GRADE: Probing Knowledge Gaps in LLMs through Gradient Subspace Dynamics

Test-time reasoning has become a significant field of study since the introduction of chain-of-thought reasoning in large language models (LLMs). However, the mechanisms of this reasoning process are still under-explored -- from the same input prompt, and even the same partial solution, LLMs can produce varied answers if sampled multiple times. We propose to leverage question-asking as an inference-time intervention that articulates information about the model's hidden state. To achieve that, we present a student-teacher setting where a student asks questions to a teacher. We train a probe on the student's hidden state before and after asking a question and find it is predictive of the trajectory's final correctness, even before generating the teacher's answer. This suggests there is a meaningful signal from the self-diagnosis that occurs during question generation rather than information transfer from the teacher. We then frame question-asking as a sequential decision problem, using this probe as a quality score, and define a gating policy to ask questions that maximize likelihood of correctness. We find that the success of question-asking as an intervention is largely dependent on the model's self-consistency. Our empirical results show a gap between detection and recovery; while our gating policy captures model correctness and uncertainty, interventions are equally likely to harm correct trajectories as they are to recover incorrect ones. This gap between diagnosis and correction has broader implications on language models' capacity for self-refinement under uncertainty.

Data attribution methods using gradient similarity are widely used to analyze and select training data for large language models, but what gradient similarity actually measures is debated. Some interpret it as identifying task-relevant skills, while other work reports that surface form is the main factor. We resolve this debate for supervised fine-tuning examples by varying task and answer format independently. Specifically, we render benchmarks in different answer formats, such that datasets can share a task without a format or a format without a task. We find that gradient alignment follows the answer format, as benchmark pairs sharing an answer format align strongly (disattenuated cosine near 0.4), while same benchmarks rendered with different answer format classes show no alignment (near 0.0). We demonstrate that this ordering holds from the earliest pretraining checkpoints through post-training, and across model scales and families. We then analyze the released selections of LESS, a gradient-based data selection method for instruction tuning, and find that each target's selections over-represent the target's own answer format. Hence, we demonstrate that gradient-based attribution methods track format similarity more than task semantics, meaning that such methods, as well as the semantic interpretation of the gradient, should be tested on data where answer format and task vary independently for greater robustness and reliability.

Optimization-based latent reasoning improves large language model outputs by optimizing instance-specific continuous states at test time while keeping model parameters frozen. Existing methods, however, typically connect these states to the reasoning trajectory through decoded tokens, making sequence-level credit assignment indirect and obscuring how latent updates shape subsequent reasoning. We introduce GradCuit (gradient through circuit), which inserts optimizable latent states at a selected Transformer layer between the hidden representations of the prompt and the generated continuation. Causal self-attention provides every continuation-token log-probability with a differentiable path to every preceding latent state through the remaining Transformer blocks, enabling reward-weighted gradients from the entire continuation to be assigned directly to the latents. Across five instruction-tuned backbones, three reasoning benchmarks, and two answer formats, GradCuit achieves an average accuracy of 64.5%, outperforming chain-of-thought prompting by 6.6 percentage points and the strongest competing method by 2.4 points. GradCuit also demonstrates greater robustness: across seven learning-rate settings, it consistently outperforms LatentSeek while reducing the standard deviation of accuracy from 1.53 to 0.82, and even its random-walk variant remains competitive with LatentSeek. For interpretability, token-level gradient attribution reveals that latent influence concentrates on reasoning-connector tokens, while layer analysis identifies early-to-middle Transformer layers as the most effective optimization space. By directly optimizing internal reasoning from outcome feedback, GradCuit opens a new axis of robust and interpretable test-time scaling, where LLMs adapt how they reason rather than merely regenerate, sample, or rerank outputs.