Understanding how the test risk scales with model complexity is a central question in machine learning. Classical theory is challenged by the learning curves observed for large over-parametrized deep networks. Capacity measures based on parameter count typically fail to account for these empirical observations. To tackle this challenge, we consider norm-based capacity measures and develop our study for random features based estimators, widely used as simplified theoretical models for more complex networks. In this context, we provide a precise characterization of how the estimator's norm concentrates and how it governs the associated test error. Our results show that the predicted learning curve admits a phase transition from under-to over-parameterization, but no double descent behavior. This confirms that more classical U-shaped behavior is recovered considering appropriate capacity measures based on models norms rather than size. From a technical point of view, we leverage deterministic equivalence as the key tool and further develop new deterministic quantities which are of independent interest.

Transformers have demonstrated remarkable in-context learning (ICL) capabilities, adapting to new tasks by simply conditioning on demonstrations without parameter updates. Compelling empirical and theoretical evidence suggests that ICL, as a general-purpose learner, could outperform task-specific models. However, it remains unclear to what extent the transformers optimally learn in-context compared to principled learning algorithms. To investigate this, we employ a meta ICL framework in which each prompt defines a distinctive regression task whose target function is drawn from a hierarchical distribution, requiring inference over both the latent model class and task-specific parameters.

We present *Fractional Diffusion Bridge Models* (FDBM), a novel generative diffusion bridge framework driven by an approximation of the rich and non-Markovian fractional Brownian motion (fBM). Real stochastic processes exhibit a degree of memory effects (correlations in time), long-range dependencies, roughness and anomalous diffusion phenomena that are not captured in standard diffusion or bridge modeling due to the use of Brownian motion (BM). As a remedy, leveraging a recent Markovian approximation of fBM (MA-fBM), we construct FDBM that enable tractable inference while preserving the non-Markovian nature of fBM. We prove the existence of a coupling-preserving generative diffusion bridge and leverage it for future state prediction from paired training data. We then extend our formulation to the Schrödinger bridge problem and derive a principled loss function to learn the unpaired data translation. We evaluate FDBM on both tasks: predicting future protein conformations from aligned data, and unpaired image translation. In both settings, FDBM achieves superior performance compared to the Brownian baselines, yielding lower root mean squared deviation (RMSD) of C$_\alpha$ atomic positions in protein structure prediction and lower Fréchet Inception Distance (FID) in unpaired image translation.

In this paper, we develop a novel algorithm for constructing time-uniform, asymptotic confidence sequences for quantiles under local differential privacy (LDP). The procedure combines dynamically chained parallel stochastic gradient descent (P-SGD) with a randomized response mechanism, thereby guaranteeing privacy protection while simultaneously estimating the target quantile and its variance. A strong Gaussian approximation for the proposed estimator yields asymptotically anytime-valid confidence sequences whose widths obey the law of the iterated logarithm (LIL). Moreover, the method is fully online, offering high computational efficiency and requiring only $\mathcal{O}(\kappa)$ memory, where $\kappa$ denotes the number of chains and is much smaller than the sample size. Rigorous mathematical proofs and extensive numerical experiments demonstrate the theoretical soundness and practical effectiveness of the algorithm.

Semi-Infinite Programming (SIP) has emerged as a powerful framework for modeling problems with infinite constraints, however, its theoretical development in the context of nonconvex and large-scale optimization remains limited. In this paper, we investigate a class of nonconvex min-max optimization problems with nonconvex infinite constraints, motivated by applications such as adversarial robustness and safety-constrained learning. We propose a novel inexact dynamic barrier primal-dual algorithm and establish its convergence properties. Specifically, under the assumption that the squared infeasibility residual function satisfies the Lojasiewicz inequality with exponent $\theta \in (0,1)$, we prove that the proposed method achieves $\mathcal{O}(\epsilon^{-3})$, $\mathcal{O}(\epsilon^{-6\theta})$, and $\mathcal{O}(\epsilon^{-3\theta/(1-\theta)})$ iteration complexities to achieve an $\epsilon$-approximate stationarity, infeasibility, and complementarity slackness, respectively. Numerical experiments on robust multitask learning with task priority further illustrate the practical effectiveness of the algorithm.

