Refined Analysis of Entropy-Regularized Actor-Critic

Actor-critic methods depend on learned critics, but critic quality is often evaluated only indirectly through return, temporal-difference error, or value loss. Critic complexity is introduced as an additional diagnostic and intervention dimension for actor-critic reinforcement learning. The analysis uses spectral effective-rank entropy, a rank-like summary of the singular-value distributions of critic weight matrices, to assess critic model complexity. Across TD3 and PPO experiments, critic complexity is tracked together with return and Monte Carlo value-estimation bias. The results show that critic complexity is measurable throughout training and is systematically associated with training behavior, while also making clear that the relationship is heterogeneous across algorithms, tasks, and hyperparameters. A direct complexity-control intervention is then evaluated by adding a spectral-entropy penalty to the critic loss. This intervention reliably changes the targeted spectral quantity, demonstrating that critic complexity can be controlled rather than only observed. Return effects are treated as task-dependent evidence rather than as a general performance claim, because overall complexity-control results vary.

Mixture policies theoretically offer greater flexibility than unimodal policies in continuous action reinforcement learning, but the practical benefits of this complexity remain elusive. Mixture policies are notably absent from most state-of-the-art algorithms, raising a fundamental question: Is the added representational overhead useful? We show that increased flexibility can theoretically enhance solution quality and entropy robustness. Yet standard algorithms like SAC do not leverage these advantages. A core issue is the lack of a low-variance reparameterization trick for mixtures, a luxury Gaussian policies enjoy. We propose a marginalized reparameterization (MRP) estimator to address this, proving it offers lower variance than the standard likelihood-ratio (LR) approach. Our experiments across Gym MuJoCo, DeepMind Control Suite, and MetaWorld show that MRP mixture policies significantly outperform their LR ones, and reach parity (sometimes better) with Gaussian counterparts. In addition, we do find several cases where MRP mixture policies exhibit clear empirical advantages. In this paper, we provide a clearer understanding of the trade-offs involved, elevating MRP mixture policies from theoretical curiosity to a practical tool.

In this paper, we establish last-iterate convergence rates for off-policy actor--critic methods in reinforcement learning. In particular, under a single-loop, single-timescale implementation and a broad class of policy updates, including approximate policy iteration and natural policy gradient methods, we prove the first $\tilde{\mathcal{O}}(ε^{-2})$ sample complexity guarantee for finding an $ε$-optimal policy under minimal assumptions, namely, the existence of a policy that induces an irreducible Markov chain. This stands in stark contrast to the existing literature, where an $\tilde{\mathcal{O}}(ε^{-2})$ sample complexity is achieved only through nested-loop updates and/or under strong, algorithm-dependent assumptions on the policies, such as uniform mixing and uniform exploration. Technically, to address the challenges posed by the coupled update equations arising from the single-loop implementation, as well as the potentially unbounded iterates induced by off-policy learning, our analysis is based on a coupled Lyapunov drift framework. Specifically, we establish a geometric convergence rate for the actor and an $\tilde{\mathcal{O}}(1/T)$ convergence rate for the critic, and combine the two Lyapunov drift inequalities through a cross-domination property. We believe this analytical framework is of independent interest and may be applicable to other coupled iterative algorithms with unbounded