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We propose theadaptive hybrid particle-grid flow mapmethod, a novel flow-map approach that leverages Lagrangian particles to simultaneously transport impulse and guide grid adaptation, introducing a fully adaptive flow map-based fluid simulation framework. The core idea of our method is to maintain flow-map trajectories separately on grid nodes and particles: the grid-based representation tracks long-range flow maps at a coarse spatial resolution, while the particle-based representation tracks both long and short-range flow maps, enhanced by their gradients, at a fine resolution. This hybrid Eulerian-Lagrangian flow-map representation naturally enables adaptivity for both advection and projection steps. We implement this method inCirrus, a GPU-based fluid simulation framework designed for octree-like adaptive grids enhanced with particle trackers. The efficacy of our system is demonstrated through numerical tests and various simulation examples, achieving up to 512 × 512 × 2048 effective resolution on an RTX 4090 GPU. We achieve a 1.5 to 2× speedup with our GPU optimization over the Particle Flow Map method on the same hardware, while the adaptive grid implementation offers efficiency gains of one to two orders of magnitude by reducing computational resource requirements. The source code has been made publicly available at: https://wang-mengdi.github.io/proj/25-cirrus/. 

Reinforcement Learning from Human Feedback (RLHF) has become the predominant approach for language model (LM) alignment. At its core, RLHF uses a margin-based loss for preference optimization, specifying ideal LM behavior only by the difference between preferred and dispreferred responses. In this paper, we identify a common pitfall of margin-based methods -- the under-specification of ideal LM behavior on preferred and dispreferred responses individually, which leads to two unintended consequences as the margin increases: (1) The probability of dispreferred (e.g., unsafe) responses may increase, resulting in potential safety alignment failures. (2) The probability of preferred responses may decrease, even when those responses are ideal. We demystify the reasons behind these problematic behaviors: margin-based losses couple the change in the preferred probability to the gradient of the dispreferred one, and vice versa, often preventing the preferred probability from increasing while the dispreferred one decreases, and thus causing a synchronized increase or decrease in both probabilities. We term this effect, inherent in margin-based objectives, gradient entanglement. Formally, we derive conditions for general margin-based alignment objectives under which gradient entanglement becomes concerning: the inner product of the gradients of preferred and dispreferred log-probabilities is large relative to the individual gradient norms. We theoretically investigate why such inner products can be large when aligning language models and empirically validate our findings. Empirical implications of our framework extend to explaining important differences in the training dynamics of various preference optimization algorithms, and suggesting potential algorithm designs to mitigate the under-specification issue of margin-based methods and thereby improving language model alignment. 

ABSTRACT Diffusion models, a powerful and universal generative artificial intelligence technology, have achieved tremendous success and opened up new possibilities in diverse applications. In these applications, diffusion models provide flexible high-dimensional data modeling, and act as a sampler for generating new samples under active control towards task-desired properties. Despite the significant empirical success, theoretical underpinnings of diffusion models are very limited, potentially slowing down principled methodological innovations for further harnessing and improving diffusion models. In this paper, we review emerging applications of diffusion models to highlight their sample generation capabilities under various control goals. At the same time, we dive into the unique working flow of diffusion models through the lens of stochastic processes. We identify theoretical challenges in analyzing diffusion models, owing to their complicated training procedure and interaction with the underlying data distribution. To address these challenges, we overview several promising advances, demonstrating diffusion models as an efficient distribution learner and a sampler. Furthermore, we introduce a new avenue in high-dimensional structured optimization through diffusion models, where searching for solutions is reformulated as a conditional sampling problem and solved by diffusion models. Lastly, we discuss future directions about diffusion models. The purpose of this paper is to provide a well-rounded exposure for stimulating forward-looking theories and methods of diffusion models. 

We study offline multitask representation learning in reinforcement learning (RL), where a learner is provided with an offline dataset from different tasks that share a common representation and is tasked to learn the shared representation. We theoretically investigate offline multitask low-rank RL, and propose a new algorithm called MORL for offline multitask representation learning. Furthermore, we examine downstream RL in reward-free, offline and online scenarios, where a new task is introduced to the agent that shares the same representation as the upstream offline tasks. Our theoretical results demonstrate the benefits of using the learned representation from the upstream offline task instead of directly learning the representation of the low-rank model.