Large Language Models (LLMs) as autonomous agents are increasingly tasked with solving complex, long-horizon problems. Aligning these agents via preference-based offline methods like Direct Preference Optimization (DPO) is a promising direction, yet it faces a critical granularity mismatch. Trajectory-level DPO provides a signal that is too coarse for precise credit assignment, while step-level DPO is often too myopic to capture the value of multi-step behaviors. To resolve this challenge, we introduce Hierarchical Preference Learning (HPL), a hierarchical framework that optimizes LLM agents by leveraging preference signals at multiple, synergistic granularities. While HPL incorporates trajectory- and step-level DPO for global and local policy stability, its core innovation lies in group-level preference optimization guided by a dual-layer curriculum. Our approach first decomposes expert trajectories into semantically coherent action groups and then generates contrasting suboptimal groups to enable preference learning at a fine-grained, sub-task level. Then, instead of treating all preference pairs equally, HPL introduces a curriculum scheduler that organizes the learning process from simple to complex. This curriculum is structured along two axes: the group length, representing sub-task complexity, and the sample difficulty, defined by the reward gap between preferred and dispreferred action groups. Experiments on three challenging agent benchmarks show that HPL outperforms existing state-of-the-art methods. Our analyses demonstrate that the hierarchical DPO loss effectively integrates preference signals across multiple granularities, while the dual-layer curriculum is crucial for enabling the agent to solve a wide range of tasks, from simple behaviors to complex multi-step sequences.

A central question in both cognitive science and artificial intelligence is how humans and artificial systems can acquire competencies for language and motor command in a co-developmental manner, despite having access to only limited learning experiences. This question is exemplified in human infants, who achieve remarkable generalization with sparse input. This is a stark contrast to large-scale models which rely on massive training corpora, to reach similar capabilities. This raises the issue of what mechanisms enable such efficient developmental learning. From the perspective of developmental psychology, infants acquire language through rich interaction with their embodied environments. T omasello's "verb-island" hypothesis argues that children initially learn verbs in specific, isolated contexts before generalizing across broader linguistic structures (1). He also emphasized the importance of embodiment in language acquisition, suggesting that grounding linguistic symbols in sensorimotor experiences is fundamental to language learning (2).

Many machine learning algorithms rely on iterative updates of uncertainty representations, ranging from variational inference and expectation-maximization, to reinforcement learning, continual learning, and multi-agent learning. In the presence of imprecision and ambiguity, credal sets -- closed, convex sets of probability distributions -- have emerged as a popular framework for representing imprecise probabilistic beliefs. Under such imprecision, many learning problems in imprecise probabilistic machine learning (IPML) may be viewed as processes involving successive applications of update rules on credal sets. This naturally raises the question of whether this iterative process converges to stable fixed points -- or, more generally, under what conditions on the updating mechanism such fixed points exist, and whether they can be attained. We provide the first analysis of this problem and illustrate our findings using Credal Bayesian Deep Learning as a concrete example. Our work demonstrates that incorporating imprecision into the learning process not only enriches the representation of uncertainty, but also reveals structural conditions under which stability emerges, thereby offering new insights into the dynamics of iterative learning under imprecision.

The Busemann function has recently found much interest in a variety of geometric machine learning problems, as it naturally defines projections onto geodesic rays of Riemannian manifolds and generalizes the notion of hyperplanes. As several sources of data can be conveniently modeled as probability distributions, it is natural to study this function in the Wasserstein space, which carries a rich formal Riemannian structure induced by Optimal Transport metrics. In this work, we investigate the existence and computation of Busemann functions in Wasserstein space, which admits geodesic rays. We establish closed-form expressions in two important cases: one-dimensional distributions and Gaussian measures. These results enable explicit projection schemes for probability distributions on $\mathbb{R}$, which in turn allow us to define novel Sliced-Wasserstein distances over Gaussian mixtures and labeled datasets. We demonstrate the efficiency of those original schemes on synthetic datasets as well as transfer learning problems.

We study the problem of auditing the fairness of a given classifier under partial feedback, where true labels are available only for positively classified individuals, (e.g., loan repayment outcomes are observed only for approved applicants). We introduce a novel cost model for acquiring additional labeled data, designed to more accurately reflect real-world costs such as credit assessment, loan processing, and potential defaults. Our goal is to find optimal fairness audit algorithms that are more cost-effective than random exploration and natural baselines. In our work, we consider two audit settings: a black-box model with no assumptions on the data distribution, and a mixture model, where features and true labels follow a mixture of exponential family distributions. In the black-box setting, we propose a near-optimal auditing algorithm under mild assumptions and show that a natural baseline can be strictly suboptimal. In the mixture model setting, we design a novel algorithm that achieves significantly lower audit cost than the black-box case. Our approach leverages prior work on learning from truncated samples and maximum-a-posteriori oracles, and extends known results on spherical Gaussian mixtures to handle exponential family mixtures, which may be of independent interest. Moreover, our algorithms apply to popular fairness metrics including demographic parity, equal opportunity, and equalized odds. Empirically, we demonstrate strong performance of our algorithms on real-world fair classification datasets like Adult Income and Law School, consistently outperforming natural baselines by around 50% in terms of audit cost.

