# Generalization Error Bounds for Noisy, Iterative Algorithms via Maximal Leakage

## TL;DR

This paper introduces an information-theoretic approach using maximal leakage to analyze the generalization error of iterative, noisy algorithms like SGLD, providing explicit bounds and insights on noise and update functions.

## Contribution

It develops a semi-closed form bound on maximal leakage for noisy iterative algorithms, linking update function properties and noise to generalization performance.

## Key findings

- Bound on maximal leakage with Gaussian noise and bounded update functions
- Explicit tight bounds for various scenarios
- Insights on optimal noise choice for minimizing leakage

## Abstract

We adopt an information-theoretic framework to analyze the generalization behavior of the class of iterative, noisy learning algorithms. This class is particularly suitable for study under information-theoretic metrics as the algorithms are inherently randomized, and it includes commonly used algorithms such as Stochastic Gradient Langevin Dynamics (SGLD). Herein, we use the maximal leakage (equivalently, the Sibson mutual information of order infinity) metric, as it is simple to analyze, and it implies both bounds on the probability of having a large generalization error and on its expected value. We show that, if the update function (e.g., gradient) is bounded in $L_2$-norm and the additive noise is isotropic Gaussian noise, then one can obtain an upper-bound on maximal leakage in semi-closed form. Furthermore, we demonstrate how the assumptions on the update function affect the optimal (in the sense of minimizing the induced maximal leakage) choice of the noise. Finally, we compute explicit tight upper bounds on the induced maximal leakage for other scenarios of interest.