The speed of sequential asymptotic learningwhanncar/slides/sses19.pdf · The speed of sequential asymptotic learning WadeHann-Caruthers1,VadimV.Martynov1,OmerTamuz1 1California Institute

The speed of sequential asymptotic learning

Wade Hann-Caruthers1, Vadim V. Martynov1, Omer Tamuz1

1California Institute of Technology

August 2019

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 1 / 14

Introduction


Introduction


Introduction


Learning from signals













!!





!!







s



!!s







P(ω | I , s)



P(ω | I , s) =P(s | I , ω)P(I , ω)

P(s | I )



P(ω | I , s) =P(s | I , ω)P(I , ω)

P(s | I )=

P(s |ω)P(I , ω)

P(s | I ).



Fixing some ω0,

P(ω | I , s)

P(ω0 | I , s)=

P(s |ω)P(I , ω)

P(s |ω0)P(I , ω0).



logP(ω | I , s)

P(ω0 | I , s)= log

P(s |ω)P(I , ω)

P(s |ω0)P(I , ω0)



logP(ω | I , s)

P(ω0 | I , s)= log

P(I , ω)

P(I , ω0)+ log

P(s |ω)

P(s |ω0)



logP(ω | I , s)

P(ω0 | I , s)= log

P(ω | I )P(ω0 | I )

+ logP(s |ω)

P(s |ω0)



logP(ω | I , s)

P(ω0 | I , s)= log

P(ω | I )P(ω0 | I )

+ logP(s |ω)

P(s |ω0)

Define

Lω1,ω2(J) = logP(ω1 | J)

P(ω2 | J).



Lω,ω0(I , s) = Lω,ω0(I ) + logP(s |ω)

P(s |ω0)

Define

Lω1,ω2(J) = logP(ω1 | J)

P(ω2 | J).


Observational learning













!!





!!







!!



!!??







!!


Observational Learning



Lω,ω0(I , a) = Lω,ω0(I ) + logP(a |ω, I )P(a |ω0, I )

.




.

For all a′,

E(u(a, ω)− u(a′, ω) | I , s) ≥ 0




.

For all a′, ∑ω

P(ω | I , s)(u(a, ω)− u(a′, ω)) ≥ 0




.

For all a′, ∑ω

P(ω | I , s)

P(ω0 | I , s)(u(a, ω)− u(a′, ω)) ≥ 0




.

For all a′, ∑ω

eLω,ω0 (I ,s)(u(a, ω)− u(a′, ω)) ≥ 0




.

For all a′, ∑ω

eLω,ω0 (I )+log P(s |ω)

P(s |ω0) (u(a, ω)− u(a′, ω)) ≥ 0




.

For all a′, ∑ω

P(s |ω)

P(s |ω0)eLω,ω0 (I )(u(a, ω)− u(a′, ω)) ≥ 0


Model

Agents: indexed by t ∈ 1, 2, . . . State of world: ω ∈ Ω = −1,+1Prior: π ∈ ∆(Ω)

Actions: at ∈ A = −1,+1Utility: u(a, θ) = 1(a = θ)

Private signals: st i.i.d. ∼ Fθ ∈ ∆(S)


Model

Agents: indexed by t ∈ 1, 2, . . .

State of world: ω ∈ Ω = −1,+1Prior: π ∈ ∆(Ω)




Model

Agents: indexed by t ∈ 1, 2, . . . State of world: ω ∈ Ω = −1,+1

Prior: π ∈ ∆(Ω)




Model





Model


Actions: at ∈ A = −1,+1

Utility: u(a, θ) = 1(a = θ)



Model





Model





Background

How much does society know?Beliefs of fictitious outside observerIt = a1, . . . , at−1

Public beliefspt = P(θ = +1 | It)


Background

How much does society know?

Beliefs of fictitious outside observerIt = a1, . . . , at−1



Background

How much does society know?Beliefs of fictitious outside observer

It = a1, . . . , at−1



Background




Background


Public beliefs

pt = P(θ = +1 | It)


Background




Background

Fact(pt) is a martingale.

Fact (Martingale Convergence Theorem)(pt) converges to some random variable p∞.

