Theoretically understanding stochastic gradient descent (SGD) in overparameterized models has led to the development of several optimization algorithms that are widely used in practice today. Recent work by Zou et al. [2021] provides sharp rates for SGD optimization in linear regression using constant learning rate, both with and without tail iterate averaging, based on a bias-variance decomposition of the risk. In our work, we provide a simplified analysis recovering the same bias and variance bounds provided in [Zou et al., 2021] based on simple linear algebra tools, bypassing the requirement to manipulate operators on positive semi-definite (PSD) matrices. We believe our work makes the analysis of SGD on linear regression very accessible and will be helpful in further analyzing mini-batching and learning rate scheduling, leading to improvements in the training of realistic models. We use bolded small letters for vectors and bolded capital letters for matrices, and we use 1 for a vector of ones.

Federated learning allows each client to keep its data locally when training machine learning models in a distributed setting. Significant recent research established the requirements that the input must satisfy in order to guarantee convergence of the training loop. This line of work uses averaging as the aggregation rule for the training models. In particular, we are interested in whether federated learning is robust to Byzantine behavior, and observe and investigate a tradeoff between the average/centroid and the validity conditions from distributed computing. We show that the various validity conditions alone do not guarantee a good approximation of the average. Furthermore, we show that reaching good approximation does not give good results in experimental settings due to possible Byzantine outliers. Our main contribution is the first lower bound of $\min\{\frac{n-t}{t},\sqrt{d}\}$ on the centroid approximation under box validity that is often considered in the literature, where $n$ is the number of clients, $t$ the upper bound on the number of Byzantine faults, and $d$ is the dimension of the machine learning model. We complement this lower bound by an upper bound of $2\min\{n,\sqrt{d}\}$, by providing a new analysis for the case $n

Existing analysis of Local (Stochastic) Gradient Descent for heterogeneous objectives requires stepsizes $η\leq 1/K$ where $K$ is the communication interval, which ensures monotonic decrease of the objective. In contrast, we analyze Local Gradient Descent for logistic regression with separable, heterogeneous data using any stepsize $η> 0$. With $R$ communication rounds and $M$ clients, we show convergence at a rate $\mathcal{O}(1/ηK R)$ after an initial unstable phase lasting for $\widetilde{\mathcal{O}}(ηK M)$ rounds. This improves upon the existing $\mathcal{O}(1/R)$ rate for general smooth, convex objectives. Our analysis parallels the single machine analysis of~\cite{wu2024large} in which instability is caused by extremely large stepsizes, but in our setting another source of instability is large local updates with heterogeneous objectives.

In this paper, we develop a wide class Mirror Descent (MD) algorithms, which play a key role in machine learning. For this purpose we formulated the constrained optimization problem, in which we exploits the Bregman divergence with the Tempesta multi-parametric deformation logarithm as a link function. This link function called also mirror function defines the mapping between the primal and dual spaces and is associated with a very-wide (in fact, theoretically infinite) class of generalized trace-form entropies. In order to derive novel MD updates, we estimate generalized exponential function, which closely approximates the inverse of the multi-parametric Tempesta generalized logarithm. The shape and properties of the Tempesta logarithm and its inverse-deformed exponential functions can be tuned by several hyperparameters. By learning these hyperparameters, we can adapt to distribution or geometry of training data, and we can adjust them to achieve desired properties of MD algorithms. The concept of applying multi-parametric logarithms allow us to generate a new wide and flexible family of MD and mirror-less MD updates.

The non-convexity of the artificial neural network (ANN) training landscape brings inherent optimization difficulties. While the traditional back-propagation stochastic gradient descent (SGD) algorithm and its variants are effective in certain cases, they can become stuck at spurious local minima and are sensitive to initializations and hyperparameters. Recent work has shown that the training of an ANN with ReLU activations can be reformulated as a convex program, bringing hope to globally optimizing interpretable ANNs. However, naively solving the convex training formulation has an exponential complexity, and even an approximation heuristic requires cubic time. In this work, we characterize the quality of this approximation and develop two efficient algorithms that train ANNs with global convergence guarantees. The first algorithm is based on the alternating direction method of multiplier (ADMM). It solves both the exact convex formulation and the approximate counterpart. Linear global convergence is achieved, and the initial several iterations often yield a solution with high prediction accuracy. When solving the approximate formulation, the per-iteration time complexity is quadratic. The second algorithm, based on the "sampled convex programs" theory, solves unconstrained convex formulations and converges to an approximately globally optimal classifier. The non-convexity of the ANN training landscape exacerbates when adversarial training is considered. We apply the robust convex optimization theory to convex training and develop convex formulations that train ANNs robust to adversarial inputs. Our analysis explicitly focuses on one-hidden-layer fully connected ANNs, but can extend to more sophisticated architectures.

