On the Validation of Gibbs Algorithms: Training Datasets, Test Datasets and their Aggregation

TL;DR

This paper analytically characterizes the dependence of Gibbs algorithms on training data, providing explicit formulas for sensitivity and insights into dataset aggregation and generalization performance.

Contribution

It introduces a closed-form sensitivity analysis of Gibbs algorithms and explores dataset aggregation effects on their generalization capabilities.

Findings

Explicit expressions linking training and test errors of GAs.

Sensitivity of GAs to training data characterized in closed form.

Connection established between Jeffrey's divergence and generalization metrics.

Abstract

The dependence on training data of the Gibbs algorithm (GA) is analytically characterized. By adopting the expected empirical risk as the performance metric, the sensitivity of the GA is obtained in closed form. In this case, sensitivity is the performance difference with respect to an arbitrary alternative algorithm. This description enables the development of explicit expressions involving the training errors and test errors of GAs trained with different datasets. Using these tools, dataset aggregation is studied and different figures of merit to evaluate the generalization capabilities of GAs are introduced. For particular sizes of such datasets and parameters of the GAs, a connection between Jeffrey's divergence, training and test errors is established.