$\nabla$-Reasoner: LLM Reasoning via Test-Time Gradient Descent in Latent Space

TL;DR

The paper introduces $ abla$-Reasoner, a novel test-time gradient-based optimization framework for LLMs that improves reasoning accuracy and efficiency by integrating differentiable optimization into decoding.

Contribution

It proposes Differentiable Textual Optimization (DTO) for on-the-fly policy refinement using gradient signals, shifting from search-based to gradient-based inference methods.

Findings

Achieves over 20% accuracy improvement on mathematical reasoning benchmarks.

Reduces model calls by 10-40% compared to baselines.

Provides theoretical insights linking gradient descent in sample space to reinforcement learning.

Abstract

Scaling inference-time compute for Large Language Models (LLMs) has unlocked unprecedented reasoning capabilities. However, existing inference-time scaling methods typically rely on inefficient and suboptimal discrete search algorithms or trial-and-error prompting to improve the online policy. In this paper, we propose $\nabla$-Reasoner, an iterative generation framework that integrates differentiable optimization over token logits into the decoding loop to refine the policy on the fly. Our core component, Differentiable Textual Optimization (DTO), leverages gradient signals from both the LLM's likelihood and a reward model to refine textual representations. $\nabla$-Reasoner further incorporates rejection sampling and acceleration design to robustify and speed up decoding. Theoretically, we show that performing inference-time gradient descent in the sample space to maximize reward is dual to aligning an LLM policy via KL-regularized reinforcement learning. Empirically, $\nabla$-Reasoner achieves over 20% accuracy improvement on a challenging mathematical reasoning benchmark, while reducing number of model calls by approximately 10-40% compared to strong baselines. Overall, our work introduces a paradigm shift from zeroth-order search to first-order optimization at test time, offering a cost-effective path to amplify LLM reasoning.