A Diffusion Analysis of Policy Gradient for Stochastic Bandits

Tor Lattimore's new research paper, 'A Diffusion Analysis of Policy Gradient for Stochastic Bandits,' provides a rigorous mathematical framework for understanding how reinforcement learning algorithms perform in classic decision-making problems. The work analyzes the continuous-time diffusion approximation of policy gradient methods applied to k-armed stochastic bandits—a fundamental problem where an agent must repeatedly choose between k options with unknown reward distributions. The paper proves that with a carefully chosen learning rate η = O(Δ²/log(n)), where Δ represents the minimum gap between optimal and suboptimal arms and n is the time horizon, the algorithm achieves regret bounded by O(k log(k) log(n)/η). This represents a significant theoretical advancement in understanding the convergence properties of policy gradient methods.

Beyond establishing this upper bound, the research makes a crucial contribution by constructing a specific problem instance that reveals fundamental limitations. The paper demonstrates that for certain bandit configurations with only logarithmically many arms, the regret becomes linear in the time horizon unless the learning rate satisfies η = O(Δ²). This establishes a precise mathematical threshold that separates efficient learning from poor performance. The 17-page paper, submitted to arXiv in March 2026, provides both theoretical guarantees and concrete counterexamples that will influence how researchers design and analyze reinforcement learning algorithms for sequential decision-making tasks.