Alternative Function Approximation Parameterizations for Solving Games: An Analysis of $f$-Regression Counterfactual Regret Minimization
This work provides incremental improvements for researchers in game theory and reinforcement learning by theoretically and empirically comparing parameterizations to optimize regret bounds.
The paper tackles the problem of solving imperfect information games by analyzing alternative policy parameterizations in regression counterfactual regret minimization, finding that ReLU performs better with small approximation error while softmax excels with large error.
Function approximation is a powerful approach for structuring large decision problems that has facilitated great achievements in the areas of reinforcement learning and game playing. Regression counterfactual regret minimization (RCFR) is a simple algorithm for approximately solving imperfect information games with normalized rectified linear unit (ReLU) parameterized policies. In contrast, the more conventional softmax parameterization is standard in the field of reinforcement learning and yields a regret bound with a better dependence on the number of actions. We derive approximation error-aware regret bounds for $(Φ, f)$-regret matching, which applies to a general class of link functions and regret objectives. These bounds recover a tighter bound for RCFR and provide a theoretical justification for RCFR implementations with alternative policy parameterizations ($f$-RCFR), including softmax. We provide exploitability bounds for $f$-RCFR with the polynomial and exponential link functions in zero-sum imperfect information games and examine empirically how the link function interacts with the severity of the approximation. We find that the previously studied ReLU parameterization performs better when the approximation error is small while the softmax parameterization can perform better when the approximation error is large.