--Personalized Federated Learning (PFL) enables clients to collaboratively train personalized models tailored to their individual objectives, addressing the challenge of model generalization in traditional Federated Learning (FL) due to high data heterogeneity. However, existing PFL methods often require increased communication rounds to achieve the desired performance, primarily due to slow training caused by the use of first-order optimization, which has linear convergence. Additionally, many of these methods increase local computation because of the additional data fed into the model during the search for personalized local models. One promising solution to this slow training is second-order optimization, known for its quadratic convergence. However, employing it in PFL is challenging due to the Hessian matrix and its inverse. In this paper, we propose pFedSOP, which efficiently utilizes second-order optimization in PFL to accelerate the training of personalized models and enhance performance with fewer communication rounds. Our approach first computes a personalized local gradient update using the Gompertz function-based normalized angle between local and global gradient updates, incorporating client-specific global information. We then use a regularized Fisher Information Matrix (FIM), computed from this personalized gradient update, as an approximation of the Hessian to update the personalized models. This FIM-based second-order optimization speeds up training with fewer communication rounds by tackling the challenges with exact Hessian and avoids additional data being fed into the model during the search for personalized local models. Extensive experiments on heterogeneously partitioned image classification datasets with partial client participation demonstrate that pFedSOP outperforms state-of-the-art FL and PFL algorithms.

Differential Privacy (DP) is a widely adopted technique, valued for its effectiveness in protecting the privacy of task-specific datasets, making it a critical tool for large language models. However, its effectiveness in Multimodal Large Language Models (MLLMs) remains uncertain. Applying Differential Privacy (DP) inherently introduces substantial computation overhead, a concern particularly relevant for MLLMs which process extensive textual and visual data. Furthermore, a critical challenge of DP is that the injected noise, necessary for privacy, scales with parameter dimensionality, leading to pronounced model degradation; This trade-off between privacy and utility complicates the application of Differential Privacy (DP) to complex architectures like MLLMs. To address these, we propose Dual-Priv Pruning, a framework that employs two complementary pruning mechanisms for DP fine-tuning in MLLMs: (i) visual token pruning to reduce input dimensionality by removing redundant visual information, and (ii) gradient-update pruning during the DP optimization process. This second mechanism selectively prunes parameter updates based on the magnitude of noisy gradients, aiming to mitigate noise impact and improve utility. Experiments demonstrate that our approach achieves competitive results with minimal performance degradation. In terms of computational efficiency, our approach consistently utilizes less memory than standard DP-SGD. While requiring only 1.74% more memory than zeroth-order methods which suffer from severe performance issues on A100 GPUs, our method demonstrates leading memory efficiency on H20 GPUs. To the best of our knowledge, we are the first to explore DP fine-tuning in MLLMs. Our code is coming soon.

Understanding how the brain learns may be informed by studying biologically plausible learning rules. These rules, often approximating gradient descent learning to respect biological constraints such as locality, must meet two critical criteria to be considered an appropriate brain model: (1) good neuroscience task performance and (2) alignment with neural recordings. While extensive research has assessed the first criterion, the second remains underexamined. Employing methods such as Procrustes analysis on well-known neuroscience datasets, this study demonstrates the existence of a biologically plausible learning rule -- namely e-prop, which is based on gradient truncation and has demonstrated versatility across a wide range of tasks -- that can achieve neural data similarity comparable to Backpropagation Through Time (BPTT) when matched for task accuracy. Our findings also reveal that model architecture and initial conditions can play a more significant role in determining neural similarity than the specific learning rule. Furthermore, we observe that BPTT-trained models and their biologically plausible counterparts exhibit similar dynamical properties at comparable accuracies. These results underscore the substantial progress made in developing biologically plausible learning rules, highlighting their potential to achieve both competitive task performance and neural data similarity.

Zeroth-order methods are extensively used in machine learning applications where gradients are infeasible or expensive to compute, such as black-box attacks, reinforcement learning, and language model fine-tuning. Existing optimization theory focuses on convergence to an arbitrary stationary point, but less is known on the implicit regularization that provides a fine-grained characterization on which particular solutions are finally reached. We show that zeroth-order optimization with the standard two-point estimator favors solutions with small trace of Hessian, which is widely used in previous work to distinguish between sharp and flat minima. We further provide convergence rates of zeroth-order optimization to approximate flat minima for convex and sufficiently smooth functions, where flat minima are defined as the minimizers that achieve the smallest trace of Hessian among all optimal solutions. Experiments on binary classification tasks with convex losses and language model fine-tuning support our theoretical findings.

