We present a fine-grained theoretical analysis of the performance gap between reinforcement learning from human feedback (RLHF) and direct preference optimization (DPO) under a representation gap. Our study decomposes this gap into two sources: an explicit representation gap under exact optimization and an implicit representation gap under finite samples. In the exact optimization setting, we characterize how the relative capacities of the reward and policy model classes influence the final policy qualities. We show that RLHF, DPO, or online DPO can outperform one another depending on type of model mis-specifications. Notably, online DPO can outperform both RLHF and standard DPO when the reward and policy model classes are isomorphic and both mis-specified. In the approximate optimization setting, we provide a concrete construction where the ground-truth reward is implicitly sparse and show that RLHF requires significantly fewer samples than DPO to recover an effective reward model, highlighting a statistical advantage of two-stage learning. Together, these results provide a comprehensive understanding of the performance gap between RLHF and DPO under various settings, and offer practical insights into when each method is preferred.

We show that reinforcement learning with verifiable rewards (RLVR) can elicit strong mathematical reasoning in certain language models even with spurious rewards that have little, no, or outright negative correlation with the correct answer. For example, RLVR training with GRPO improves MATH-500 performance for Qwen2.5-Math-7B in absolute points by 21.4% using randomly assigned rewards, nearly matching the 29.1% gained with ground truth rewards. To explain this counterintuitive observation, we show that GRPO exhibits a clipping bias arising from the clip term, which can amplify high-prior behaviors learned during pre-training even without informative rewards. As a case study, we identify one such high-prior behavior for Qwen2.5-Math models, which we term code reasoning---reasoning in code without actual code execution; code reasoning frequency increases from 65% to over 90% with spurious rewards. However, the presence of such amplifiable behaviors is highly model-dependent. In practice, spurious rewards that are effective for Qwen models often fail to produce gains for other model families, such as Llama3 or OLMo2. Our results highlight the importance of validating RL methods across diverse models rather than relying on a single de facto choice: large performance gains can arise on Qwen models even from random rewards that do not reflect genuine capability improvements.

We introduce Reinforcement Learning (RL) with Adaptive Verifiable Environments (RLVE), an approach using verifiable environments that procedurally generate problems and provide algorithmically verifiable rewards, to scale up RL for language models (LMs). RLVE enables each verifiable environment to dynamically adapt its problem difficulty distribution to the policy model's capabilities as training progresses. In contrast, static data distributions often lead to vanishing learning signals when problems are either too easy or too hard for the policy. To implement RLVE, we create RLVE-Gym, a large-scale suite of 400 verifiable environments carefully developed through manual environment engineering. Using RLVE-Gym, we show that environment scaling, i.e., expanding the collection of training environments, consistently improves generalizable reasoning capabilities. RLVE with joint training across all 400 environments in RLVE-Gym yields a 3.37% absolute average improvement across six reasoning benchmarks, starting from one of the strongest 1.5B reasoning LMs. By comparison, continuing this LM's original RL training yields only a 0.49% average absolute gain despite using over 3x more compute. We will release our code publicly.

Cold-start personalization requires inferring preferences from minimal interaction when no user-specific historical data is available. The space of possible preferences is vast, yet users care about only a sparse subset and rarely articulate them upfront; combined with limited interaction budgets, this makes preference elicitation challenging. Our key insight is that preferences exhibit predictable structure across populations; e.g., users who want detailed explanations often also value worked examples. We propose PEP (Preference Elicitation with Priors), a principled system decomposition framework for cold-start personalization: learning a structured world model of preference correlations offline using latent variables, then performing Bayesian inference online without retraining. Even simple belief model instantiations (e.g., linear regression) substantially outperform end-to-end RL. Across medical, mathematical, social, and commonsense reasoning, PEP achieves 80.8% alignment with ground-truth user preferences versus 68.5% for RL, requires 3-5× fewer interactions, and adapts twice as often. Our contribution is a principled decomposition of cold-start personalization that makes Bayesian preference elicitation practical at scale for LLM systems.

