Differentiable Reasoning

Reinforcement Learning with Verifiable Rewards (RLVR) has emerged as a dominant paradigm for enhancing Large Language Models (LLMs) reasoning, yet its reliance on external verifiers limits its scalability. Recent findings suggest that RLVR primarily functions by eliciting latent capabilities, motivating the development of verifier-free algorithms. However, in such settings, standard methods like Group Relative Policy Optimization face a critical challenge: destructive gradient variance that often leads to training collapse. To address this issue, we introduce **Verifier-Independent Curriculum Reinforcement Learning (VI-CuRL)**, a framework that leverages the model's intrinsic confidence to construct a curriculum independent from external verifiers. By prioritizing high-confidence samples, VI-CuRL effectively manages the bias-variance trade-off, specifically targeting the reduction of *action* and *problem variance*. We provide a rigorous theoretical analysis, proving that our estimator guarantees asymptotic unbiasedness. Empirically, VI-CuRL promotes stability and consistently outperforms verifier-dependent/independent baselines across math and general reasoning benchmarks with/without verifiers.

In practice, the reasoning strategy often matters more than the model. Rather than calling a fixed large model once per input, a well-designed strategy breaks the problem into steps and uses the right model for each, achieving better accuracy at lower cost. Existing methods have tried to automate strategy design by searching over a fixed set of strategy templates, but the templates themselves limit what reasoning patterns can be discovered. We propose Strategist, a meta-agent that designs a high-performing inference strategy for a given task. Each strategy is a structured executable topology that defines how agents plan, interact, verify, and route across models. Given a task and a small development set, Strategist proposes candidate strategies, runs them on real samples, and revises them against measured accuracy and cost. Every strategy and its components are reusable, forming an evolving library that grows with each new task and compounds what future tasks can build from. On 30 benchmarks, Strategist matches or beats the strongest baselines with an average of 8.5 absolute percentage points higher accuracy at 48% lower inference cost.

Test-time compute has emerged as a major approach to improving the capabilities of Large Language Models (LLMs). However, existing test-time reasoning paradigms rely heavily on externally imposed control, either through fixed reasoning programs or through costly expansion in constrained search spaces, limiting both generalization and efficiency. We propose State of Thought (SoT), a new reasoning paradigm that enables endogenous reasoning in LLMs, with the model's internal reasoning state governing how reasoning unfolds. Concretely, SoT extracts a compact dynamics-geometric state from the model's internal information transfer and selectively activates historical reasoning support useful under the current reasoning state, framing reasoning as a state-conditioned process over evidence rather than an externally prescribed token chain. Across quantitative (1.29x), general (1.62x), symbolic and code (1.72x), long-context (2.63x), and multimodal (1.08x) reasoning tasks on 4 models with 20 datasets, SoT consistently improves task accuracy while reducing tokens by 69.0% and latency by 48.7%, supporting endogenous state-driven reasoning as a more generalizable and efficient alternative.

Omnidirectional or 360° cameras provide embodied AI agents with a holistic, wide field-of-view (FoV) perception of the 3D structure of their surroundings. This capability has generated significant interest in applying Multi-modal Large Language Models (MLLMs) to omnidirectional spatial reasoning. However, most MLLMs are primarily trained on conventional 2D perspective images and therefore struggle with the severe distortions and wrap-around discontinuities introduced by spherical geometry. As a result, enabling MLLMs to generalize effectively to non-Euclidean 3D spaces without retraining remains an open challenge. In this paper, we propose SphMind, a novel, training-free, and plug-and-play framework designed to bridge this gap. The central idea of SphMind is to decouple semantic perception from geometric reasoning. Instead of requiring MLLMs to learn complex spherical geometric principles internally, the framework preserves their strong semantic understanding while handling geometric reasoning externally. To achieve this, we introduce a Spherical Harmonics-based Spatial Graph (SHSG) that models spatial relationships using equivariant transformations on the sphere. We further integrate this with Inference-Time Geometric Grounding (IGG), a model-agnostic closed-loop optimization process that aligns the internal representations of MLLMs with spherical geometric constraints during inference. Extensive experiments on three benchmark datasets demonstrate the effectiveness of SphMind. Without any additional training, the framework achieves more than 21.4% average improvement in directional reasoning on MP3D and Stanford2D-3D, outperforms all prompt-engineering baselines by 8.7% on the real-world ODI-Bench dataset, and achieves 5.9× higher rotational invariance under panorama rotations compared to existing baselines, all without dataset-specific tuning. We also validate the framework on real-world in-the-wild captures, where SphMind successfully resolves directional reasoning queries that baseline vision-language models fail to answer correctly.

