We introduce Autoregressive Retrieval Augmentation (AR-RAG), a novel paradigm that enhances image generation by autoregressively incorporating k-nearest neighbor retrievals at the patch level. Unlike prior methods that perform a single, static retrieval before generation and condition the entire generation on fixed reference images, AR-RAG performs context-aware retrievals at each generation step, using prior-generated patches as queries to retrieve and incorporate the most relevant patch-level visual references, enabling the model to respond to evolving generation needs while avoiding limitations (e.g., over-copying, stylistic bias, etc.) prevalent in existing methods. To realize AR-RAG, we propose two parallel frameworks: (1) Distribution-Augmentation in Decoding (DAiD), a training-free plug-and-use decoding strategy that directly merges the distribution of model-predicted patches with the distribution of retrieved patches, and (2) Feature-Augmentation in Decoding (FAiD), a parameter-efficient fine-tuning method that progressively smooths the features of retrieved patches via multi-scale convolution operations and leverages them to augment the image generation process. We validate the effectiveness of AR-RAG on widely adopted benchmarks, including Midjourney-30K, GenEval and DPG-Bench, demonstrating significant performance gains over state-of-the-art image generation models.

As large language models (LLMs) increasingly underpin technological advancements, the privacy of their training data emerges as a critical concern. Differential Privacy (DP) serves as a rigorous mechanism to protect this data, yet its integration via Differentially Private Stochastic Gradient Descent (DP-SGD) introduces substantial challenges, primarily due to the complexities of per-sample gradient clipping.

We study the implicit bias of flatness / low (loss) curvature and its effects on generalization in two-layer overparameterized ReLU networks with multivariate inputs---a problem well motivated by the minima stability and edge-of-stability phenomena in gradient-descent training. Existing work either requires interpolation or focuses only on univariate inputs. This paper presents new and somewhat surprising theoretical results for multivariate inputs. On two natural settings (1) generalization gap for flat solutions, and (2) mean-squared error (MSE) in nonparametric function estimation by stable minima, we prove upper and lower bounds, which establish that while flatness does imply generalization, the resulting rates of convergence necessarily deteriorate exponentially as the input dimension grows. This gives an exponential separation between the flat solutions compared to low-norm solutions (i.e., weight decay), which are known not to suffer from the curse of dimensionality. In particular, our minimax lower bound construction, based on a novel packing argument with boundary-localized ReLU neurons, reveals how flat solutions can exploit a kind of neural shattering where neurons rarely activate, but with high weight magnitudes. This leads to poor performance in high dimensions. We corroborate these theoretical findings with extensive numerical simulations. To the best of our knowledge, our analysis provides the first systematic explanation for why flat minima may fail to generalize in high dimensions.

Stochastic gradient descent (SGD) with polynomially decaying step sizes has long underpinned theoretical analyses, yielding a broad spectrum of statistically attractive guarantees. Yet in practice, such schedules find rare use due to their prohibitively slow convergence, revealing a persistent gap between theory and empirical performance. In this paper, we introduce a unified framework that quantifies the uncertainty of online SGD under arbitrary learning rate choices. In particular, we provide the first comprehensive convergence characterizations for two widely used but theoretically under-examined schemes--cyclical learning rates and linear decay to zero. Our results not only explain the observed behavior of these schedules but also facilitate principled tools for statistical inference and algorithm design. All theoretical findings are corroborated by extensive simulations across diverse settings.

Markov chain Monte Carlo (MCMC) methods are foundational algorithms for Bayesian inference and probabilistic modeling. However, most MCMC algorithms are inherently sequential and their time complexity scales linearly with the sequence length. Previous work on adapting MCMC to modern hardware has therefore focused on running many independent chains in parallel. Here, we take an alternative approach: we propose algorithms to evaluate MCMC samplers in parallel across the chain length. To do this, we build on recent methods for parallel evaluation of nonlinear recursions that formulate the state sequence as a solution to a fixed-point problem and solve for the fixed-point using a parallel form of Newton's method. We show how this approach can be used to parallelize Gibbs, Metropolis-adjusted Langevin, and Hamiltonian Monte Carlo sampling across the sequence length. In several examples, we demonstrate the simulation of up to hundreds of thousands of MCMC samples with only tens of parallel Newton iterations. Additionally, we develop two new parallel quasi-Newton methods to evaluate nonlinear recursions with lower memory costs and reduced runtime. We find that the proposed parallel algorithms accelerate MCMC sampling across multiple examples, in some cases by more than an order of magnitude compared to sequential evaluation.

