Minimax strategy for prediction with expert advice under stochastic - - PowerPoint PPT Presentation

Minimax strategy for prediction with expert advice under stochastic assumptions

Wojciech Kot lowski

Pozna´ n University of Technology, Poland

Learning Faster from Easy Data II NIPS 2015 Workshop

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Prediction with expert advice

In trials t = 1, 2, . . . , T: – Algorithm predicts with wt ∈ ∆K. – Loss vector ℓt ∈ [0, 1]K is revealed. – Algorithm incurs loss wt · ℓt. Regret of a strategy ω = (w1, . . . , wT ): R =

• t

wt · ℓt − min

k

• t≤T

ℓt,k

LT,k

Goal: find ω minimizing the worst-case regret over all sequences.

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Prediction with expert advice

In trials t = 1, 2, . . . , T: – Algorithm predicts with wt ∈ ∆K. – Loss vector ℓt ∈ [0, 1]K is revealed. – Algorithm incurs loss wt · ℓt. Regret of a strategy ω = (w1, . . . , wT ): R =

• t

wt · ℓt − min

k

• t≤T

ℓt,k

LT,k

Goal: find ω minimizing the worst-case regret over all sequences. Too pessimistic: minimax ω has the same regret on all sequences!

2 / 3

Prediction with expert advice

In trials t = 1, 2, . . . , T: – Algorithm predicts with wt ∈ ∆K. – Loss vector ℓt ∈ [0, 1]K is revealed. – Algorithm incurs loss wt · ℓt. Regret of a strategy ω = (w1, . . . , wT ): R =

• t

wt · ℓt − min

k

• t≤T

ℓt,k

LT,k

Goal: find ω minimizing the worst-case regret over all sequences. Too pessimistic: minimax ω has the same regret on all sequences! Drop the minimax principle? Drop the worst-case assumptions?

2 / 3

Prediction with expert advice

In trials t = 1, 2, . . . , T: – Algorithm predicts with wt ∈ ∆K. – Loss vector ℓt ∈ [0, 1]K is revealed. – Algorithm incurs loss wt · ℓt. Regret of a strategy ω = (w1, . . . , wT ): R =

• t

wt · ℓt − min

k

• t≤T

ℓt,k

LT,k

Goal: find ω minimizing the worst-case regret over all sequences. Too pessimistic: minimax ω has the same regret on all sequences! Drop the minimax principle? Drop the worst-case assumptions?

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Stochastic setting: optimal strategy for “easy” data

Assumption: Each expert k = 1, . . . , K generates losses i.i.d. from a fixed distribution Pk. Goal: Find the minimax strategy ω w.r.t. all choices of distributions P = (P1, . . . , Pk).

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Stochastic setting: optimal strategy for “easy” data

Assumption: Each expert k = 1, . . . , K generates losses i.i.d. from a fixed distribution Pk. Goal: Find the minimax strategy ω w.r.t. all choices of distributions P = (P1, . . . , Pk). Minimax in terms of what?

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Stochastic setting: optimal strategy for “easy” data

Assumption: Each expert k = 1, . . . , K generates losses i.i.d. from a fixed distribution Pk. Goal: Find the minimax strategy ω w.r.t. all choices of distributions P = (P1, . . . , Pk). Minimax in terms of what? Expected regret: Reg(ω, P) = E

t wt · ℓt − mink LT,k

• Redundancy:

Red(ω, P) = E

t wt · ℓt

• − mink E [LT,k]

Excess risk: Risk(ω, P) = E [wT · ℓT ] − mink E [ℓT,k]

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Stochastic setting: optimal strategy for “easy” data

Assumption: Each expert k = 1, . . . , K generates losses i.i.d. from a fixed distribution Pk. Goal: Find the minimax strategy ω w.r.t. all choices of distributions P = (P1, . . . , Pk). Minimax in terms of what? Expected regret: Reg(ω, P) = E

t wt · ℓt − mink LT,k

• Redundancy:

Red(ω, P) = E

t wt · ℓt

• − mink E [LT,k]

Excess risk: Risk(ω, P) = E [wT · ℓT ] − mink E [ℓT,k] We give a strategy ω∗, which is minimax with respect to all three measures simultaneously: sup

P

R(ω∗, P) = inf

ω sup P

R(ω, P), where R is either Reg, Red, or Risk.

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