Discrete diffusion models have recently become competitive with autoregressive models for language modeling, even outperforming them on reasoning tasks requiring planning and global coherence, but diffusion requires more computation at inference time. We trace this trade-off to a key mechanism: diffusion models are trained to jointly predict all unknown tokens simultaneously, including those that will not actually be decoded in the current step. Ablating this joint prediction yields faster inference but degrades performance, revealing that accurate prediction at the decoded position relies on joint reasoning about the undecoded tokens. We interpret these undecoded positions as latent tokens and introduce a method for modulating their number, achieving a smooth tradeoff between inference speed and sample quality. Furthermore, we demonstrate that latent tokens can be introduced into autoregressive models through an auxiliary multi-token prediction objective, yielding substantial improvements on the same reasoning tasks where they have traditionally struggled. Our results suggest that latent tokens--arising from jointly predicting multiple unknown positions--represent a general mechanism for improving performance on tasks requiring global coherence or lookahead.

Post-training quantization (PTQ) is widely used to deploy large language models efficiently, but its effect on reasoning models is not well understood. Across math, coding, and science QA, we find that aggressive PTQ reduces accuracy while increasing chain-of-thought (CoT) length. Surprisingly, we show that in up to 52% of the quantized models' failures, models reach the right answer in intermediate reasoning steps but do not output it as a final answer. To understand why quantization leads to this increase in overthinking errors, we measure the token-level KL divergence between quantized and full-precision output distributions. Positions with high KL divergence correlate strongly with high next-token entropy, and at these positions quantized models disproportionately sample overthinking markers such as “wait", “but", and “alternatively". We show that simply introducing a training-free logit penalty on a curated set of overthinking markers can reduce CoT length by 12--23% while preserving or improving accuracy across 5 models (1.5B--32B parameters), 3 quantization methods, and 5 benchmarks, yielding the best Pareto frontier of accuracy against reasoning cost. Overthinking errors produced by quantized models are particularly reduced by up to 58%.

Chain-of-Thought reasoning has enabled large language models to achieve substantial performance gains on complex tasks. However, these gains come at the cost of dramatically increased token consumption. This raises a fundamental question: is every token in the reasoning trace equally valuable? We present a diagnostic and optimization framework grounded in a key empirical finding: the value of tokens within a CoT reasoning sequence is highly non-uniform, and this non-uniformity can be effectively characterized by token-level log probability signals. We show that normalized log probability helps distinguish core tokens, which carry structural and decisive reasoning content, from redundant tokens, which are exploratory, low-confidence filler that contributes less directly to the final answer. Building on these findings, we formulate the TokenProbe framework around two empirical findings and one claim: findings identify token value inequality first and then establish TokenProbe as a core-token proxy, and the claim introduces an efficient GRPO objective positing that selectively compressing redundant tokens can yield Pareto improvements in the accuracy–token efficiency space. Empirically, our method preserves reasoning quality while reducing the token usage by 76% of the baseline. Under matched reasoning-length budgets, we show that it can even outperform strong flagship baselines like Gemini-3.1-Pro.

Large Language Models (LLMs) have achieved remarkable success in complex reasoning tasks through Chain-of-Thought (CoT) prompting. However, these models often exhibit "computational overthinking," generating redundant reasoning steps that increase latency and cost without improving accuracy. Recent studies suggest that CoT trajectories can be significantly pruned, yet existing methods often rely on forcing a static thinking budget, heuristic filtering, sub-optimal early exit via classification, or expensive re-training. In this paper, we introduce OS-Pruner, a lightweight plug-in framework that formulates chain-of-thought pruning as an optimal stopping problem. Given a reasoning prefix, OS-Pruner learns whether further reasoning is worth its token cost by optimizing an explicit utility that trades off final-answer accuracy against generated length. Our novel formulation enables the model to dynamically assess the sufficient point of termination for a reasoning chain. OS-Pruner is designed to be lightweight during both training and inference, and to provide users with fine-grained control over the reasoning-effort vs. accuracy trade-off. On diverse reasoning benchmarks and base models, OS-Pruner achieves 20-60\% reduction in generation length with minimal accuracy sacrifice.