Recent advances in large language models (LLMs) have enabled breakthroughs in mathematical discovery, exemplified by AlphaEvolve, a closed-source system that evolves programs to improve bounds on open problems. However, it relies on ensembles of frontier LLMs to achieve new bounds and is a pure inference system that models cannot internalize the evolving strategies. We introduce ThetaEvolve, an open-source framework that simplifies and extends AlphaEvolve to efficiently scale both in-context learning and Reinforcement Learning (RL) at test time, allowing models to continually learn from their experiences in improving open optimization problems. ThetaEvolve features a single LLM, a large program database for enhanced exploration, batch sampling for higher throughput, lazy penalties to discourage stagnant outputs, and optional reward shaping for stable training signals, etc. ThetaEvolve is the first evolving framework that enable a small open-source model, like DeepSeek-R1-0528-Qwen3-8B, to achieve new best-known bounds on open problems (circle packing and first auto-correlation inequality) mentioned in AlphaEvolve. Besides, across two models and four open tasks, we find that ThetaEvolve with RL at test-time consistently outperforms inference-only baselines, and the model indeed learns evolving capabilities, as the RL-trained checkpoints demonstrate faster progress and better final performance on both trained target task and other unseen tasks. We will release our code publicly.

Being able to cooperate with new people is an important component of many economically valuable AI tasks, from household robotics to autonomous driving. However, generalizing to novel humans requires training on data that captures the diversity of human behaviors. Adversarial training is a promising method that allows dynamic data generation and ensures that agents are robust. It creates a feedback loop where the agent's performance influences the generation of new adversarial data, which can be used immediately to train the agent. However, adversarial training is difficult to apply in a cooperative task; how can we train an adversarial cooperator? We propose a novel strategy that combines a pretrained generative model to simulate valid cooperative agent policies with adversarial training to maximize regret. We call our method GOAT: Generative Online Adversarial Training. In this framework, the GOAT dynamically searches the latent space of the generative model for coordination strategies where the learning policy, the Cooperator agent, underperforms. GOAT enables better generalization by exposing the Cooperator to various challenging interaction scenarios. We maintain realistic coordination strategies by keeping the generative model frozen, thus avoiding adversarial exploitation. We evaluate GOAT with real human partners, and the results demonstrate state of the art performance on the Overcooked benchmark, highlighting its effectiveness in generalizing to diverse human behaviors.

Score matching has become a central training objective in modern generative modeling, particularly in diffusion models, where it is used to learn high-dimensional data distributions through the estimation of score functions. Despite its empirical success, the theoretical understanding of the optimization behavior of score matching, particularly in over-parameterized regimes, remains limited. In this work, we study gradient descent for training over-parameterized models to learn a single Gaussian distribution. Specifically, we use a student model with $n$ learnable parameters, motivated by the structure of a Gaussian mixture model, and train it on data generated from a single ground-truth Gaussian using the population score matching objective. We analyze the optimization dynamics under multiple regimes. When the noise scale is sufficiently large, we prove a global convergence result for gradient descent, which resembles the known behavior of gradient EM in over-parameterized settings. In the low-noise regime, we identify the existence of a stationary point, highlighting the difficulty of proving global convergence in this case. Nevertheless, we show convergence under certain initialization conditions: when the parameters are initialized to be exponentially small, gradient descent ensures convergence of all parameters to the ground truth. We further give an example where, without the exponentially small initialization, the parameters may not converge to the ground truth. Finally, we consider the case of random initialization, where parameters are sampled from a Gaussian distribution far from the ground truth. We prove that, with high probability, only one parameter converges while the others diverge to infinity, yet the loss still converges to zero with a $1/\tau$ rate, where $\tau$ is the number of iterations. We also establish a nearly matching lower bound on the convergence rate in this regime. This is the first work to establish global convergence guarantees for Gaussian mixtures with at least three components under the score matching framework.

Recent advances in deep learning optimization have unveiled two intriguing phenomena under large learning rates: Edge of Stability (EoS) and Progressive Sharpening (PS), challenging classical Gradient Descent (GD) analyses. Current research approaches, using either generalist frameworks or minimalist examples, face significant limitations in explaining these phenomena. This paper advances the minimalist approach by introducing a two-layer network with a two-dimensional input, where one dimension is relevant to the response and the other is irrelevant. Through this model, we rigorously prove the existence of progressive sharpening and self-stabilization under large learning rates, and establish non-asymptotic analysis of the training dynamics and sharpness along the entire GD trajectory. Besides, we connect our minimalist example to existing works by reconciling the existence of a well-behaved ``stable set" between minimalist and generalist analyses, and extending the analysis of Gradient Flow Solution sharpness to our two-dimensional input scenario. These findings provide new insights into the EoS phenomenon from both parameter and input data distribution perspectives, potentially informing more effective optimization strategies in deep learning practice.

