Delightful Gradients Accelerate Corner Escape

This work revisits standard policy gradient methods used on restricted policy classes, which are known to get stuck in suboptimal critical points. We identify an important cause for this phenomenon to be that the policy gradient is itself fundamentally myopic, i.e. it only improves the policy based on the one-step $Q$-function. In this work, we propose a generalized $k$-step policy gradient method that couples the randomness within a $k$-step time window and can escape the myopic local optima in MDPs with restricted policy classes. We show this new method is theoretically guaranteed to converge to a solution that is exponentially close in performance to the optimal deterministic policy with respect to $k$. Further, we show projected gradient descent and mirror descent with this $k$-step policy gradient can achieve this exponential guarantee in $O(\frac{1}{T})$ iterations, despite only assuming smoothness and differentiability of the value function. This will provide near optimal solutions to previously elusive applications like state aggregation and partially observable cooperative multi-agent settings. Moreover, our bounds avoid the ubiquitous distribution mismatch factors $||d_μ^{π^*} / d_μ^π||_\infty$ and $||d_μ^{π^*} / μ||_\infty$ enabling the $k$-step policy gradient method to escape suboptimal critical points that emerge from poor exploration in fully observable settings.

This paper studies the policy gradient update for a multi-arm bandit problem in diffusion environment that is described by a stochastic differential equation (SDE) under the continuous-time reinforcement learning framework by Wang et al. (2020), Jia and Zhou (2022b). With the logit parameterization for the stochastic policy, we show that it converges almost surely to the optimal arm under an arbitrary constant learning rate. Furthermore, we derive the non-asymptotic regret upper bound when the constant learning rate is below a time-invariant threshold; and the regret bound has order $O(\log T)$. We improve the analysis in Lattimore (2026a) for the same SDE by constructing a novel Lyapunov function and demonstrate the transparency of analyzing policy gradient using the tools in SDEs. In addition, the same Lyapunov function is also helpful in analyzing the discrete-time policy gradient algorithm.

We study the global convergence of policy gradient for infinite-horizon entropy-regularized Markov decision processes (MDPs) with continuous state and action spaces. We consider log-linear softmax policies with linear function approximation, which extend the tabular softmax parameterization while retaining a tractable policy class. Under $Q^π_τ$-realizability for the regularized state-action value function, we first establish a non-uniform Polyak--Łojasiewicz (PŁ) inequality. The non-uniformity arises through degeneracy of constants associated with the policy geometry, namely the Fisher information matrix or an uncentered feature covariance matrix. We then identify two feature regimes under which this non-uniform constant can be bounded along the gradient flow. For full-affine-span features, we prove radial unboundedness of the KL regularizer and show that the smallest eigenvalue of the Fisher information matrix remains bounded below by an initialization-dependent positive constant. For simplex-valued features, we prove an analogous radial unboundedness result in the subspace orthogonal to the all-ones vector and obtain a uniform lower bound for the smallest eigenvalue of the uncentered covariance matrix. These results imply global linear convergence of the regularized objective along the gradient flow, i.e. suboptimality decaying as $\mathcal{O}(e^{-Ct})$ for some $C>0$. Our analysis extends the global convergence theory of entropy-regularized softmax policy gradient beyond the tabular setting of Agarwal et al. (2020); Bhandari and Russo (2024); Mei et al. (2020).