Fact (BHW ’92)If signals are boundedly informative, all agents take the wrong action withpositive probability

Fact (SS ’00)If signals are unboundedly informative, only finitely many agents take thewrong action.


Background






Background






Background






Background






Speed of learning

Suppose signals are unboundedly informative.

QuestionHow fast does learning occur?−! At what rate does (pt) converge?


Speed of learning




Speed of learning


QuestionHow fast does learning occur?

−! At what rate does (pt) converge?


Speed of learning




Dynamics


Dynamics

Define

`t = logpt

1− pt


Dynamics

Define

`t = logpt

1− pt

Then

`t


Dynamics

Define

`t = logpt

1− pt

Then

`t = logP(θ = +1 | It)P(θ = −1 | It)


Dynamics

Define

`t = logpt

1− pt

Then

`t = logP(θ = +1 | It)P(θ = −1 | It)

= L+1,−1(It)


Dynamics

Define

`t = logpt

1− pt

Then

`t = logP(θ = +1 | It)P(θ = −1 | It)

= L+1,−1(It)

`t = L+1,−1(It)


Dynamics

Define

`t = logpt

1− pt

Then

`t = logP(θ = +1 | It)P(θ = −1 | It)

= L+1,−1(It)

`t = L+1,−1(It−1, at−1)


Dynamics

Define

`t = logpt

1− pt

Then

`t = logP(θ = +1 | It)P(θ = −1 | It)

= L+1,−1(It)

`t = L+1,−1(It−1) + logP(at−1 | θ = +1, It−1)

P(at−1 | θ = −1, It−1)


Dynamics

Define

`t = logpt

1− pt

Then

`t = logP(θ = +1 | It)P(θ = −1 | It)

= L+1,−1(It)

`t = `t−1 + logP(at−1 | θ = +1, It−1)

P(at−1 | θ = −1, It−1)


Dynamics

Define

`t = logpt

1− pt

Then

`t = logP(θ = +1 | It)P(θ = −1 | It)

= L+1,−1(It)

`t = `t−1 + logP(at−1 | θ = +1, It−1)

P(at−1 | θ = −1, It−1)= `t−1 + D(at−1, `t−1)


Long term dynamics

Suppose θ = +1at = +1 for all sufficiently large t

`t = `t−1 + D(+1, `t−1) for all sufficiently large t

FactFor x very large,

D(+1, x) ≈ P(

logP(s | θ = +1)

P(s | θ = −1)< −x | θ = −1

)


Long term dynamics

Suppose θ = +1

at = +1 for all sufficiently large t



D(+1, x) ≈ P(

logP(s | θ = +1)

P(s | θ = −1)< −x | θ = −1

)


Long term dynamics




D(+1, x) ≈ P(

logP(s | θ = +1)

P(s | θ = −1)< −x | θ = −1

)


Long term dynamics




D(+1, x) ≈ P(

logP(s | θ = +1)

P(s | θ = −1)< −x | θ = −1

)


Long term dynamics



Fact

For x very large,

D(+1, x) ≈ P(

logP(s | θ = +1)

P(s | θ = −1)< −x | θ = −1

)


Long term dynamics




D(+1, x) ≈ P(

logP(s | θ = +1)

P(s | θ = −1)< −x | θ = −1

)


Long term dynamics




D(+1, x) ≈ P(

logP(s | θ = +1)

P(s | θ = −1)< −x | θ = −1

)


Long term dynamics




D(+1, x) ≈ P(

logP(s | θ = +1)

P(s | θ = −1)< −x | θ = −1

)


Long term dynamics

TheoremLet f (t) be any solution to

f ′(t) = P(

logP(s | θ = +1)

P(s | θ = −1)< −f (t) | θ = −1

).

Then almost surely

limt!∞

`tf (t)

= 1.


Time to learn

Corollary`t converges sublinearly.

TheoremSublinearity is the only constraint on how quickly `t converges.

TheoremIt is possible for the expected time to learn to be finite or infinite.


Time to learn





Time to learn





Time to learn





And...

Thank you!


And...

Thank you!


Documents

The speed of sequential asymptotic learningwhanncar/slides/sses19.pdf · The speed of sequential asymptotic learning WadeHann-Caruthers1,VadimV.Martynov1,OmerTamuz1 1California Institute