Policy gradient (PG) methods are a class of effective reinforcement learning algorithms, particularly when dealing with continuous control problems. These methods learn the parameters of parametric policies via stochastic gradient ascent, typically using on-policy trajectory data to estimate the policy gradient. However, such reliance on fresh data makes them sample-inefficient. Indeed, vanilla PG methods require $O(ε^{-2})$ trajectories to reach an $ε$-approximate stationary point. A common strategy to improve efficiency is to reuse off-policy information from past iterations, such as previous gradients or trajectories. While gradient reuse has received substantial theoretical attention, leading to improved rates of $O(ε^{-3/2})$, the reuse of past trajectories remains largely unexplored from a theoretical perspective. In this work, we provide the first rigorous theoretical evidence that extensive reuse of past off-policy trajectories can significantly accelerate convergence in PG methods. We introduce a power mean correction to the multiple importance weighting estimator and propose RPG (Retrospective Policy Gradient), a PG algorithm that combines old and new trajectories for policy updates. Through a novel analysis, we show that, under established assumptions, RPG achieves a sample complexity of $\widetilde{O}(ε^{-1})$, the best known rate in the literature. We further validate empirically our approach against PG methods with state-of-the-art rates.

Multi-agent reinforcement learning, despite its popularity and empirical success, faces significant scalability challenges in large-population dynamic games. Graphon mean field games (GMFGs) offer a principled framework for approximating such games while capturing heterogeneity among players. In this paper, we propose and analyze a policy optimization framework for continuous-time, finite-horizon linear-quadratic GMFGs. Exploiting the structural properties of GMFGs, we design an efficient policy parameterization in which each player's policy is represented as an affine function of their private state, with a shared slope function and player-specific intercepts. We develop a bilevel optimization algorithm that alternates between policy gradient updates for best-response computation under a fixed population distribution, and distribution updates using the resulting policies. We prove linear convergence of the policy gradient steps to best-response policies and establish global convergence of the overall algorithm to the Nash equilibrium. The analysis relies on novel landscape characterizations over infinite-dimensional policy spaces. Numerical experiments demonstrate the convergence and robustness of the proposed algorithm under varying graphon structures, noise levels, and action frequencies.

Semi-implicit variational inference (SIVI) is a powerful framework for approximating complex posterior distributions, but training with the Kullback-Leibler (KL) divergence can be challenging due to high variance and bias in high-dimensional settings. While current state-of-the-art semi-implicit variational inference methods, particularly Kernel Semi-Implicit Variational Inference (KSIVI), have been shown to work in high dimensions, training remains moderately expensive. In this work, we propose a kernelized KL divergence estimator that stabilizes training through nonparametric smoothing. To further reduce the bias, we introduce an importance sampling correction. We provide a theoretical connection to the amortized version of the Stein variational gradient descent, which estimates the score gradient via Stein's identity, showing that both methods minimize the same objective, but our semi-implicit approach achieves lower gradient variance. In addition, our method's bias in function space is benign, leading to more stable and efficient optimization. Empirical results demonstrate that our method outperforms or matches state-of-the-art SIVI methods in both performance and training efficiency.

Distributionally robust optimization offers a compelling framework for model fitting in machine learning, as it systematically accounts for data uncertainty. Focusing on Wasserstein distributionally robust optimization, we investigate the regularized problem where entropic smoothing yields a sampling-based approximation of the original objective. We establish the convergence of the approximate gradient over a compact set, leading to the concentration of the regularized problem critical points onto the original problem critical set as regularization diminishes and the number of approximation samples increases. Finally, we deduce convergence guarantees for a projected stochastic gradient method. Our analysis covers a general machine learning situation with an unbounded sample space and mixed continuous-discrete data.

Conventional wisdom attributes the mysterious generalization abilities of overparameterized neural networks to gradient descent (and its variants). The recent volume hypothesis challenges this view: it posits that these generalization abilities persist even when gradient descent is replaced by Guess & Check (G&C), i.e., by drawing weight settings until one that fits the training data is found. The validity of the volume hypothesis for wide and deep neural networks remains an open question. In this paper, we theoretically investigate this question for matrix factorization (with linear and non-linear activation)--a common testbed in neural network theory. We first prove that generalization under G&C deteriorates with increasing width, establishing what is, to our knowledge, the first case where G&C is provably inferior to gradient descent. Conversely, we prove that generalization under G&C improves with increasing depth, revealing a stark contrast between wide and deep networks, which we further validate empirically. These findings suggest that even in simple settings, there may not be a simple answer to the question of whether neural networks need gradient descent to generalize well.

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.