Alternative Function Approximation Parameterizations for Solving Games: An Analysis of $f$-Regression Counterfactual Regret Minimization
Bibliographic record
Abstract
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.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.014 | 0.064 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.004 | 0.004 |
| Research integrity | 0.003 | 0.006 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".