Sharp concentration of uniform generalization errors in binary linear classification

We examine the concentration of uniform generalization errors around their expectation in binary linear classification problems via an isoperimetric argument. In particular, we establish Poincar\'{e} and log-Sobolev inequalities for the joint distribution of the output labels and the label-weighted input vectors, which we apply to derive concentration bounds. The derived concentration bounds are sharp up to moderate multiplicative constants by those under well-balanced labels. In asymptotic analysis, we also show that almost sure convergence of uniform generalization errors to their expectation occurs in very broad settings, such as proportionally high-dimensional regimes. Using this convergence, we establish uniform laws of large numbers under dimension-free conditions.

Federated Learning with Relative Fairness

This paper proposes a federated learning framework designed to achieve \textit{relative fairness} for clients. Traditional federated learning frameworks typically ensure absolute fairness by guaranteeing minimum performance across all client subgroups. However, this approach overlooks disparities in model performance between subgroups. The proposed framework uses a minimax problem approach to minimize relative unfairness, extending previous methods in distributionally robust optimization (DRO). A novel fairness index, based on the ratio between large and small losses among clients, is introduced, allowing the framework to assess and improve the relative fairness of trained models. Theoretical guarantees demonstrate that the framework consistently reduces unfairness. We also develop an algorithm, named \textsc{Scaff-PD-IA}, which balances communication and computational efficiency while maintaining minimax-optimal convergence rates. Empirical evaluations on real-world datasets confirm its effectiveness in maintaining model performance while reducing disparity.

Effect of Random Learning Rate: Theoretical Analysis of SGD Dynamics in Non-Convex Optimization via Stationary Distribution

We consider a variant of the stochastic gradient descent (SGD) with a random learning rate and reveal its convergence properties. SGD is a widely used stochastic optimization algorithm in machine learning, especially deep learning. Numerous studies reveal the convergence properties of SGD and its simplified variants. Among these, the analysis of convergence using a stationary distribution of updated parameters provides generalizable results. However, to obtain a stationary distribution, the update direction of the parameters must not degenerate, which limits the applicable variants of SGD. In this study, we consider a novel SGD variant, Poisson SGD, which has degenerated parameter update directions and instead utilizes a random learning rate. Consequently, we demonstrate that a distribution of a parameter updated by Poisson SGD converges to a stationary distribution under weak assumptions on a loss function. Based on this, we further show that Poisson SGD finds global minima in non-convex optimization problems and also evaluate the generalization error using this method. As a proof technique, we approximate the distribution by Poisson SGD with that of the bouncy particle sampler (BPS) and derive its stationary distribution, using the theoretical advance of the piece-wise deterministic Markov process (PDMP).

Dimension-free uniform concentration bound for logistic regression

We provide a novel dimension-free uniform concentration bound for the empirical risk function of constrained logistic regression. Our bound yields a milder sufficient condition for a uniform law of large numbers than conditions derived by the Rademacher complexity argument and McDiarmid's inequality. The derivation is based on the PAC-Bayes approach with second-order expansion and Rademacher-complexity-based bounds for the residual term of the expansion.

Non-asymptotic analysis of Langevin-type Monte Carlo algorithms

We study the Langevin-type algorithms for Gibbs distributions such that the potentials are dissipative and their weak gradients have the finite moduli of continuity. Our main result is a non-asymptotic upper bound of the 2-Wasserstein distance between the Gibbs distribution and the law of general Langevin-type algorithms based on the Liptser--Shiryaev theory and functional inequalities. We apply this bound to show that the dissipativity of the potential and the $\alpha$-H\"{o}lder continuity of the gradient with $\alpha>1/3$ are sufficient for the convergence of the Langevin Monte Carlo algorithm with appropriate control of the parameters. We also propose Langevin-type algorithms with spherical smoothing for potentials without convexity or continuous differentiability.

Benign Overfitting in Time Series Linear Model with Over-Parameterization

The success of large-scale models in recent years has increased the importance of statistical models with numerous parameters. Several studies have analyzed over-parameterized linear models with high-dimensional data that may not be sparse; however, existing results depend on the independent setting of samples. In this study, we analyze a linear regression model with dependent time series data under over-parameterization settings. We consider an estimator via interpolation and developed a theory for excess risk of the estimator under multiple dependence types. This theory can treat infinite-dimensional data without sparsity and handle long-memory processes in a unified manner. Moreover, we bound the risk in our theory via the integrated covariance and nondegeneracy of autocorrelation matrices. The results show that the convergence rate of risks with short-memory processes is identical to that of cases with independent data, while long-memory processes slow the convergence rate. We also present several examples of specific dependent processes that can be applied to our setting.