Neurosymbolic learning can use symbolic rules to provide supervision for latent concepts from weak labels, but it commonly assumes that the entities referenced by these rules are already specified. Object-centric models decompose images into slot-like representations; however, such slots are not necessarily aligned with the predicates required for symbolic reasoning. We investigate object-centric neurosymbolic learning under distant supervision, where the object-level arguments of a logic program are learned directly from images using only global task labels. We introduce DeepObjectLog, a probabilistic neurosymbolic model that integrates a slot-based perceptual encoder with a probabilistic logic layer. The encoder predicts objectness and class probabilities for candidate object representations, while the logic layer marginalizes over latent objectness and class assignments to compute the likelihood of the observed label. This formulation provides a differentiable task-level learning signal for object-centric perception without requiring per-object labels, masks, bounding boxes, or heuristic set matching. Evaluations across diverse visual reasoning tasks demonstrate that DeepObjectLog achieves superior out-of-distribution generalization to compositional, object-count, and rule shifts compared to neural object-centric and standard neurosymbolic baselines.

Reactive synthesis, the problem of automatically constructing a hardware circuit from a logical specification, is a long-standing challenge in formal verification. It is elusive for two reasons: It is algorithmically hard, and writing formal specifications by hand is notoriously difficult. In this paper, we tackle both sides of the problem. For the algorithmic side, we present a neuro-symbolic approach to reactive synthesis that couples large reasoning models with model checkers to iteratively repair a synthesized Verilog implementation via sound symbolic feedback. Our approach solves more benchmarks than the best dedicated tools in the annual synthesis competition and extends to constructing parameterized systems, a problem known to be undecidable. On the specification side, we introduce an autoformalization step that shifts the specification task from temporal logic to natural language by introducing a hand-authored dataset of natural-language specifications for evaluation. We demonstrate performance comparable to that of starting from formal specifications, establishing natural synthesis as a viable end-to-end workflow.

Large Reasoning Language Models (LRLMs) leverage long Chain-of-Thought (CoT) reasoning to solve complex tasks, yet often engage in overthinking. Specifically, they continue to generate reasoning steps that provide meaningless contribution to or even degrade the final answer, even after sufficient reasoning has been produced. Early-exit strategies are proposed to dynamically terminate meaningless or even harmful reasoning, aiming to improve both efficiency and accuracy. However, existing methods either suffer from over-truncation that degrades accuracy below vanilla CoT, or yield only spurious efficiency gains, which reduce token counts but increase wall-clock inference time. We observe that when an LRLM's reasoning path deviates into meaningless or even harmful wandering, this deviation is accompanied by an anomalous surge of high-entropy transition tokens. Building on this insight, we propose RPDI-EE, a training-free early-exit method that monitors the Reasoning Path Deviation Index (RPDI), defined as the ratio of local to global average entropy, which serves as a proxy for reasoning path deviation. By tracking this index, RPDI-EE detects and terminates likely meaningless or even harmful reasoning while preserving productive inference. Experiments across multiple benchmarks and four LRLMs of varying types and scales show that RPDI-EE achieves the largest average accuracy improvement over vanilla CoT among all tested early-exit methods, while mitigating the spurious efficiency gains of existing alternatives.

Humans effortlessly infer 3D structure, such as depth, occlusion, and spatial arrangement, from 2D images and reason about it fluidly. Multimodal large language models still struggle with such spatial reasoning. Current approaches attempt to bridge this gap by injecting explicit 3D priors at inference time or aligning features with 3D-aware teachers during training. While these strategies improve geometric perception, they typically treat 3D knowledge as static representations, rather than enabling the model to reason with 3D cues during inference. We argue that closing this gap requires internalizing 3D knowledge in a form that can participate in the model's reasoning dynamics. To this end, we introduce \textbf{Latent Spatial Reasoning (LSR)}, a framework that distills geometric knowledge into \emph{latent spatial tokens}: geometry‑aware representations that the model actively queries and conditions on to guide its spatial reasoning process. Through hierarchical distillation, LSR transfers 3D knowledge from foundation models, aligns it with the vision-language space, and trains the model to interleave spatial and linguistic reasoning, all without external 3D modules at inference time. Experiments on diverse spatial reasoning benchmarks demonstrate state-of-the-art performance.

