Data attribution methods aim to answer useful counterfactual questions like "what would a ML model's prediction be if it were trained on a different dataset?" However, estimation of data attribution models through techniques like empirical influence or "datamodeling" remains very computationally expensive. This causes a critical trust issue: if only a few computationally rich parties can obtain data attributions, how can resource-constrained parties trust that the provided attributions are indeed "good," especially when they are used for important downstream applications (e.g., data pricing)? In this paper, we address this trust issue by proposing an interactive verification paradigm for data attribution. An untrusted and computationally powerful Prover learns data attributions, and then engages in an interactive proof with a resource-constrained Verifier.

However, a prevalent issue is "superficial self-reflection", where models fail to robustly verify their own outputs. We introduce RISE (Reinforcing Reasoning with Self-Verification), a novel online RL framework designed to tackle this. RISE explicitly and simultaneously trains an LLM to improve both its problemsolving and self-verification abilities within a single, integrated RL process. The core mechanism involves leveraging verifiable rewards from an outcome verifier to provide on-the-fly feedback for both solution generation and self-verification tasks. In each iteration, the model generates solutions, then critiques its own onpolicy generated solutions, with both trajectories contributing to the policy update. Extensive experiments on diverse mathematical reasoning benchmarks show that RISE consistently improves model's problem-solving accuracy while concurrently fostering strong self-verification skills. Our analyses highlight the advantages of online verification and the benefits of increased verification compute.

Large language models (LLMs) are often prompted with multi-level directives, such as system instructions and user queries, that imply a hierarchy of authority. Yet models frequently fail to enforce this structure, especially in multi-step reasoning where errors propagate across intermediate steps. Existing methods rely on oracle completions but lack verifiable reward signals or intermediate traces, limiting their applicability. We introduce a unified supervision framework that embeds programmatically verifiable checkers into synthesized instruction-conflict instances. Each instance pairs a compliance directive with a conflicting one, along with an executable verifier that deterministically checks output adherence. This enables alignment without oracle labels or reasoning traces, supporting both instruction-tuned and reasoning models. The framework is instantiated via a synthesis pipeline that includes unittest-based validation, LLM-assisted repair, and a probabilistic analysis of cleaning reliability. Fine-tuning on the resulting data improves instruction hierarchy adherence and boosts safety robustness, generalizing to adversarial safety benchmarks without task-specific supervision. This highlights verifiable supervision as a scalable foundation for robust alignment.

We study protocols for verifying approximate optimality of strategies in multiarmed bandits and normal-form games. As the number of actions available to each player is often large, we seek protocols where the number of queries to the utility oracle is sublinear in the number of actions. We prove that such verification is possible for sufficiently smooth strategies that do not put too much probability mass on any specific action and provide protocols for verifying that a smooth policy for a multi-armed bandit is close to optimal. Our verification protocols require provably fewer arm queries than learning. Furthermore, we show how to use cryptographic tools to reduce the communication cost of our protocols. We complement our protocol by proving a nearly tight lower bound on the query complexity of verification in our settings. As an application, we use our bandit verification protocol to build a protocol for verifying approximate optimality of a strong smooth Nash equilibrium, with sublinear query complexity.

Test-time scaling (TTS) has proven effective in enhancing the reasoning capabilities of large language models (LLMs). Verification plays a key role in TTS, simultaneously influencing (1) reasoning performance and (2) compute efficiency, due to the quality and computational cost of verification. In this work, we challenge the conventional paradigms of verification, and make the first attempt toward systematically investigating the impact of verification granularity--that is, how frequently the verifier is invoked during generation, beyond verifying only the final output or individual generation steps. To this end, we introduce Variable Granularity Search (VG-Search), a unified algorithm that generalizes beam search and Best-of-N sampling via a tunable granularity parameter g. Extensive experiments with VG-Search under varying compute budgets, generator-verifier configurations, and task attributes reveal that dynamically selecting g can improve the compute efficiency and scaling behavior. Building on these findings, we propose adaptive VG-Search strategies that achieve accuracy gains of up to 3.1% over Beam Search and 3.6% over Best-of-N, while reducing FLOPs by over 52%. Our code is avaiblae at github.com/hmarkc/VG-Search.

