Fine-Tuning on Self-Generated and Reward-Weighted Data: Learning Dynamics, Convergence Rates, and Benefits of Off-Policyness

2026-09-29

AI Summary

This study addresses the lack of theoretical foundations for off-policy self-generated data fine-tuning in large model post-training, where convergence is difficult to guarantee when the sampling distribution is updated infrequently. To this end, this work proposes RE(S), a unified framework that formulates the optimization as a staged KL-divergence minimization process and provides rigorous analysis by integrating multi-armed bandits with a generalized REINFORCE algorithm under a Softmax policy. The authors prove that global convergence is achievable for any fixed S, establishing a tight O(1/T) convergence rate. Furthermore, they reveal a distinct advantage of off-policy learning: under weak initialization, appropriately increasing S helps escape local traps and significantly accelerates convergence, thereby breaking the conventional reliance on strict on-policy training.

Abstract

We study the learning dynamics of fine-tuning a policy model on self-generated and reward-weighted data, with particular focus on a generalized version of REINFORCE -- referred to as RE(S) -- that updates the rollout distribution once every $S \ge 1$ gradient steps. Prior work in bandits and reinforcement learning has developed rich theory for policy gradient methods, and on-policy sampling (i.e., a small $S$, ideally $1$) is often viewed as crucial to their success; yet in prominent application like post-training large language models, reward-guided self-training has proved to be effective even when the rollout distribution is updated infrequently, but theoretical understanding remains limited for the convergence properties of these off-policy methods. To bridge these gaps, we develop a unified theory for RE(S) that covers the full spectrum of $S \ge 1$: it can be interpreted as a stage-wise optimization process, where each stage takes $S$ gradient steps for minimizing the Kullback-Leibler distance to a fixed reward-weighted rollout distribution. For multi-arm bandits with softmax policies, our in-depth analysis and numerical experiments reveal three key findings: (1) for any fixed $S$, RE(S) enjoys global convergence to the optimal policy as the number of rollout distribution updates $B = \lfloor T / S \rfloor \rightarrow \infty$, where $T$ denotes the number of gradient steps; (2) we prove tight two-sided bounds showing that the suboptimality gap of RE(S) achieves an asymptotic $Θ(1 / T)$ convergence rate, while $S$ only affects the length of a burn-in phase; (3) when initialized at a weak policy with a small optimal-action probability, RE(1) gets trapped around suboptimal policies for a long period, whereas RE(S) with a suitable $S$ avoids the detour and achieves significantly faster convergence to the global optimum, highlighting the benefits of off-policyness in this case.

Problem

• fine-tuning
• reward-weighted data
• learning dynamics
• convergence rate
• off-policyness

Innovation

• Off-policy learning
• Reward-weighted fine-tuning
• Policy gradient convergence
• REINFORCE
• Learning dynamics