That is bread-expensive and human-intelligence-cheap (once you come up with the plan at least). But humans had lots of intelligence and not much bread. And if later we automate a task like this, before we have computers that can act very similarly to brains, then the alternate procedure will tend to be one that replaces human thought with something that actually is cheap at the time, such as metal.

I think a lot of this is that to deal with a given problem you can either use flexible intelligence in the moment, or you can have an inflexible system that happens to be just what you need. Often you will start out using the flexible intelligence, because being flexible it is useful for lots of things, so you have some sitting around for everything, whereas you don’t have an inflexible system that happens to be just what you need. But if a problem seems to be happening a lot, it can become worth investing the up-front cost of getting the ideal tool, to free up your flexible intelligence again.



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The Born Rule is Time-Symmetric

2 ноября, 2020 - 02:24
Published on November 1, 2020 11:24 PM GMT

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src: local('MathJax_Vector Bold'), local('MathJax_Vector-Bold')} @font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax_Vector'); font-weight: bold} @font-face {font-family: MJXc-TeX-vec-Bw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Vector-Bold.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Vector-Bold.otf') format('opentype')} of a particle in state ">|ψ⟩ with eigenvalues {λ1,λ2,…} is measured then the probability of yielding a particular eigenvalue λi equals |">|⟨λ|ψ⟩|.

In quantum mechanics, time-reversal is performed by complex conjugation ψ→ψ∗. If we apply time-reversal to the Born rule then all the math stays the same except |">|⟨λ|ψ⟩| becomes |">|⟨λ|ψ∗⟩|.

The mathematics for time-reversing the Born rule is straightforward complex algebra. I use the variable V to indicate an arbitrary vector of complex values {v1,v2,…}.

&=&\sum_i|v_i|^2 \\ &=&\sum_i|v_i^*|^2 \\ &=&\left \\ \end{eqnarray*} ">⟨V|V⟩=∑i|vi|2=∑i|v∗i|2=⟨V∗|V∗⟩

Consider the particular case ψ=V.

&=&\left<\psi^*|\psi^*\right> \\ (\left<\psi|\psi\right>)^*&=&(\left<\psi^*|\psi^*\right>)^* \\ \left|\psi^*\right>\left<\psi^*\right|&=&\left|\psi\right>\left<\psi\right| \\ \left<\lambda|\psi^*\right>\left<\psi^*|\lambda\right>&=&\left<\lambda|\psi\right>\left<\psi|\lambda\right> \\ |\left<\lambda|\psi^*\right>|^2&=&|\left<\lambda|\psi\right>|^2 \\ |\left<\lambda|\psi\right>|^2&=&|\left<\lambda|\psi^*\right>|^2 \end{eqnarray*} ">⟨ψ|ψ⟩=⟨ψ∗|ψ∗⟩(⟨ψ|ψ⟩)∗=(⟨ψ∗|ψ∗⟩)∗|ψ∗⟩⟨ψ∗|=|ψ⟩⟨ψ|⟨λ|ψ∗⟩⟨ψ∗|λ⟩=⟨λ|ψ⟩⟨ψ|λ⟩|⟨λ|ψ∗⟩|2=|⟨λ|ψ⟩|2|⟨λ|ψ⟩|2=|⟨λ|ψ∗⟩|2

But what, intuitively, does it mean?

Alternate Histories

We traditionally think of the multiverse as several possible futures branching off of the present.

But we have just shown that the arrow can be reversed. All possible futures are also possible pasts.

Distinguishing past from future is a tricky problem when the set of possible futures equals the set of possible pasts.

The physics establishment sidesteps this quandary by defining time to progress in the direction of increasing entropy.

Entropy

An ontology O is an associative operator that buckets a set of microstates m={mi} into a smaller set of macrostates M={Mi}.

|M|\\ O:m\to M \end{eqnarray*}">|m|>|M|O:m→M

The entropy S of a macrostate Mi equals the natural log of the number of microstates in its bucket.

S(Mi)=ln|O−1Mi|

The entropy S of a microstate mi equals the natural log of the number of microstates in its bucket.

S(mi)=ln|O−1Omi|

The Random Walk of Time

The only thing you directly experience is this moment right now. The past and future are inferences. They are not empirical observations.

You are a classical being. Therefore your experience of this moment is a degenerate macrostate. A degenerate macrostate is a macrostate with more than one microstate.

We can treat the multiverse as a graph of microstates. Edges connect microstates to their immediate pasts and futures. The edges are bidirectional because the Born rule is time-symmetric.

A random walk along this graph (almost always) moves in the direction of increasing entropy until you approach the heat death of the universe.

We have dissolved time in a way that adds up to normalcy. This theory, which I call The Theory of Entropic Time, predicts that localized quantum fields will maximize proper time and that is exactly what we observe.



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Why are deaths not increasing with infections in the US?

2 ноября, 2020 - 01:43
Published on November 1, 2020 10:43 PM GMT

According to COVID act now, cases are now up to 100k a day, from an average of 40k/day in early October. Two weeks ag case numbers were already up by 50%. Meanwhile the change in deaths has been modest, still around 10%. This would require a three week lag for deaths to be proportional to cases. What might be happening?



