# Distribution-Dependent Analysis of Gibbs-ERM Principle

## TL;DR

This paper provides distribution-dependent bounds on the excess risk of Gibbs-ERM learners, linking it to the effective dimension of the problem and extending understanding of learning in large hypothesis spaces.

## Contribution

It introduces the first analysis connecting Gibbs-ERM excess risk bounds to the effective dimension, improving upon previous distribution-agnostic results.

## Key findings

- Excess risk is controlled by the effective dimension of the Hessian at local minima.
- Distribution-dependent bounds can be tighter than previous results.
- Analysis extends from local minima to global excess risk bounds.

## Abstract

Gibbs-ERM learning is a natural idealized model of learning with stochastic optimization algorithms (such as Stochastic Gradient Langevin Dynamics and ---to some extent--- Stochastic Gradient Descent), while it also arises in other contexts, including PAC-Bayesian theory, and sampling mechanisms. In this work we study the excess risk suffered by a Gibbs-ERM learner that uses non-convex, regularized empirical risk with the goal to understand the interplay between the data-generating distribution and learning in large hypothesis spaces. Our main results are distribution-dependent upper bounds on several notions of excess risk. We show that, in all cases, the distribution-dependent excess risk is essentially controlled by the effective dimension $\mathrm{tr}\left(\boldsymbol{H}^{\star} (\boldsymbol{H}^{\star} + \lambda \boldsymbol{I})^{-1}\right)$ of the problem, where $\boldsymbol{H}^{\star}$ is the Hessian matrix of the risk at a local minimum. This is a well-established notion of effective dimension appearing in several previous works, including the analyses of SGD and ridge regression, but ours is the first work that brings this dimension to the analysis of learning using Gibbs densities. The distribution-dependent view we advocate here improves upon earlier results of Raginsky et al. (2017), and can yield much tighter bounds depending on the interplay between the data-generating distribution and the loss function. The first part of our analysis focuses on the localized excess risk in the vicinity of a fixed local minimizer. This result is then extended to bounds on the global excess risk, by characterizing probabilities of local minima (and their complement) under Gibbs densities, a results which might be of independent interest.