The speed of sequential asymptotic learningwhanncar/slides/sses19.pdf · The speed of sequential...

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

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 2 / 14

Introduction

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 2 / 14

Introduction

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 2 / 14

Learning from signals

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 3 / 14

Learning from signals

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 3 / 14

Learning from signals

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 3 / 14

Learning from signals

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 3 / 14

Learning from signals

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 3 / 14

Learning from signals

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 3 / 14

Learning from signals

!!

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 3 / 14

Learning from signals

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 3 / 14

Learning from signals

!!

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 3 / 14

Learning from signals

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 3 / 14

Learning from signals

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 3 / 14

Learning from signals

s

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 3 / 14

Learning from signals

!!s

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 3 / 14

Learning from signals

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 3 / 14

Learning from signals

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 3 / 14

Learning from signals

P(ω | I , s)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 3 / 14

Learning from signals

P(ω | I , s) =P(s | I , ω)P(I , ω)

P(s | I )

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 3 / 14

Learning from signals

P(ω | I , s) =P(s | I , ω)P(I , ω)

P(s | I )=

P(s |ω)P(I , ω)

P(s | I ).

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 3 / 14

Learning from signals

Fixing some ω0,

P(ω | I , s)

P(ω0 | I , s)=

P(s |ω)P(I , ω)

P(s |ω0)P(I , ω0).

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 3 / 14

Learning from signals

logP(ω | I , s)

P(ω0 | I , s)= log

P(s |ω)P(I , ω)

P(s |ω0)P(I , ω0)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 3 / 14

Learning from signals

logP(ω | I , s)

P(ω0 | I , s)= log

P(I , ω)

P(I , ω0)+ log

P(s |ω)

P(s |ω0)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 3 / 14

Learning from signals

logP(ω | I , s)

P(ω0 | I , s)= log

P(ω | I )P(ω0 | I )

+ logP(s |ω)

P(s |ω0)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 3 / 14

Learning from signals

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).

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 3 / 14

Learning from signals

Lω,ω0(I , s) = Lω,ω0(I ) + logP(s |ω)

P(s |ω0)

Define

Lω1,ω2(J) = logP(ω1 | J)

P(ω2 | J).

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 3 / 14

Observational learning

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 4 / 14

Observational learning

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 4 / 14

Observational learning

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 4 / 14

Observational learning

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 4 / 14

Observational learning

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 4 / 14

Observational learning

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 4 / 14

Observational learning

!!

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 4 / 14

Observational learning

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 4 / 14

Observational learning

!!

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 4 / 14

Observational learning

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 4 / 14

Observational learning

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 4 / 14

Observational learning

!!

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 4 / 14

Observational learning

!!??

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 4 / 14

Observational learning

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 4 / 14

Observational learning

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 4 / 14

Observational learning

!!

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 4 / 14

Observational Learning

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 5 / 14

Observational Learning

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

.

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 5 / 14

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

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 5 / 14

Observational Learning

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

.

For all a′, ∑ω

P(ω | I , s)(u(a, ω)− u(a′, ω)) ≥ 0

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 5 / 14

Observational Learning

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

.

For all a′, ∑ω

P(ω | I , s)

P(ω0 | I , s)(u(a, ω)− u(a′, ω)) ≥ 0

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 5 / 14

Observational Learning

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

.

For all a′, ∑ω

eLω,ω0 (I ,s)(u(a, ω)− u(a′, ω)) ≥ 0

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 5 / 14

Observational Learning

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

.

For all a′, ∑ω

eLω,ω0 (I )+log P(s |ω)

P(s |ω0) (u(a, ω)− u(a′, ω)) ≥ 0

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 5 / 14

Observational Learning

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

.

For all a′, ∑ω

P(s |ω)

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

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 5 / 14

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)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 6 / 14

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)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 6 / 14

Model

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

Prior: π ∈ ∆(Ω)

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

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

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 6 / 14

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)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 6 / 14

Model

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

Actions: at ∈ A = −1,+1

Utility: u(a, θ) = 1(a = θ)

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

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 6 / 14

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)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 6 / 14

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)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 6 / 14

Background

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

Public beliefspt = P(θ = +1 | It)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 7 / 14

Background

How much does society know?

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

Public beliefspt = P(θ = +1 | It)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 7 / 14

Background

How much does society know?Beliefs of fictitious outside observer

It = a1, . . . , at−1

Public beliefspt = P(θ = +1 | It)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 7 / 14

Background

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

Public beliefspt = P(θ = +1 | It)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 7 / 14

Background

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

Public beliefs

pt = P(θ = +1 | It)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 7 / 14

Background

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

Public beliefspt = P(θ = +1 | It)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 7 / 14

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.

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 8 / 14

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.

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 8 / 14

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.

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 8 / 14

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.

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 8 / 14

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.

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 8 / 14

Speed of learning

Suppose signals are unboundedly informative.

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

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 9 / 14

Speed of learning

Suppose signals are unboundedly informative.

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

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 9 / 14

Speed of learning

Suppose signals are unboundedly informative.

QuestionHow fast does learning occur?

−! At what rate does (pt) converge?

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 9 / 14

Speed of learning

Suppose signals are unboundedly informative.

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

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 9 / 14

Dynamics

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 10 / 14

Dynamics

Define

`t = logpt

1− pt

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 10 / 14

Dynamics

Define

`t = logpt

1− pt

Then

`t

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 10 / 14

Dynamics

Define

`t = logpt

1− pt

Then

`t = logP(θ = +1 | It)P(θ = −1 | It)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 10 / 14

Dynamics

Define

`t = logpt

1− pt

Then

`t = logP(θ = +1 | It)P(θ = −1 | It)

= L+1,−1(It)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 10 / 14

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)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 10 / 14

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)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 10 / 14

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)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 10 / 14

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)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 10 / 14

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)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 10 / 14

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

)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 11 / 14

Long term dynamics

Suppose θ = +1

at = +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

)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 11 / 14

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

)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 11 / 14

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

)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 11 / 14

Long term dynamics

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

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

Fact

For x very large,

D(+1, x) ≈ P(

logP(s | θ = +1)

P(s | θ = −1)< −x | θ = −1

)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 11 / 14

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

)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 11 / 14

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

)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 11 / 14

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

)

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 11 / 14

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.

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 12 / 14

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.

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 13 / 14

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.

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 13 / 14

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.

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 13 / 14

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.

Wade Hann-Caruthers, Vadim V. Martynov, Omer Tamuz (California Institute of Technology)The speed of sequential asymptotic learning August 2019 13 / 14

And...

Thank you!

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

And...

Thank you!

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