Direct Preference Optimization (DPO) has emerged as a stable, scalable, and efficient solution for language model alignment.Despite its empirical success, the optimization properties, particularly the impact of samplers on its convergence rates, remain under-explored. In this paper, we provide a rigorous analysis of DPO's convergence rates with different sampling strategies under the exact gradient setting, revealing a surprising separation: uniform sampling achieves $\textbf{linear}$ convergence, while our proposed online sampler achieves $\textbf{quadratic}$ convergence. We further adapt the sampler to practical settings by incorporating posterior distributions and logit mixing, demonstrating improvements over previous methods. For example, it outperforms vanilla DPO by over $7.4$% on Safe-RLHF dataset. Our results not only offer insights into the theoretical understanding of DPO but also pave the way for further algorithm designs.

We study deployment-efficient reward-free exploration with linear function approximation, where the goal is to explore a linear Markov Decision Process (MDP) without revealing the reward function, while minimizing the number of distinct policies implemented during learning. By ``deployment efficient'', we mean algorithms that require few policies deployed during exploration -- crucial in real-world applications where such deployments are costly or disruptive. We design a novel reinforcement learning algorithm that achieves near-optimal deployment efficiency for linear MDPs in the reward-free setting, using at most $H$ exploration policies during execution (where $H$ is the horizon length), while maintaining sample complexity polynomial in feature dimension and horizon length. Unlike previous approaches with similar deployment efficiency guarantees, our algorithm's sample complexity is independent of the reachability or explorability coefficients of the underlying MDP, which can be arbitrarily small and lead to unbounded sample complexity in certain cases -- directly addressing an open problem from prior work. Our technical contributions include a data-dependent method for truncating state-action pairs in linear MDPs, efficient offline policy evaluation and optimization algorithms for these truncated MDPs, and a careful integration of these components to implement reward-free exploration with linear function approximation without sacrificing deployment efficiency.

While an image is worth more than a thousand words, only a few provide crucial information for a given task and thus should be focused on. In light of this, ideal text-to-image (T2I) retrievers should prioritize specific visual attributes relevant to queries. To evaluate current retrievers on handling attribute-focused queries, we build COCO-Facet, a COCO-based benchmark with 9,112 queries about diverse attributes of interest. We find that CLIP-like retrievers, which are widely adopted due to their efficiency and zero-shot ability, have poor and imbalanced performance, possibly because their image embeddings focus on global semantics and subjects while leaving out other details. Notably, we reveal that even recent Multimodal Large Language Model (MLLM)-based, stronger retrievers with a larger output dimension struggle with this limitation. Hence, we hypothesize that retrieving with *general* image embeddings is suboptimal for performing such queries. As a solution, we propose to use *promptable* image embeddings enabled by these multimodal retrievers, which boost performance by highlighting required attributes. Our pipeline for deriving such embeddings generalizes across query types, image pools, and base retriever architectures. To enhance real-world applicability, we offer two acceleration strategies: Pre-processing promptable embeddings and using linear approximations. We show that the former yields a 15\% improvement in Recall@5 when prompts are predefined, while the latter achieves an 8\% improvement when prompts are only available during inference.

Personalizing large language models (LLMs) to accommodate diverse user preferences is essential for enhancing alignment and user satisfaction. Traditional reinforcement learning from human feedback (RLHF) approaches often rely on monolithic value representations, limiting their ability to adapt to individual preferences. We introduce a novel framework that leverages low-rank preference modeling to efficiently learn and generalize user-specific reward functions. By representing reward functions in a low-dimensional subspace and modeling individual preferences as weighted combinations of shared basis functions, our approach avoids rigid user categorization while enabling scalability and few-shot adaptation. We validate our method on multiple preference datasets, demonstrating superior generalization to unseen users and improved accuracy in preference prediction tasks.

The current state-of-the-art video generative models can produce commercial-grade videos with highly realistic details. However, they still struggle to coherently present multiple sequential events in the stories specified by the prompts, which is foreseeable an essential capability for future long video generation scenarios. For example, top T2V generative models still fail to generate a video of the short simple story "how to put an elephant into a refrigerator." While existing detail-oriented benchmarks primarily focus on fine-grained metrics like aesthetic quality and spatial-temporal consistency, they fall short of evaluating models' abilities to handle event-level story presentation. To address this gap, we introduce StoryEval, a story-oriented benchmark specifically designed to assess text-to-video (T2V) models' story-completion capabilities. StoryEval features 423 prompts spanning 7 classes, each representing short stories composed of 2-4 consecutive events. We employ Vision-Language Models, such as GPT-4o and LLaVA-OV-Chat-72B, to verify the completion of each event in the generated videos, applying a unanimous voting method to enhance reliability. Our methods ensure high alignment with human evaluations, and the evaluation of 11 models reveals its challenge, with none exceeding an average story-completion rate of 50%. StoryEval provides a new benchmark for advancing T2V models and highlights the challenges and opportunities in developing next-generation solutions for coherent storydriven video generation. Project website is available at https://ypwang61.github.io/project/StoryEval.

