Unregularized Convergence of Single-Loop, Entropy-Regularized Natural Actor-Critic
The paper analyzes a single-loop, entropy-regularized Natural Actor-Critic algorithm under compatible linear function approximation, bridging the gap between theoretical double-loop methods and practical single-loop implementations. An uncentered critic is introduced to maintain stable critic tracking even as the policy approaches determinism and the Fisher information matrix degenerates. Two optimization regimes are analyzed: a Stochastic Regime using joint Lyapunov recurrence, and a Determinis
Analysis
TL;DR
- The paper analyzes a single-loop, entropy-regularized Natural Actor-Critic algorithm under compatible linear function approximation, bridging the gap between theoretical double-loop methods and practical single-loop implementations.
- An uncentered critic is introduced to maintain stable critic tracking even as the policy approaches determinism and the Fisher information matrix degenerates.
- Two optimization regimes are analyzed: a Stochastic Regime using joint Lyapunov recurrence, and a Deterministic Regime pivoting to Policy Mirror Descent to handle the collapse of Euclidean geometry.
- An Exponential Translation mechanism maps the regularized optimality gap to the unregularized one, achieving accelerated convergence rates of $\tilde{\mathcal{O}}(T_{total}^{-1})$ in the Stochastic Regime and $\tilde{\mathcal{O}}(T_{total}^{-2/3})$ (average) / $\tilde{\mathcal{O}}(T_{total}^{-1/3})$ (last) in the Deterministic Regime.
- In the tabular setting with a positive Minimal Action Gap, the algorithm surpasses the $\mathcal{O}(T_{total}^{-1/2})$ worst-case statistical barrier, achieving $\tilde{\mathcal{O}}(T_{total}^{-2/3})$ average-iterate convergence.
Why It Matters
This work directly addresses a critical theoretical gap: while entropy regularization is ubiquitous in practice for stabilizing policy gradient methods, its ability to accelerate convergence toward the unregularized optimal policy has lacked rigorous single-loop analysis. For AI practitioners, this provides theoretical justification for the empirical success of single-loop entropy-regularized actor-critic methods and offers concrete convergence guarantees that inform hyperparameter selection, particularly around temperature tuning.
Technical Details
- Algorithm: Single-loop, entropy-regularized Natural Actor-Critic with compatible linear function approximation and an uncentered critic to handle Fisher matrix degeneracy near deterministic policies.
- Stochastic Regime: Actor and critic updates are fused into a joint Lyapunov recurrence analysis, yielding $\tilde{\mathcal{O}}(T_{total}^{-1})$ convergence up to approximation-error terms.
- Deterministic Regime: The analysis pivots to a Policy Mirror Descent framework to circumvent the collapse of Euclidean geometry as the policy becomes deterministic, achieving $\tilde{\mathcal{O}}(T_{total}^{-2/3})$ for the average iterate and $\tilde{\mathcal{O}}(T_{total}^{-1/3})$ for the last iterate.
- Exponential Translation Mechanism: Exploits a positive Minimal Action Gap in the unregularized MDP to map the regularized gap to the unregularized gap with only an exponentially decaying error tail, enabling the accelerated rates.
- Tabular Setting: The positive-action-gap analysis breaks the $\mathcal{O}(T_{total}^{-1/2})$ worst-case statistical barrier, achieving $\tilde{\mathcal{O}}(T_{total}^{-2/3})$ average-iterate rate.
Industry Insight
- The theoretical guarantees validate the widespread practical use of single-loop entropy-regularized actor-critic methods (e.g., as used in modern RL libraries), providing a foundation for principled temperature scheduling rather than heuristic tuning.
- The Exponential Translation mechanism offers a novel analytical tool that could be adapted to other regularized RL settings where bridging regularized and unregularized objectives is critical, such as in safe exploration or reward-shaping scenarios.
- The distinction between Stochastic and Deterministic regimes suggests that practitioners should expect different convergence behaviors as policies become more peaked, and average-iterate aggregation should be preferred over last-iterate output in deterministic-leaning settings.
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