A Unified Analysis of Importance-Corrected Momentum Actor-Critic over a Markovian Trajectory

In this paper, we study the role of the critic in actor--critic for entropy-regularized, finite, discounted environments. We establish that, when the critic is exact, using the latter as a baseline is a variance-reduction method in a strong sense. In this case, actor--critic with stochastic gradients matches the sample complexity of deterministic policy gradient, reaching an $ε$-optimal regularized value with $\tilde{O}(\log(1/ε))$ samples. In practice, the critic is learned alongside the actor: the variance of the actor update is then influenced by the critic's variance and bias. Specifically, when the critic has a sufficiently small error, the variance reduction and rapid convergence are preserved. This suggests to learn the critic first, keeping it up to date after each actor update, underscoring the crucial role of accurate critic estimation in actor--critic methods.

Significant research effort has been directed in recent years towards establishing both asymptotic and non-asymptotic convergence guarantees for two-timescale actor--critic algorithms, where the actor recursion is run on a slower timescale than the critic recursion. This work derives a uniform all-time concentration bound for the actor--critic algorithm with function approximation in the long-run average-reward setting. This bound helps us analyze the behavior of the actor parameter with high probability. We show that, after some finite time, the actor parameter enters a safe region and remains within it thereafter with high probability. Specifically, with probability at least $1-ε_1-ε_2$, the actor error $\Vert θ_k-θ^{*}\Vert$ is $O\left(\frac{n_0^{3/4}}{k}\frac{1}{\sqrt{ε_2}}+\left(\frac{1}{n_0}\right)^{1/4}\log^{1/4}\left(\frac{1}{ε_1}\right)+\left(\frac{1}{n_0}\right)^{1/4}\right)$ for all $k\geq n_0$ and sufficiently large $n_0$. We also present experimental results demonstrating that the aforementioned actor error diminishes with the number of actor-parameter updates.

While entropy regularization is widely used to stabilize and accelerate Natural Policy Gradient methods, its ability to yield faster convergence rates for the unregularized objective remains underexplored. Existing analyses often rely on double-loop architectures and invoke a linear entropy penalty. To bridge the gap between theory and practice, we analyze a single-loop, entropy-regularized Natural Actor-Critic algorithm under compatible linear function approximation. By training an uncentered critic, our critic tracking can remain stable even as the training policy approaches determinism and the Fisher information matrix degenerates. We focus on two primary regimes for the optimization landscape: a Stochastic Regime, where we fuse coupled actor-critic updates into a joint Lyapunov recurrence, and a Deterministic Regime, where we pivot to a Policy Mirror Descent framework to circumvent the collapse of Euclidean geometry. By exploiting a positive Minimal Action Gap in the unregularized Markov decision process, we introduce an Exponential Translation mechanism that maps the regularized gap to the unregularized one up to an exponentially decaying tail. By tuning the fixed temperature, our algorithm achieves accelerated unregularized convergence rates, up to approximation-error terms: $\tilde{\mathcal{O}}(T_{total}^{-1})$ in the Stochastic Regime, and $\tilde{\mathcal{O}}(T_{total}^{-2/3})$ for the average iterate alongside $\tilde{\mathcal{O}}(T_{total}^{-1/3})$ for the last iterate in the Deterministic Regime. Here, $T_{total}$ denotes the total number of stochastic critic updates (or Monte Carlo rollouts). Furthermore, in the tabular setting, our positive-action-gap analysis yields a $\tilde{\mathcal{O}}(T_{total}^{-2/3})$ average-iterate rate, surpassing the $\mathcal{O}(T_{total}^{-1/2})$ worst-case statistical barrier that applies without a positive action margin.