Long-range geophysical forecasts are fundamentally limited by chaotic dynamics and numerical errors. While data assimilation can mitigate these issues, classical variational smoothers require computationally expensive tangent-linear and adjoint models. Conversely, recent efficient latent filtering methods often enforce weak trajectory-level constraints and assume fixed observation grids. To bridge this gap, we propose Latent Ensemble Variational Data Assimilation (LEVDA), an ensemble-space variational smoother that operates in the low-dimensional latent space of a pretrained differentiable neural dynamics surrogate. By performing four-dimensional ensemble-variational (4DEnVar) optimization within an ensemble subspace, LEVDA jointly assimilates states and unknown parameters without the need for adjoint code or auxiliary observation-to-latent encoders. Leveraging the fully differentiable, continuous-in-time-and-space nature of the surrogate, LEVDA naturally accommodates highly irregular sampling at arbitrary spatiotemporal locations. Across three challenging geophysical benchmarks, LEVDA matches or outperforms state-of-the-art latent filtering baselines under severe observational sparsity while providing more reliable uncertainty quantification. Simultaneously, it achieves substantially improved assimilation accuracy and computational efficiency compared to full-state 4DEnVar.

Universal Transformers (UTs) show strong performance on abstract reasoning tasks such as ARC-AGI, yet the specific sources of their performance gains remain underexplored. We conduct systematic ablations over UT variants and find that performance improvements on abstract reasoning are driven primarily by (i) the recurrent inductive bias induced by parameter sharing across depth and (ii) strong nonlinearity inside the Transformer block, rather than by elaborate architectural heuristics. Motivated by this finding, we propose the Universal Reasoning Model (URM). First, we propose ConvSwiGLU, which augments the feed-forward block with a channel-wise short convolution to effectively enhance nonlinearity of UT. Second, we adopt Truncated Backpropagation Through Loops (TBPTL) to eliminate noise from early loop when training with many loops, by freezing the gradients of early loops. Our approach substantially improves abstract reasoning performance, achieving 57.5% pass@1 on ARC-AGI 1 and 16.0% pass@1 on ARC-AGI 2. More importantly, we pretrain a 8-billion-parameter large language model based on the URM architecture from scratch, and the results show that URM still substantially outperforms the baseline across all reasoning tasks.

Large Language Models (LLMs) demonstrate remarkable reasoning capabilities, yet the internal mechanisms driving these multi-step processes remain opaque. Existing mechanistic interpretability approaches often rely on superficial text-pattern co-occurrences or capture only short-term effects, struggling to trace the long-horizon, outcome-driven influence of internal components such as neurons or sparse features in multi-step reasoning. In this paper, we introduce Integrated Policy Gradient (IPG), a novel, training-free framework that brings policy-based attribution into mechanistic interpretability for LLM reasoning. Grounded in *outcome-oriented* and *sequential-influence-aware* principles, it backpropagates outcome-level, non-differentiable behavioral signals (e.g., reasoning outcome correctness) through entire inference trajectories, utilizing path integration to isolate outcome-relevant mediators at component level for reliable attribution. This provides a linear approximation of Causal Mediation Analysis (CMA) in reasoning. Empirical evaluations demonstrate that IPG achieves precise localization and reveals mechanistic insights by identifying sparse, concentrated subsets of internal components associated with specific reasoning behaviors. Our method also enables modulation of reasoning capabilities, offering a powerful and reliable framework for understanding LLM reasoning.

Action Quality Assessment (AQA) for diving aims to quantify the execution quality of diving sequences and has long been a prominent research topic. However, existing diving AQA methods typically regress solely the final diving score, leading to poor interpretability. To address this challenge, we propose Diving-R1, the first attempt to leverage a Multimodal Large Language Model (MLLM) for interpretable diving AQA, enabling the integration of comprehensive information, thereby supporting progressive reasoning and traceable assessment. We construct a novel dataset, DivingThink, as the foundation, which contains progressively structured reasoning chains that follow judging logic and explicitly describe the execution quality of each diving sub-action, providing clear evidence for score determination. Furthermore, we establish a new benchmark, DivingInterp, which evaluates the performance of MLLMs on interpretable diving AQA from multiple complementary perspectives, including score estimation, textual similarity, and reasoning quality. Additionally, we carefully design a three-stage training paradigm, combined with a dedicated composite reward function, which gradually relaxes the annotation requirements for the training corpus at each stage to incrementally enrich the diversity of the training data, achieving continuous improvement. Extensive experiments demonstrate the superiority of Diving-R1 over existing MLLMs in interpretable diving AQA. The dataset and code will be available on GitHub.

