Regret Minimization in Stochastic Games

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Regret Minimization in Stochastic Games. Shie Mannor and Nahum Shimkin Technion, Israel Institute of Technology Dept. of Electrical Engineering. Introduction. Modeling of a dynamic decision process as a stochastic game: Non stationarity of the environment

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Regret Minimization in Stochastic Games

Shie Mannor and Nahum Shimkin

Technion, Israel Institute of Technology

Dept. of Electrical Engineering

UAI 2000

Introduction
• Modeling of a dynamic decision process as a stochastic game:
• Non stationarity of the environment
• Environments are not (necessarily) hostile
• Looking for the best possible strategy in light of the environment’s actions.

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Repeated Matrix Games
• The sets of single stage strategies P and Q are simplical.
• Rewards are defined by a reward matrix G: r(p,q)=pGq
• Reward criteria - average reward

Need not converge –stationarity is not

assumed

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Regret for Repeated Matrix Games
• Suppose by time t, average reward is , opponent empirical strategy is qt.
• The regret is defined as:
• A policy is called regret minimizing if:

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Regret minimization for repeated matrix games
• Such policies do exist (Hannan, 56)
• A proof using Approachability theory (Blackwell, 56)
• Also for games with partial observation (Auer et al. ,1995 ; Rustichini, 1999)

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Stochastic Games
• Formal Model:

S={1,…,s} state space

A=A(s) actions of Regret minimizing player, P1

B=B(s) actions of the “environment”, P2

r - reward function, r(s,a,b)

P - transition kernel, P(s`|s,a,b)

• Expected average for pP, qQ is r(p,q)
• Single state recurrence assumption

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Bayes Reward in Strategy Space
• For every stationary strategy qQ, the Bayes reward is defined as:
• Problems:
• P2’s strategy is not completely observed
• P1’s observations may depends on the strategies of both players

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Bayes Reward in State-Action Space
• Let psb be the observed frequency of P2’s action b and state s.
• A natural estimate of q is:

The associated Bayes envelope is:

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Approachability Theory
• A standard tool in the theory of repeated matrix games (Blackwell, 1956)
• For a game with vector reward and average reward
• A set is approachable by P1 with a policy s if:
• Was extended to recurrent stochastic games (Shimkin and Shwartz, 1993)

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The Convex Bayes Envelope
• In general BE is not approachable.
• Define CBE=co(BE), that is

where is the lower convex hull

of

Theorem: CBE is approachable.

(val is the value of the game)

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Single Controller Games

Theorem: Assume that P2 alone controls the transitions, i.e.

then BE itself is approachable.

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An Application to Prediction with Expert Advice
• Given a channel and a set of experts
• At each time epoch each expert states his prediction of the next symbol and P1 has to choose his prediction, 
• Then a letter  appears in the channel and P1 receives his prediction reward r(, )
• Problem can be formulated as stochastic game, P2 stands for all experts and the channel

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(0,0,0)

r(a,b)

r=0

0

0

(k-1,k,k)

(k,k,k)

Expert recommendation

Prediction Example (cont’)

Theorem: P1 has a zero regret strategy.

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a=1

P=0.99

P=0.99

P=0.99

r=b

S1

r=b

S0

a=0

B(0)=B(1)={-1,1}

P=0.99

An example in which BE is not approachable

It can be proved that BE for the

above game is not approachable

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Example (cont’)
• In r*(q) space the envelopes are:

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Open questions
• Characterization of minimal approachable sets in reward-state-actions space
• On-line learning schemes for stochastic games with unknown parameters
• Other ways of formulating optimality with respect to observed state action frequencies

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Conclusions
• The problem of regret minimization for stochastic games was considered
• The proposed solution concept, CBE, is based on convexification of the Bayes envelope in the natural state action space.
• The concept of CBE ensures an average reward that is higher than value when the opponent is sub optimal

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Regret Minimization in Stochastic Games

Shie Mannor and Nahum Shimkin

Technion, Israel Institute of Technology

Dept. of Electrical Engineering

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Approachability Theory
• Let m(p,q) be the average vector valued reward in a game when P1 and P2 play p and q
• Define
• Theorem [Blackwell 56]: A convex set C is approachable if and only if for every qQ
• Extended to stochastic games (Shimkin and Shwartz, 1993)

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A related Vector Valued Game
• Define the following vector valued game:
• If in state s action b is played by P2 and a reward r is gained then the vector valued mt :

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