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Zixuan Xie

Zixuan Xie contributes to research discovery and scholarly infrastructure.

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Machine Learning Artificial Intelligence Computation and Language Computational Engineering, Finance, and Science Computer Science and Game Theory Logic in Computer Science math.OC

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preprint2026arXiv

Almost Sure Convergence Rates of Stochastic Approximation and Reinforcement Learning via a Poisson-Moreau Drift

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 a harmonic learning rate $O(n^{-1})$, we obtain an almost sure convergence rate arbitrarily close to $o(n^{-1})$, which we argue is a strong result because it is close to the optimal rate $O(n^{-1}\log\log n)$ given by the law of the iterated logarithm (for a special case of i.i.d. noise). 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.

preprint2026arXiv

Beyond Linear Attention: Softmax Transformers Implement In-Context Reinforcement Learning

In-context reinforcement learning (ICRL) studies agents that, after pretraining, adapt to new tasks by conditioning on additional context without parameter updates. Existing theoretical analyses of ICRL largely rely on linear attention, which replaces the softmax function in the standard attention with an identity mapping. This paper provides the first theoretical understanding of ICRL without making the unrealistic linear attention simplification. In particular, we consider the standard softmax attention used in practice. We show that, with certain parameters, the layerwise forward pass of a Transformer with such softmax attention is equivalent to iterative updates of a weighted softmax temporal difference (TD) learning algorithm. Here, weighted softmax TD is a new RL algorithm that performs policy evaluation in kernel space and adopts both linear TD and tabular TD as special cases. We also prove that under a certain contraction condition, the policy evaluation error decays as the number of layers grows, with the identified parameters above. Finally, we prove that those parameters are a global minimizer of a pretraining loss, explaining their emergence in our numerical experiments.

preprint2026arXiv

Convergence and Emergence of In-Context Reinforcement Learning with Chain of Thought

In-context reinforcement learning (ICRL) refers to the ability of RL agents to adapt to new tasks at inference time without parameter updates by conditioning on additional context. Recent empirical studies further demonstrate that Chain-of-Thought (CoT) generation can amplify this ICRL capability. This paper is the first to provide a theoretical understanding on how CoT interacts with ICRL. We conduct our analysis in a policy evaluation setup with linear Transformer. We prove that with specific Transformer parameters, the CoT generation process is equivalent to repeatedly executing temporal difference learning updates. Additionally, we provide finite sample convergence analysis showing that the policy evaluation error decreases geometrically with CoT length and eventually saturates at a statistical floor determined by the context length. We also prove that the desired Transformer parameters are a global minimizer of the pretraining loss, providing a theoretical understanding on the empirical emergence of those parameters.

preprint2026arXiv

From Feedback Loops to Policy Updates: Reinforcement Fine-Tuning for LLM-Based Alpha Factor Discovery

Modern quantitative trading increasingly relies on systematic models to extract predictive signals from large-scale financial data, where alpha factor discovery plays a central role in transforming market observations into tradable signals. Recent LLM-based methods have shown promise in automating factor generation, but most of them still rely on prompt-level generation--evaluation--feedback loops for iterative optimization. As the loop becomes longer, repeatedly appended historical candidates and feedback can cause context explosion, increase inference cost, dilute useful information, and introduce feedback drift. Moreover, these methods often depend on very large LLMs whose stable generation preferences may lead to structurally similar expressions, redundant candidates, and search stagnation. To address these limitations, we propose \textsc{QuantEvolver}, a self-evolving alpha factor discovery framework based on reinforcement fine-tuning. Instead of accumulating feedback in the prompt, \textsc{QuantEvolver} converts executable quantitative evaluation into policy updates, enabling a Miner LLM to internalize historical optimization experience through parameter learning. Specifically, \textsc{QuantEvolver} constructs high-quality seed factors, builds diverse seed--time-window training tasks, generates executable Factor DSL expressions, evaluates them through Regime Backtest, and optimizes the Miner LLM with Diversity-Complementarity Reward. During training, high-quality factors are continuously accumulated in a Mined Factor Database, which serves as the final discovered factor library. Extensive experiments on three realistic market benchmarks demonstrate the effectiveness of \textsc{QuantEvolver}, which consistently improves the primary evaluation metric of each task over existing LLM-based alpha factor discovery baselines, produces higher-quality and more complementary factor pools.

preprint2026arXiv

MathlibPR: Pull Request Merge-Readiness Benchmark for Formal Mathematical Libraries

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.

preprint2026arXiv

Offline Two-Player Zero-Sum Markov Games with KL Regularization

We study the problem of learning Nash equilibria in offline two-player zero-sum Markov games. While existing approaches often rely on explicit pessimism to address distribution shift, we show that KL regularization alone suffices to stabilize learning and guarantee convergence. We first introduce Regularized Offline Sequential Equilibrium (ROSE), a theoretical framework that achieves a fast $\widetilde{\mathcal{O}}(1/n)$ convergence rate under \textit{unilateral concentrability}, improving over the standard $\widetilde{\mathcal{O}}(1/\sqrt{n})$ rates in unregularized settings. We then propose Sequential Offline Self-play Mirror Descent (SOS-MD), a practical model-free algorithm based on least-squares value estimation and iterative self-play updates. We prove that the last iterate of SOS-MD attains the same $\widetilde{\mathcal{O}}(1/n)$ statistical rate up to a vanishing optimization error of order $\widetilde{\mathcal{O}}(1/\sqrt{T})$ in the number of self-play iterations $T$.

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