Gradient clipping is increasingly important in centralized learning (CL) and federated learning (FL). Many works focus on its optimization properties under strong assumptions involving Gaussian noise and standard smoothness. However, practical machine learning tasks often only satisfy weaker conditions, such as heavy-tailed noise and $(L_0, L_1)$-smoothness. To bridge this gap, we propose a high-probability analysis for clipped Stochastic Gradient Descent (SGD) under these weaker assumptions. Our findings show a better convergence rate than existing ones can be achieved, and our high-probability analysis does not rely on the bounded gradient assumption. Moreover, we extend our analysis to FL, where a gap remains between expected and high-probability convergence, which the naive clipped SGD cannot bridge. Thus, we design a new \underline{Fed}erated \underline{C}lipped \underline{B}atched \underline{G}radient (FedCBG) algorithm, and prove the convergence and generalization bounds with high probability for the first time. Our analysis reveals the trade-offs between the optimization and generalization performance. Extensive experiments demonstrate that \methodname{} can generalize better to unseen client distributions than state-of-the-art baselines.

With origins in game-theory, probabilistic values like Shapley values, Banzhaf values, and semi-values have emerged as a central tool in explainable AI. They are used for feature attribution, data attribution, data valuation, and more. Since all of these values require exponential time to compute exactly, research has focused on efficient approximation methods using two techniques: Monte Carlo sampling and linear regression formulations. In this work, we present a new way of combining both of these techniques. Our approach is more flexible than prior algorithms, allowing for linear regression to be replaced with any function family whose probabilistic values can be computed efficiently. This allows us to harness the accuracy of tree-based models like XGBoost, while still producing unbiased estimates. From experiments across eight datasets, we find that our methods give state-of-the-art performance for estimating probabilistic values. For Shapley values, the error of our methods is up to $6\times$ lower than Permutation SHAP (the most popular Monte Carlo method), $2.75\times$ lower than Kernel SHAP (the most popular linear regression method), and $1.75\times$ lower than Leverage SHAP (the prior state-of-the-art Shapley value estimator). For more general probabilistic values, we can obtain error up to $60\times$ lower than prior work.

Understanding the dynamics of optimization algorithms in deep learning has become increasingly critical, especially as models grow in scale and complexity. Despite the empirical success of stochastic gradient descent (SGD) and its variants in finding solutions that generalize well, the precise mechanisms underlying this generalization remain poorly understood. A particularly intriguing aspect of this phenomenon is the bias of optimization algorithms towards certain types of minima--often flatter or simpler--especially in overparameterized regimes. While prior works have associated flatness of the loss landscape with better generalization, tools to mechanistically connect data, optimization algorithms, and the nature of the resulting minima are still limited. For instance, methods like Sharpness-Aware Minimization (SAM) have shown practical gains by explicitly promoting flatness, but lack a unified theoretical framework explaining their influence across different data structures and model architectures. In this work, we introduce a comprehensive linear stability analysis framework to dissect the behavior of optimization algorithms--SGD, random perturbations, and SAM--in neural networks, focusing particularly on two-layer ReLU models. Our approach is built upon a novel coherence measure that captures the interaction between data geometry and gradient similarity, providing new insights into why and how certain solutions are favored.

We study the dynamics of stochastic gradient descent (SGD) for a class of sequence models termed Sequence Single-Index (SSI) models, where the target depends on a single direction in input space applied to a sequence of tokens. This setting generalizes classical single-index models to the sequential domain, encompassing simplified one-layer attention architectures. We derive a closed-form expression for the population loss in terms of a pair of sufficient statistics capturing semantic and positional alignment, and characterize the induced high-dimensional SGD dynamics for these coordinates. Our analysis reveals two distinct training phases: escape from uninformative initialization and alignment with the target subspace, and demonstrates how the sequence length and positional encoding influence convergence speed and learning trajectories. These results provide a rigorous and interpretable foundation for understanding how sequential structure in data can be beneficial for learning with attention-based models.

We study online estimation for high-dimensional generalized linear models with streaming data. First, for the non-distributed setting, we propose a gradient-enhanced surrogate loss that approximates the cumulative loss using only historical summaries, which modifies and improves upon the existing renewable estimation approach for the same model in the high-dimensional setting, and removes the batch-number constraint in previous studies. We then extend the method to distributed streaming data under the master-client architecture, where batches are partitioned across sites and only summaries (gradient vectors) are exchanged. Instead of directing applying the popular method of Jordan et al. (2019) to the surrogate quadratic loss, our adjusted approach does not require the clients to compute the full surrogate loss. We derive non-asymptotic error bounds under the high-dimensional scaling, without the stringent constraint on the number of batches in the previous studies. Simulation results under linear and logistic models, together with a real-data application, show improved accuracy over existing renewable estimators.