Pre-trained stable diffusion models (SD) have shown great advances in visual correspondence. In this paper, we investigate the capabilities of Diffusion Transformers (DiTs) for accurate dense correspondence. Distinct from SD, DiTs exhibit a critical phenomenon in which very few feature activations exhibit significantly larger values than others, known as massive activations, leading to uninformative representations and significant performance degradation for DiTs. The massive activations consistently concentrate at very few fixed dimensions across all image patch tokens, holding little local information. We analyze these dimension-concentrated massive activations and uncover that their concentration is inherently linked to the Adaptive Layer Normalization (AdaLN) in DiTs. Building on these findings, we propose the Diffusion Transformer Feature (DiTF), a training-free AdaLN-based framework that extracts semantically discriminative features from DiTs. Specifically, DiTF leverages AdaLN to adaptively localize and normalize massive activations through channel-wise modulation. Furthermore, a channel discard strategy is introduced to mitigate the adverse effects of massive activations. Experimental results demonstrate that our DiTF outperforms both DINO and SD-based models and establishes a new state-of-the-art performance for DiTs in different visual correspondence tasks (e.g., with +9.4\% on Spair-71k and +4.4\% on AP-10K-C.S.).

Enhancing the reasoning capabilities of large language models effectively using reinforcement learning (RL) remains a crucial challenge. Existing approaches primarily adopt two contrasting advantage estimation granularities: token-level methods (e.g., PPO) aim to provide fine-grained advantage signals but suffer from inaccurate estimation due to difficulties in training an accurate critic model. On the other extreme, trajectory-level methods (e.g., GRPO) solely rely on a coarse-grained advantage signal from the final reward, leading to imprecise credit assignment. To address these limitations, we propose Segment Policy Optimization (SPO), a novel RL framework that leverages segment-level advantage estimation at an intermediate granularity, achieving a better balance by offering more precise credit assignment than trajectory-level methods and requiring fewer estimation points than token-level methods, enabling accurate advantage estimation based on Monte Carlo (MC) without a critic model. SPO features three components with novel strategies: (1) flexible segment partition; (2) accurate segment advantage estimation; and (3) policy optimization using segment advantages, including a novel probability-mask strategy.

Adaptive cooperation in multi-agent reinforcement learning (MARL) requires policies to express homogeneous, specialised, or mixed behaviours, yet achieving this adaptivity remains a critical challenge. While parameter sharing (PS) is standard for efficient learning, it notoriously suppresses the behavioural diversity required for specialisation. This failure is largely due to cross-agent gradient interference, a problem we find is surprisingly exacerbated by the common practice of . Existing remedies typically add complexity through altered objectives, manual preset diversity levels, or sequential updates -- raising a fundamental question: We propose a solution built on a key insight: an agent-conditioned hypernetwork can generate agent-specific parameters and observation-and agent-conditioned gradients, directly countering the interference from coupling agent IDs with observations.

We propose Heterogeneous Swarms, an algorithm to design multi-LLM systems by jointly optimizing model roles and weights. We represent multi-LLM systems as directed acyclic graphs (DAGs) of LLMs with topological message passing for collaborative generation. Given a pool of LLM experts and a utility function, Heterogeneous Swarms employs two iterative steps: role-step and weight-step. For role-step, we interpret model roles as learning a DAG that specifies the flow of inputs and outputs between LLMs. Starting from a swarm of random continuous adjacency matrices, we decode them into discrete DAGs, call the LLMs in topological order, evaluate on the utility function (e.g.

A core difficulty lies in precisely characterizing and strategically manipulating the overall reward gap configuration, that is, the deliberate design of how to shape these reward gaps within each preference pair across the data. To address this, we introduce Topic-level Preference Rewriting (TPR), a novel framework designed for the systematic optimization of reward gap configuration. Through selectively replacing semantic topics within VLM responses with model's own resampled candidates for targeted rewriting, TPR can provide topic-level control over fine-grained semantic details. This precise control enables advanced data curation strategies, such as progressively adjusting the difficulty of rejected responses, thereby sculpting an effective reward gap configuration that guides the model to overcome challenging hallucinations. Comprehensive experiments demonstrate TPR achieves state-of-the-art performance on multiple hallucination benchmarks, outperforming previous methods by an average of $\sim$20%. Notably, it significantly reduces hallucinations by up to 93% on ObjectHal-Bench, and also exhibits superior data efficiency towards robust and cost-effective VLM alignment.