On the Convergence of Self-Improving Online LLM Alignment

2026-06-30

AI Summary

This work addresses the lack of theoretical convergence guarantees in self-improving online alignment of large language models, which arises from distributional shift. To tackle this issue, the authors propose SAIL-RevKL, a method that reformulates the original bilevel optimization into a single-level framework and incorporates a reverse KL divergence regularizer to improve the optimization landscape. They establish, for the first time, that the resulting objective satisfies the Polyak–Łojasiewicz condition over a bounded parameter space, thereby ensuring global convergence with near-linear sample complexity. Empirical evaluations demonstrate that SAIL-RevKL significantly outperforms the original SAIL algorithm on both MuJoCo benchmarks and large language model alignment tasks, achieving consistent improvements in both performance and stability.

Abstract

The Self-Improving Alignment (SAIL) algorithm addresses distribution shift by reducing a bilevel formulation of the problem to an efficient, single-level method. Empirically, SAIL has demonstrated strong performance on this task. However, a formal analysis of its convergence properties has been lacking. We identify a key theoretical challenge: the standard SAIL objective function is not guaranteed to be strongly concave due to unfavorable properties of its Hessian. To address this limitation, we propose a regularized objective, SAIL-RevKL, which incorporates a reverse Kullback-Leibler (KL) divergence penalty to improve the optimization landscape. Our central theoretical contribution is to prove that this regularized objective satisfies the Polyak-Lojasiewicz (PL) condition within a bounded parameter space. We establish global convergence guarantees, achieving a near-linear sample complexity. We further validate the effectiveness and stability of SAIL-RevKL through empirical evaluations, demonstrating that it outperforms the vanilla SAIL on both MuJoCo benchmarks and LLM alignment tasks.

Problem

• convergence
• self-improving alignment
• distribution shift
• strong concavity
• optimization landscape

Innovation

• SAIL-RevKL
• Polyak-Lojasiewicz condition
• reverse KL divergence
• convergence analysis
• LLM alignment