Range-partition entropy is a structural complexity measure used in entropy-sensitive computational geometry. We introduce Differentiable Entropy Regularization (DER), a gradient-optimizable surrogate for the entropy of learned geometric partitions. The paper makes three main contributions. First, we separate the normalized entropy optimized by DER, denoted H_norm(S), from the runtime-scaled entropy H_work(S) = n · H_norm(S) that appears in algorithmic work bounds. Second, for a fixed margin-separated halfspace arrangement, we prove a self-contained soft-to-hard approximation theorem: the soft halfspace-cell entropy converges to the corresponding hard sign-pattern entropy at a rate controlled by margin, temperature, and the number of separators. A connection to optimal range-partition entropy holds when the fixed partition is admissible for the geometric problem and near-optimal in the admissible partition class. Third, we evaluate DER as learned geometric preprocessing for SciPy/Qhull-based convex-hull and Delaunay pipelines. On the evaluated 2D tasks, DER gives up to 4×–5× solver-side speedups and smaller end-to-end gains after preprocessing, with convex-hull area error below 0.2%. We also include a secondary transfer study in ViT-style attention, where the same regularizer induces structured sparsity and improves empirical efficiency. The central contribution is not the entropy formula alone, which resembles soft clustering, but the range-aware differentiable construction, the explicit approximation analysis, and the geometry-first training pipeline.

The efficient computation of parametric solution sensitivities is a key challenge in the integration of learning-enhanced methods with nonlinear model predictive control (MPC), as their availability is crucial for many learning algorithms. This paper discusses the computation of solution sensitivities of general nonlinear programs (NLPs) using the implicit function theorem (IFT) and smoothed optimality conditions treated in interior-point methods (IPM). We detail sensitivity computation within a sequential quadratic programming (SQP) method which employs an IPM for the quadratic subproblems. Previous works presented in the machine learning community are limited to convex or unconstrained formulations, or lack an implementation for efficient sensitivity evaluation. The publication is accompanied by an efficient open-source implementation within the `acados` framework, providing both forward and adjoint sensitivities for general optimal control problems, achieving speedups exceeding 3x over the state-of-the-art solvers `mpc.pytorch` and `cvxpygen`.

We introduce a framework for differentiable exact learning of algorithms. The framework couples a state-tracking neural controller to an unbounded external environment and trains on policy-trajectory observations (PTOs) -- traces that record the environment's evolution under an expert's actions without exposing the expert's internal state. This supervision regime sidesteps Gold's impossibility barrier for input-output recursive learning while avoiding the unrealistic internal-state access required by prior provably correct approaches. As controllers, we propose Differentiable Finite-State Transducers (DFSTs), a minimalist multilinear model family that contains no nonlinearities and admits log-parallel training via prefix scans. Trained on tiny datasets, DFST and RNN controllers achieve error-free generalization on binary and decimal addition and multiplication to operands thousands of times longer than the training examples. Moreover, we provide strong empirical proof for exact learning of universal computation. Finally, we develop an extraction procedure that recovers compact finite-state transducers from trained controllers. We prove under a geometric clustering assumption that for binary addition the extracted transducer exactly replicates the expert policy on all valid inputs -- certifying differentiable exact learning of the algorithm.

Human coordination often relies on the ability to influence the beliefs of others through strategic action. In multi-agent reinforcement learning, opponent shaping attempts to replicate this influence, though existing methods typically operate within an opponent's parameter, policy, or value space. Meanwhile, belief-manipulation techniques in hidden-role games often rely on hard-coded objectives, such as deception or belief saturation. We propose Differentiable Belief-based Opponent Shaping (D-BOS), a first-order method that treats each observer's belief as the shaped opponent state and differentiates through $k$-step softmax-Bayes belief dynamics. Rather than explicitly rewarding deceptive or cooperative behavior, our method treats the belief state as the target for shaping. This allows the optimal strategy to emerge naturally from the environment's reward structure. This belief-space formulation provides an opponent-shaping signal by differentiating through opponent belief updates, and naturally extends to multiple observers by aggregating gradients over their individual inferred belief trajectories. Empirically, D-BOS outperforms PPO and BBM in hidden-role games, with the largest gains in mixed-motive settings.