We show that reinforcement learning with verifiable reward using one training example (1-shot RLVR) is effective in incentivizing the math reasoning capabilities of large language models (LLMs). Applying RLVR to the base model Qwen2.5-Math-1.5B, we identify a single example that elevates model performance on MATH500 from 36.0\% to 73.6\% (8.6\% improvement beyond format correction), and improves the average performance across six common mathematical reasoning benchmarks from 17.6\% to 35.7\% (7.0\% non-format gain). This result matches the performance obtained using the 1.2k DeepScaleR subset (MATH500: 73.6\%, average: 35.9\%), which contains the aforementioned example. Furthermore, RLVR with only two examples even slightly exceeds these results (MATH500: 74.8\%, average: 36.6\%). Similar substantial improvements are observed across various models (Qwen2.5-Math-7B, Llama3.2-3B-Instruct, DeepSeek-R1-Distill-Qwen-1.5B), RL algorithms (GRPO and PPO), and different math examples. In addition, we identify some interesting phenomena during 1-shot RLVR, including cross-category generalization, increased frequency of self-reflection, and sustained test performance improvement even after the training accuracy has saturated, a phenomenon we term \textit{post-saturation generalization}. Moreover, we verify that the effectiveness of 1-shot RLVR primarily arises from the policy gradient loss, distinguishing it from the "grokking" phenomenon. We also show the critical role of promoting exploration (e.g., by incorporating entropy loss with an appropriate coefficient) in 1-shot RLVR training. We also further discuss related observations about format correction, label robustness and prompt modification. These findings can inspire future work on RLVR efficiency and encourage a re-examination of recent progress and the underlying mechanisms in RLVR. Our code, models, and data are open source at https://github.com/ypwang61/One-Shot-RLVR.

This work investigates stepsize-based acceleration of gradient descent with anytime convergence guarantees. For smooth (non-strongly) convex optimization, we propose a stepsize schedule that allows gradient descent to achieve convergence guarantees of O(T- 2 log2 rho/1+log2 rho) approximate to O(T-1.119) for any stopping time T, where rho = root 2 + 1 is the silver ratio and the stepsize schedule is predetermined without prior knowledge of the stopping time. This result provides an affirmative answer to a COLT open problem (Kornowski and Shamir, 2024) regarding whether stepsize-based acceleration can yield anytime convergence rates of o(T-1). We further extend our theory to yield anytime convergence guarantees of exp(-Omega(T/kappa(0.893))) for smooth and strongly convex optimization, with kappa being the condition number.

We consider the gap-dependent regret bounds for episodic MDPs. We show that the Monotonic Value Propagation (MVP) algorithm achieves a variance-aware gap-dependent regret bound of $$\tilde{O}\left(\left(\sum_{\Delta_h(s,a)>0} \frac{H^2 \log K \land \mathtt{Var}_{\max}^{\text{c}}}{\Delta_h(s,a)} +\sum_{\Delta_h(s,a)=0}\frac{ H^2 \land \mathtt{Var}_{\max}^{\text{c}}}{\Delta_{\mathrm{min}}} + SAH^4 (S \lor H) \right) \log K\right),$$ where $H$ is the planning horizon, $S$ is the number of states, $A$ is the number of actions, and $K$ is the number of episodes. Here, $\Delta_h(s,a) =V_h^* (a) - Q_h^* (s, a)$ represents the suboptimality gap and $\Delta_{\mathrm{min}} := \min_{\Delta_h (s,a) > 0} \Delta_h(s,a)$. The term $\mathtt{Var}_{\max}^{\text{c}}$ denotes the maximum conditional total variance, calculated as the maximum over all $(\pi, h, s)$ tuples of the expected total variance under policy $\pi$ conditioned on trajectories visiting state $s$ at step $h$. $\mathtt{Var}_{\max}^{\text{c}}$ characterizes the maximum randomness encountered when learning any $(h, s)$ pair. Our result stems from a novel analysis of the weighted sum of the suboptimality gap and can be potentially adapted for other algorithms. To complement the study, we establish a lower bound of $$\Omega \left( \sum_{\Delta_h(s,a)>0} \frac{H^2 \land \mathtt{Var}_{\max}^{\text{c}}}{\Delta_h(s,a)}\cdot \log K\right),$$ demonstrating the necessity of dependence on $\mathtt{Var}_{\max}^{\text{c}}$ even when the maximum unconditional total variance (without conditioning on $(h, s)$) approaches zero.

