$\nabla$-Reasoner: LLM Reasoning via Test-Time Gradient Descent in Latent Space
This work provides a more cost-effective method for improving LLM reasoning capabilities for users and developers by shifting from discrete search to first-order optimization at test time.
This paper introduces $\nabla$-Reasoner, an iterative generation framework that uses differentiable optimization over token logits to refine LLM policies during decoding. It achieves over 20% accuracy improvement on a mathematical reasoning benchmark while reducing model calls by 10-40% compared to strong baselines.
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.