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October 2020 gwern.net newsletter

2 ноября, 2020 - 00:38
Published on November 1, 2020 9:38 PM GMT



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Cowering To Genocide: Uighur Persecution And The World’s Last Hope

2 ноября, 2020 - 00:01
Published on November 1, 2020 9:01 PM GMT

If you could stop a genocide-would you? 

In western China, right now, more than a million men, women, and children sit in reeducation camps. Mothers separated from children, forced marriages, forced sterilizations-Margaret Atwood’s fiction made a reality. 

What’s to be done? Have we learned nothing? From Rwanda, from the Holocaust, from the Holomodor? The cat’s out of the bag, and the End of history is over. Finally, we can acknowledge that the future is not just greyed out liberal democracy, 2% a year GDP growth, easing ourselves into decadence. In reality, our dark 20th century never really left us after all. 

Modernity and the belief in bad people are incompatible.  The enlightenment has swept conflict under the rug and right out the Overton window. Hobbes, banished from the lexicon. To even conceive a world with people who are themselves, bad, is a heretical assumption. To understand the aspects of human nature that are deeply problematic is to understand the true nature of reality. American culture lulled to sleep by narcotics and video games, our senses dulled by ultra-palatable foods and 2-hour delivery. In some sense, things are better than they ever have been before; in another sense, they’re much worse. Something is rotten in the state of Denmark. Americans are depressed, fat, and lead lives bereft of meaning. We are reduced to larping real revolutionary change on the streets, with bats, shields, and batons. A simulacra of revolution, a cry for meaning in a world gone mad. 

I recall a cool October night in 2018. Joe Biden spoke softly to a crowd I was a part of. Bathed in dim light, he rambled. He wove soft, slow, soliloquies on American Greatness. He remarked on the magnificence of our landmass. That we are surrounded by vast oceans, blessed by great mineral reserves, possessing unrivalled food production capability, the mightiest navy the world has ever seen. The Zeihan-Esque supposition? Geography is destiny. Our geopolitical rivals? Nothing to worry about. China-why worry about China…                                    “c’mon man.” 

This theory is seductive because it is easy. It’s a denial of agency-the idea that nothing can be done to challenge us. This makes our position precarious. You can’t fix problems you can’t see. If, like Biden says-we’re the greatest country in the world, why can’t we stop Uighur persecution in Xinjiang? The reason, I posit here, is that for all our technical capability, we lack the will and imagination to see it through. 

In the farthest reaches of eastern North Carolina sits a massive dune of sand. Jockey’s Ridge rises out of the warm waters of the Pamlico Sound. Here, a century ago, two men from Ohio made history with the first powered flight.  Wilbur and Orville Wright were bicycle mechanics-yet they were the first to fly. Imagine the expectations for a bicycle mechanic today. Produce a plane-are you kidding me? That’s for the experts, the technocrats. The institutionalization of science, of technology, of politics, has eroded our capability to achieve the American greatness we, as a people, are destined for. Science, once  a process of obtaining truth-has become a bloated bureaucratic institution, a tool for politicians. . When our ruling class says “trust science” they don’t mean “trust the process of inquiry”, they mean “trust our experts.” Somehow, we have gone from a culture that encourages bicycle mechanics to invent an airplane to American institutions that can no longer prevent planes many times more advanced from diving straight into the ground. (Really like this bit.)

The belief, the conceit, is that only experts can manipulate reality. But while experts have gotten us into this mess, they won’t get us out. This is not to denigrate experts, but the bureaucracies in which they work have become sick-sick enough that our planes fall out of the sky, and our factories that produce life-saving PPE have been shipped off our continent long ago. Their bureaucracies and their selection mechanisms have become corrupt, and sclerotic. What is to be done?

Stopping a genocide. 

American military might is now too far behind to stop the genocide. Our carriers can be shot out of the water, with advanced surface to surface Chinese missiles. Tariffs won’t work either-just look at North Korea. but with a little imagination-a plan from me, a 26-year-old technologist emerges. 

First, we need to visualize the problem as a chessboard. Each player has strategic pieces-some are stronger than others. Here, we want to understand how to incentivize the PRC to close the camps and allow Xinjiang a level of autonomy it rightly deserves. Bludgeoning them into submission won’t work. Attacking the PRC, whether rhetorically or physically will get us nowhere. The only way for us to avert a genocide is to alter the incentive structure, in such a way that it is easy and graceful for the PRC to change course. Attacking will just lead to path dependence for the CCP. 

First, our strengths. Currently, America’s most powerful institutions are our worldwide entertainment and corporations. Hollywood, the NBA, Disney. These institutions have soft power that matters to a lot of people overseas. When I worked in China, I was always amazed at how popular the NBA was. The finals played on almost every TV and billboard I saw. With over 600 million viewers in China, the NBA is a cultural behemoth. It is important to note that China is a rising power, with a burgeoning middle class. This middle class finally has time to consume entertainment-and what a luxury it is. 

