Establishing almost sure convergence rates for stochastic approximation and reinforcement learning under Markovian noise is a fundamental theoretical challenge. We make progress towards this challenge for a class of stochastic approximation algorithms whose expected updates are contractive, a setting that arises in many reinforcement learning algorithms such as $Q$-learning and linear temporal difference learning. Specifically, for a power-law learning rate $O(n^{-η})$ with $η\in (1/2, 1)$, we obtain an almost sure convergence rate arbitrarily close to $o(n^{1 - 2η})$. For the harmonic learning rate $α_n=α/(n+1)$, the almost sure squared-error rate is $o(n^{-ζ})$ for every $ζ<\min\{1,2(1-κ)α\}$, where $κ$ is the contraction factor. Thus exponents arbitrarily close to one are available when $2(1-κ)α\geq1$. This approaches the $n^{-1}$ polynomial exponent of classical law-of-the-iterated-logarithm benchmarks. Key to our analysis is a novel Lyapunov drift construction that applies a Poisson-equation based correction for Markovian noise to the well-established Moreau-envelope smoothing for the contractive mapping.
The ecosystem of Lean and Mathlib has become the de facto standard for large language model (LLM) assisted formal reasoning with remarkable successes in recent years. Those successes, however, only consume Mathlib as an essential dependency but do not directly contribute to it. In the meantime, the growth of Mathlib has recently been bottlenecked by the review process, which requires human reviewers to judge whether proposed pull requests (PRs) follow the Mathlib's conventions and are worth integrating as part of a shared mathematical infrastructure. This leads to our central question: can LLMs help review Mathlib PRs? To this end, we introduce MathlibPR, a benchmark built from real Mathlib4 PR histories. We further propose a staged evaluation protocol and use it to evaluate both LLM models (e.g., DeepSeek, Qwen, Goedel, and Kimina) and LLM agents (e.g., Codex and Claude Code). Surprisingly, both LLM models and LLM agents struggle to distinguish merge-ready PRs from build-passing PRs that were revised or never merged. By turning Mathlib PR histories into a supervised signal, MathlibPR provides a step toward reviewer assistants and reward models that could help evaluate PRs and steer LLMs toward producing merge-ready Mathlib contributions.
Learning rate is a critical component of reinforcement learning (RL). This work uses global and local clocks to distinguish two types of learning rates. The former is of the standard form $α_t$ that depends only on the time step $t$ (i.e., a global clock). The latter is of the form $α_{ν(S_t, t)}$, where $ν(s, t)$ counts the number of visits to state $s$ until time $t$ (i.e., a local clock). In discounted RL, an RL algorithm that is convergent with a local clock is always also convergent with a global clock, and vice versa. We are not aware of any counterexample. The key contribution of this work is to show that this nice correspondence breaks down in average-reward RL. Specifically, we construct a counterexample showing that although differential temporal difference (TD) learning is convergent with a local clock, it can diverge with a global clock. We empirically observe this divergence in our experiments and also prove that the associated ODE of differential TD with a global clock is unstable. This counterexample closes the open problem in Wan et al. [2021], Blaser et al. [2026].