Gradient descent dynamics on the deep matrix factorization problem is extensively studied as a simplified theoretical model for deep neural networks. Although the convergence theory for two-layer matrix factorization is well-established, no global convergence guarantee for general deep matrix factorization under random initialization has been established to date. To address this gap, we provide a polynomial-time global convergence guarantee for randomly initialized gradient descent on four-layer matrix factorization, given certain conditions on the target matrix and a standard balanced regularization term. Our analysis employs new techniques to show saddle-avoidance properties of gradient decent dynamics, and extends previous theories to characterize the change in eigenvalues of layer weights.

Conventional language model (LM) safety alignment relies on a reactive, disjoint procedure: attackers exploit a static model, followed by defensive fine-tuning to patch exposed vulnerabilities. This sequential approach creates a mismatch – attackers overfit to obsolete defenses, while defenders perpetually lag behind emerging threats. To address this, we propose Self-RedTeam, an online self-play reinforcement learning algorithm where an attacker and defender agent co-evolve through continuous interaction. We cast safety alignment as a two-player zero-sum game, where a single model alternates between attacker and defender roles – generating adversarial prompts and safeguarding against them – while a reward LM adjudicates outcomes. This enables dynamic co-adaptation. Grounded in the game-theoretic framework of zero-sum games, we establish a theoretical safety guarantee which motivates the design of our method: if self-play converges to a Nash Equilibrium, the defender will reliably produce safe responses to any adversarial input. Empirically, Self-RedTeam uncovers more diverse attacks (+21.8

Recent research has shown that CLIP models struggle with visual reasoning tasks that require grounding compositionality, understanding spatial relationships, or capturing fine-grained details. One natural hypothesis is that the CLIP vision encoder does not embed essential information for these tasks. However, we find that this is not always the case: The encoder gathers query-relevant visual information, while CLIP fails to extract it. In particular, we show that another branch of Vision-Language Models (VLMs), Generative Multimodal Large Language Models (MLLMs), achieve significantly higher accuracy than CLIP in many of these tasks using the same vision encoder and weights, indicating that these Generative MLLMs perceive more -- as they extract and utilize visual information more effectively. We conduct a series of controlled experiments and reveal that their success is attributed to multiple key design choices, including patch tokens, position embeddings, and prompt-based weighting. On the other hand, enhancing the training data alone or applying a stronger text encoder does not suffice to solve the task, and additional text tokens offer little benefit. Interestingly, we find that fine-grained visual reasoning is not exclusive to generative models trained by an autoregressive loss: When converted into CLIP-like encoders by contrastive finetuning, these MLLMs still outperform CLIP under the same cosine similarity-based evaluation protocol. Our study highlights the importance of VLM architectural choices and suggests directions for improving the performance of CLIP-like contrastive VLMs.

Learning Gaussian Mixture Models (GMMs) is a fundamental problem in statistics and machine learning, with the Expectation-Maximization (EM) algorithm and its popular variant gradient EM being arguably the most widely used algorithms in practice. In the exact-parameterized setting, where both the ground truth GMM and the learning model have the same number of components m, a vast line of work has aimed to establish rigorous recovery guarantees for EM. However, global convergence has only been proven for the case of m=2, and EM is known to fail to recover the ground truth when m≥ 3. In this paper, we consider the over-parameterized setting, where the learning model uses n>m components to fit an m-component ground truth GMM. In contrast to the exact-parameterized case, we provide a guarantee for convergence to the globally optimal solution (the ground truth) for gradient EM. Specifically, for any well-separated GMMs, we prove that with only mild over-parameterization n = Ω(mlog m), randomly initialized gradient EM converges to the ground truth with polynomial time and samples. Our analysis proceeds in two stages and introduces a suite of novel tools for Gaussian Mixture analysis to study the dynamics of gradient EM and characterize the geometric landscape of the likelihood loss. This is the first global convergence and recovery result for EM or Gradient EM beyond the special case of m=2.