America has a monopoly on the best basketball played anywhere. It’s quality is unsurpassed. Paragraph 1.1 of the Wikipedia page on Olympic basketball is aptly entitled “American Dominance.” A monopoly gives you market power. You can squash competitors, and you can force people to obey your will. In this way, the NBA has power, and I propose that it use it to solve a genocide. 

The plan is quite simple. Adam Silver contacts the CCP, and requests that they start treating Uighurs decently, with a list of specific demands- let everyone go, and start over as if nothing ever happened. There is no need for this to be violent: everyone who was involved (police officers, bureaucrats, administrators) will be given new positions elsewhere and taken care of appropriately. The camps will be demolished, and each individual sent home. Mr Silver will do this respectfully, there is no need to be ugly, and without disrespect to the CCP’s power in mainland China.

If they refuse, the commissioner’s next move will be to offer alternatives. A full-scale PR nightmare. We will publicly embrace the CCP, and Chinese people. No hate for either (or Xi himself), but we will pack our stands with Uighur refugees, and activists. We will put #freeuighurs on the court at every game, and we will continue to do this until the situation resolves to our liking. 

The CCP can try to ban the NBA, but it will face serious backlash. The party has experienced this before. When things are very unpopular, they try to fix them. 600 million fans of the NBA want to watch basketball, and even only a minority are diehard fans, those diehard fans will make their displeasure known, and it would hardly prove expedient to throw half the country into camps if they disagree with you. 

As I like to say, “hell hath no fury like a consumer scorned.” 

The worst-case scenario here is that China really does successfully ban the league. I find this eventuality unlikely, but this is where the State Department enters. We can organize a group that writes an insurance policy for the NBA-you lose revenue, we’ll cover your costs. Putting this together, we could make it a no-lose situation for the NBA. 

This strategy, if implemented, has a high chance of success- in fact, it may have the highest chance of success that currently exists in the problem space. Tariffs don’t work, sanctions don’t work, military force won’t work, but maybe, just maybe, asking nicely, while carrying cultural clout of American basketball might.



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What is the right phrase for "theoretical evidence"?

1 ноября, 2020 - 23:43
Published on November 1, 2020 8:43 PM GMT

I mean "theoretical evidence" as something that is in contrast to empirical evidence. Alternative phrases include "inside view evidence" and "gears-level evidence".

I personally really like the phrase "gears-level evidence". What I'm trying to refer to is something like, "our knowledge of how the gears turn would imply X". However, I can't recall ever hearing someone use the phrase "gears-level evidence". On the other hand, I think I recall hearing "theoretical evidence" used before.

Here are some examples that try to illuminate what I am referring to.

Effectiveness of masks

Iirc, earlier on in the coronavirus pandemic there was empirical evidence saying that masks are not effective. However, as Zvi talked about, "belief in the physical world" would imply that they are effective.

Foxes vs hedgehogs

Consider Isaiah Berlin’s distinction between “hedgehogs” (who rely more on theories, models, global beliefs) and “foxes” (who rely more on data, observations, local beliefs).
- Blind Empiricism

Foxes place more weight on empirical evidence, hedgehogs on theoretical evidence.

Harry's dark side

HPMoR chapter 10:

Then I won't do that again! I'll be extra careful not to turn evil!

"Heard it."

Frustration was building up inside Harry. He wasn't used to being outgunned in arguments, at all, ever, let alone by a Hat that could borrow all of his own knowledge and intelligence to argue with him and could watch his thoughts as they formed. Just what kind of statistical summary do your 'feelings' come from, anyway? Do they take into account that I come from an Enlightenment culture, or were these other potential Dark Lords the children of spoiled Dark Age nobility, who didn't know squat about the historical lessons of how Lenin and Hitler actually turned out, or about the evolutionary psychology of self-delusion, or the value of self-awareness and rationality, or -

"No, of course they were not in this new reference class which you have just now constructed in such a way as to contain only yourself. And of course others have pleaded their own exceptionalism, just as you are doing now. But why is it necessary? Do you think that you are the last potential wizard of Light in the world? Why must you be the one to try for greatness, when I have advised you that you are riskier than average? Let some other, safer candidate try!"

The Sorting Hat has empirical evidence that Harry is at risk of going dark. Harry's understanding of how the gears turn in his brain makes him think that he is not actually at risk of going dark.

Instincts vs A/B tests

Imagine that you are working on a product. A/B tests are showing that option A is better, but your instincts, based on your understanding of how the gears turn, suggest that B is better.

Posting up in basketball

Over the past 5-10 years in basketball, there has been a big push to use analytics more. Analytics people hate post-ups (an approach to scoring). The data says that they are low-efficiency. 

I agree with that in a broad sense, but I believe that a specific type of posting up is very high efficiency. Namely, trying to get deep-position post seals when you have a good height-weight advantage. My knowledge of how the gears turn strongly indicates to me that this would be high efficiency offense. However, analytics people still seem to advise against this sort of offense.



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