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# Convergence Analysis of Reinforcement Learning Agents

Convergence Analysis of Reinforcement Learning Agents. Srinivas Turaga 9.912 30th March, 2004. The Learning Algorithm. The Assumptions. Players use stochastic strategies. Players only observe their reward . Players attempt to estimate the value of choosing a particular action.

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## Convergence Analysis of Reinforcement Learning Agents

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1. Convergence Analysis of Reinforcement Learning Agents Srinivas Turaga 9.912 30th March, 2004

2. The Learning Algorithm The Assumptions • Players use stochastic strategies. • Players only observe their reward. • Players attempt to estimate the value of choosing a particular action. The Algorithm • Play action i with probability Pr(i) • Observe reward r • Update value function v

3. The Learning Algorithm Payoff matrix Player 2’s choice Player 1’s choice The Algorithm Value of action i • Play action iwith probability Pr(i) • Proportional to value of action i • Observe reward r • Depends on other player’s choice jalso • Update value function v • 2 simple schemes Algorithm 2 Algorithm 1 If action i chosen: If action i not chosen: forgetting no forgetting

4. Analysis Techniques • Analysis of stochastic dynamics is hard! • So approximate: • Consider average case (deterministic) • Consider continuous time (differential equation) Random! Discrete time! Deterministic! Discrete time! Deterministic! Continuous time!

5. Results - Matching Pennies Game • Analysis shows a fixed point corresponding to the Nash equilibrium. Linear stability analysis shows marginal stability. • Simulations of stochastic algorithm and deterministic dynamics diverge to corners. • Analysis shows a stable fixed point corresponding to matching behavior. • Simulations of stochastic algorithm and deterministic dynamics converge as expected.

6. Future Directions • Validate approximation technique. • Analyze properties of more general reinforcement learners. • Consider situations with asymmetric learning rates. • Study behavior of algorithms for arbitrary payoff matrices.

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