Neural network training is inherently sensitive to initialization and the randomness induced by stochastic gradient descent. However, it is unclear to what extent such effects lead to meaningfully different networks, either in terms of the models' weights or the underlying functions that were learned. In this work, we show that during the initial "chaotic" phase of training, even extremely small perturbations reliably causes otherwise identical training trajectories to diverge-an effect that diminishes rapidly over training time. We quantify this divergence through (i) $L^2$ distance between parameters, (ii) the loss barrier when interpolating between networks, (iii) $L^2$ and barrier between parameters after permutation alignment, and (iv) representational similarity between intermediate activations; revealing how perturbations across different hyperparameter or fine-tuning settings drive training trajectories toward distinct loss minima. Our findings provide insights into neural network training stability, with practical implications for fine-tuning, model merging, and diversity of model ensembles.

Variational inference (VI) is a popular approach in Bayesian inference, that looks for the best approximation of the posterior distribution within a parametric family, minimizing a loss that is typically the (reverse) Kullback-Leibler (KL) divergence. In this paper, we focus on the following parametric family: mixtures of isotropic Gaussians (i.e., with diagonal covariance matrices proportional to the identity) and uniform weights. We develop a variational framework and provide efficient algorithms suited for this family. In contrast with mixtures of Gaussian with generic covariance matrices, this choice presents a balance between accurate approximations of multimodal Bayesian posteriors, while being memory and computationally efficient. Our algorithms implement gradient descent on the location of the mixture components (the modes of the Gaussians), and either (an entropic) Mirror or Bures descent on their variance parameters. We illustrate the performance of our algorithms on numerical experiments.

We provide an algorithm to privately generate continuous-time data (e.g. marginals from stochastic differential equations), which has applications in highly sensitive domains involving time-series data such as healthcare. We leverage the connections between trajectory inference and continuous-time synthetic data generation, along with a computational method based on mean-field Langevin dynamics. As discretized mean-field Langevin dynamics and noisy particle gradient descent are equivalent, DP results for noisy SGD can be applied to our setting. We provide experiments that generate realistic trajectories on a synthesized variation of hand-drawn MNIST data while maintaining meaningful privacy guarantees. Crucially, our method has strong utility guarantees under the setting where each person contributes data for \emph{only one time point}, while prior methods require each person to contribute their \emph{entire temporal trajectory}--directly improving the privacy characteristics by construction.

In this paper, we propose Adjusted Shuffling SARAH, a novel algorithm that integrates shuffling techniques with the well-known variance-reduced algorithm SARAH while dynamically adjusting the stochastic gradient weights in each update to enhance exploration. Our method achieves the best-known gradient complexity for shuffling variance reduction methods in a strongly convex setting. This result applies to any shuffling technique, which narrows the gap in the complexity analysis of variance reduction methods between uniform sampling and shuffling data. Furthermore, we introduce Inexact Adjusted Reshuffling SARAH, an inexact variant of Adjusted Shuffling SARAH that eliminates the need for full-batch gradient computations. This algorithm retains the same linear convergence rate as Adjusted Shuffling SARAH while showing an advantage in total complexity when the sample size is very large.

This paper investigates the sublinear regret guarantees of two non-no-regret algorithms in zero-sum games: Fictitious Play, and Online Gradient Descent with constant stepsizes. In general adversarial online learning settings, both algorithms may exhibit instability and linear regret due to no regularization (Fictitious Play) or small amounts of regularization (Gradient Descent). However, their ability to obtain tighter regret bounds in two-player zero-sum games is less understood. In this work, we obtain strong new regret guarantees for both algorithms on a class of symmetric zero-sum games that generalize the classic three-strategy Rock-Paper-Scissors to a weighted, n-dimensional regime. Under symmetric initializations of the players' strategies, we prove that Fictitious Play with any tiebreaking rule has $O(\sqrt{T})$ regret, establishing a new class of games for which Karlin's Fictitious Play conjecture holds. Moreover, by leveraging a connection between the geometry of the iterates of Fictitious Play and Gradient Descent in the dual space of payoff vectors, we prove that Gradient Descent, for almost all symmetric initializations, obtains a similar $O(\sqrt{T})$ regret bound when its stepsize is a sufficiently large constant. For Gradient Descent, this establishes the first "fast and furious" behavior (i.e., sublinear regret without time-vanishing stepsizes) for zero-sum games larger than 2x2.