Robust Principal Component Analysis (RPCA) aims to recover a low-rank structure from noisy, partially observed data that is also corrupted by sparse, potentially large-magnitude outliers. Traditional RPCA models rely on convex relaxations, such as nuclear norm and $\ell_1$ norm, to approximate the rank of a matrix and the $\ell_0$ functional (the number of non-zero elements) of another. In this work, we advocate a nonconvex regularization method, referred to as transformed $\ell_1$ (TL1), to improve both approximations. The rationale is that by varying the internal parameter of TL1, its behavior asymptotically approaches either $\ell_0$ or $\ell_1$. Since the rank is equal to the number of non-zero singular values and the nuclear norm is defined as their sum, applying TL1 to the singular values can approximate either the rank or the nuclear norm, depending on its internal parameter. We conduct a fine-grained theoretical analysis of statistical convergence rates, measured in the Frobenius norm, for both the low-rank and sparse components under general sampling schemes. These rates are comparable to those of the classical RPCA model based on the nuclear norm and $\ell_1$ norm. Moreover, we establish constant-order upper bounds on the estimated rank of the low-rank component and the cardinality of the sparse component in the regime where TL1 behaves like $\ell_0$, assuming that the respective matrices are exactly low-rank and exactly sparse. Extensive numerical experiments on synthetic data and real-world applications demonstrate that the proposed approach achieves higher accuracy than the classic convex model, especially under non-uniform sampling schemes.

Large language models (LLMs) are typically deployed under diverse memory and compute constraints. Existing approaches build model families by training each size independently, which is prohibitively expensive and provides only coarse-grained size options. In this work, we identify a novel phenomenon that we call boomerang distillation: starting from a large base model (the teacher), one first distills down to a small student and then progressively reconstructs intermediate-sized models by re-incorporating blocks of teacher layers into the student without any additional training. This process produces zero-shot interpolated models of many intermediate sizes whose performance scales smoothly between the student and teacher, often matching or surpassing pretrained or distilled models of the same size. We further analyze when this type of interpolation succeeds, showing that alignment between teacher and student through pruning and distillation is essential. Boomerang distillation thus provides a simple and efficient way to generate fine-grained model families, dramatically reducing training cost while enabling flexible adaptation across deployment environments. The code and models are available at https://github.com/dcml-lab/boomerang-distillation.

A fundamental challenge in embodied intelligence is developing expressive and compact state representations for efficient world modeling and decision making. However, existing methods often fail to achieve this balance of compactness and expressivity, yielding representations that are either overly redundant or lacking in task-critical information. We propose an unsupervised approach that learns a highly compressed two-token state representation using a lightweight encoder and a pre-trained Diffusion Transformer (DiT) decoder, capitalizing on its strong generative prior. Our representation is efficient, interpretable, and integrates seam-lessly into existing VLA-based models, improving performance by 14.3% on LIBERO and 30% in real-world task success rate with minimal inference overhead. More importantly, we find that the difference between these tokens, obtained via latent interpolation, naturally represents the motion, which can be further decoded into executable robot actions. This emergent capability reveals that our representation captures dynamics without explicit motion supervision. We name our method StaMo for its ability to learn generalizable robotic Motion from compact State representation, which is encoded from static images, challenging the prevalent dependence to learning robotic motions with complex temporal modeling and video data. Our learned representations also enhance policy co-training, outperforming prior methods by 10.4% with improved interpretability. "What we observe as static is merely dynamic equilibrium. " -- Richard Feynman, The Feynman Lectures on Physics Learning reusable and generalizable representations is a cornerstone of intelligent robotics systems.

While most autoregressive LLMs are constrained to one-by-one decoding, diffusion LLMs (dLLMs) have attracted growing interest for their potential to dramatically accelerate inference through parallel decoding. Despite this promise, the conditional independence assumption in dLLMs causes parallel decoding to ignore token dependencies, inevitably degrading generation quality when these dependencies are strong. However, existing works largely overlook these inherent challenges, and evaluations on standard benchmarks (e.g., math and coding) are not sufficient to capture the quality degradation caused by parallel decoding. To address this gap, we first provide an information-theoretic analysis of parallel decoding. We then conduct case studies on analytically tractable synthetic list operations from both data distribution and decoding strategy perspectives, offering quantitative insights that highlight the fundamental limitations of parallel decoding. Building on these insights, we propose ParallelBench, the first benchmark specifically designed for dLLMs, featuring realistic tasks that are trivial for humans and autoregressive LLMs yet exceptionally challenging for dLLMs under parallel decoding. Using ParallelBench, we systematically analyze both dLLMs and autoregressive LLMs, revealing that: (i) dLLMs under parallel decoding can suffer dramatic quality degradation in real-world scenarios, and (ii) current parallel decoding strategies struggle to adapt their degree of parallelism based on task difficulty, thus failing to achieve meaningful speedup without compromising quality. Our findings underscore the pressing need for innovative decoding methods that can overcome the current speed-quality trade-off. We release our benchmark to help accelerate the development of truly efficient dLLMs.

Large Language Models (LLMs) are widely applied in real world scenarios, but fine-tuning them comes with significant computational and storage costs. Parameter-Efficient Fine-Tuning (PEFT) methods such as LoRA mitigate these costs, but the adapted parameters are dependent on the base model and cannot be transferred across different backbones. One way to address this issue is through knowledge distillation, but its effectiveness inherently depends on training data. Recent work such as TransLoRA avoids this by generating synthetic data, but this adds complexity because it requires training an additional discriminator model. In this paper, we propose TiTok, a new framework that enables effective LoRA Transplantation through Token-level knowledge transfer. Specifically, TiTok captures task-relevant information through a contrastive excess between a source model with and without LoRA. This excess highlights informative tokens and enables selective filtering of synthetic data, all without additional models or overhead. Through experiments on three benchmarks across multiple transfer settings, our experiments show that the proposed method is consistently effective, achieving average performance gains of +4~8% compared to baselines overall.