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This research proposes a novel framework where causal softmax attention mechanisms function as repeated controllers implementing policy mirror descent, effectively bridging transformer architecture with advanced reinforcement learning theory. Empirical results demonstrate that learned actors with exact critics closely approximate the theoretical oracle, significantly outperforming existing adaptations like algorithm distillation in closed-loop control tasks. The study advances our understanding of how standard attention layers can inherently encode complex optimization dynamics without requiring architectural modifications.
PolicyAttention: Softmax Attention Implements Policy Mirror Descent for Closed-Loop Control
arXiv:2609.30500v1 Announce Type: new Abstract: Can causal softmax attention implement policy mirror descent as a repeated controller rather than a one-step algebraic identity? Negative-entropy policy mirror descent (PMD) has the statewise update $\operatorname{PMD}_\eta(\pi,Q)=\operatorname{softmax}(\log\pi+\eta Q)$. Building on the known Q-TD-PMD recursion, we construct one fixed causal-softmax actor--environment--one-step-critic protocol with explicit actor, routing, sampling, and normalization residuals, and propagate them to the policy actually returned. The construction states the finite-logit/full-support domain, the external tokenization and sampling boundary, and the mean-zero LayerNorm carrier conditions required by the normalized compilation. Separately trained pre-LN Transformers recover the target computation empirically. A frozen one-step audit model is closest to PMD among the tested fixed rules; in a preregistered five-run $S=4$ repeated-control test, the learned actor with an exact one-step critic reaches median returned-policy loss $1.052\times$ the Exact PMD oracle and retains the criterion across four no-retraining shifts. The same checkpoints with their learned critic give descriptive median $1.050\times$ the oracle (no registered margin). At $S=8$, replacing the exact critic by the learned critic raises median $T=20$ loss to $0.0225$ yet leaves the Liang--Lai and Algorithm Distillation adaptations $20.2$--$24.2\times$ higher-loss; this is a one-sided sampled-critic bound because PolicyAttention consumes 144 generative transitions per round versus 20 on-policy transitions for the adaptations. The strict 20-transition comparison remains open. At $S=8,16$, the exact-critic common-harness comparison remains $17.7$--$28.2\times$ lower-loss than those adaptations, with the information asymmetry stated locally.
Abstract:Can causal softmax attention implement policy mirror descent as a repeated controller rather than a one-step algebraic identity? Negative-entropy policy mirror descent (PMD) has the statewise update $\operatorname{PMD}_\eta(\pi,Q)=\operatorname{softmax}(\log\pi+\eta Q)$. Building on the known Q-TD-PMD recursion, we construct one fixed causal-softmax actor--environment--one-step-critic protocol with explicit actor, routing, sampling, and normalization residuals, and propagate them to the policy actually returned. The construction states the finite-logit/full-support domain, the external tokenization and sampling boundary, and the mean-zero LayerNorm carrier conditions required by the normalized compilation. Separately trained pre-LN Transformers recover the target computation empirically. A frozen one-step audit model is closest to PMD among the tested fixed rules; in a preregistered five-run $S=4$ repeated-control test, the learned actor with an exact one-step critic reaches median returned-policy loss $1.052\times$ the Exact PMD oracle and retains the criterion across four no-retraining shifts. The same checkpoints with their learned critic give descriptive median $1.050\times$ the oracle (no registered margin). At $S=8$, replacing the exact critic by the learned critic raises median $T=20$ loss to $0.0225$ yet leaves the Liang--Lai and Algorithm Distillation adaptations $20.2$--$24.2\times$ higher-loss; this is a one-sided sampled-critic bound because PolicyAttention consumes 144 generative transitions per round versus 20 on-policy transitions for the adaptations. The strict 20-transition comparison remains open. At $S=8,16$, the exact-critic common-harness comparison remains $17.7$--$28.2\times$ lower-loss than those adaptations, with the information asymmetry stated locally.
Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
arXiv:2609.30500 [cs.LG]
(or arXiv:2609.30500v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.30500 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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