Verifiers can improve language model (LM) capabilities by providing feedback or selecting the best response from a pool of generated candidates. Currently, high-quality verifiers are either unscalable (e.g., humans) or limited in utility (e.g., tools like Lean for formal proofs). While LM judges and reward models have become broadly useful as general-purpose verifiers, a significant performance gap remains between them and oracle verifiers. To help close this gap, we introduce WEAVER, a framework for designing a strong verifier by combining multiple weak, imperfect verifiers. First we find that weighted ensembles of verifiers, which typically require learning from labeled data, significantly outperform unweighted combinations due to differences in the verifiers. To reduce the dependency on labeled data, WEAVER leverages weak supervision to estimate each verifier's accuracy and combines their outputs into a unified score that better reflects true response quality.

Multi-modal large language models (MLLMs) have achieved remarkable capabilities by integrating visual perception with language understanding, enabling applications such as image-grounded dialogue, visual question answering, and scientific analysis. However, most MLLMs adopt a static inference paradigm, encoding the entire image into fixed visual tokens upfront, which limits their ability to iteratively refine understanding or adapt to context during inference. This contrasts sharply with human perception, which is dynamic, selective, and feedback-driven. In this work, we introduce a novel framework for inference-time visual token scaling that enables MLLMs to perform iterative, verifier-guided reasoning over visual content. We formulate the problem as a Markov Decision Process, involving a reasoner that proposes visual actions and a verifier--trained via multi-step Direct Preference Optimization (DPO)--that evaluates these actions and determines when reasoning should terminate. To support this, we present a new dataset, VTS, comprising supervised reasoning trajectories (VTS-SFT) and preference-labeled reasoning comparisons (VTS-DPO). Our method significantly outperforms existing approaches across diverse visual reasoning benchmarks, offering not only improved accuracy but also more interpretable and grounded reasoning processes. These results demonstrate the promise of dynamic inference mechanisms for enabling fine-grained, context-aware visual reasoning in next-generation MLLMs. Code and datasets are publicly released at https://vts-v.github.io/.

Deep Neural Networks (DNN) have emerged as an effective approach to implementing challenging subproblems. They are increasingly being used as components in critical transportation, medical, and military systems. However, like human-written software, DNNs may have flaws that can lead to unsafe system performance. To confidently deploy DNNs in such systems, strong evidence is needed that they do not contain such flaws. This has led researchers to explore the adaptation and customization of software verification approaches to the problem of neural network verification (NNV). Many dozens of NNV tools have been developed in recent years and as a field these techniques have matured to the point where realistic networks can be analyzed to detect flaws and to prove conformance with specifications. NNV tools are highly-engineered and complex may harbor flaws that cause them to produce unsound results. We identify commonalities in algorithmic approaches taken by NNV tools to define a verifier independent proof format--activation pattern tree proofs (APTP)--and design an algorithm for checking those proofs that is proven correct and optimized to enable scalable checking. We demonstrate that existing verifiers can efficiently generate APTP proofs, and that an APTPcheckersignificantly outperforms prior work on a benchmark of 16 neural networks and 400 NNV problems, and that it is robust to variation in APTP proof structure arising from different NNV tools.

Verifying the behavior of neural networks is necessary if developers are to confidently deploy them as parts of mission-critical systems. Toward this end, researchers have been actively developing a range of increasingly sophisticated and scalable neural network verifiers. However, scaling verification to large networks is challenging, at least in part due to the significant memory requirements of verification algorithms. In this paper, we propose an assume-guarantee compositional framework, CoVeNN, that is parameterized by an underlying verifier to generate a sequence of verification sub-problems to address this challenge. We present an iterative refinement-based strategy for computing assumptions that allow sub-problems to retain sufficient accuracy. An evaluation using 7 neural networks and a total of 140 property specifications demonstrates that CoVeNN can verify nearly 7 times more problems than state-of-the-art verifiers.

Multimodal agents, which integrate a controller (e.g., a vision language model) with external tools, have demonstrated remarkable capabilities in tackling complex multimodal tasks. Existing approaches for training these agents, both supervised fine-tuning and reinforcement learning, depend on extensive human-annotated taskanswer pairs and tool trajectories. However, for complex multimodal tasks, such annotations are prohibitively expensive or impractical to obtain. In this paper, we propose an iterative tool usage exploration method for multimodal agents without any pre-collected data, namely SPORT, via step-wise preference optimization to refine the trajectories of tool usage. Our method enables multimodal agents to autonomously discover effective tool usage strategies through self-exploration and optimization, eliminating the bottleneck of human annotation.