Learning from weakly dependent data under Dobrushin's condition

TL;DR

This paper extends statistical learning theory to data with complex dependencies satisfying Dobrushin's condition, providing bounds on generalization error and learning rates that are comparable to i.i.d. scenarios.

Contribution

It introduces learning and generalization bounds for dependent data on networks or spatial domains under Dobrushin's condition, using standard complexity measures.

Findings

Generalization bounds degrade only by constant factors compared to i.i.d.

Learning bounds degrade by logarithmic factors in training set size.

Standard complexity measures suffice for dependent data under Dobrushin's condition.

Abstract

Statistical learning theory has largely focused on learning and generalization given independent and identically distributed (i.i.d.) samples. Motivated by applications involving time-series data, there has been a growing literature on learning and generalization in settings where data is sampled from an ergodic process. This work has also developed complexity measures, which appropriately extend the notion of Rademacher complexity to bound the generalization error and learning rates of hypothesis classes in this setting. Rather than time-series data, our work is motivated by settings where data is sampled on a network or a spatial domain, and thus do not fit well within the framework of prior work. We provide learning and generalization bounds for data that are complexly dependent, yet their distribution satisfies the standard Dobrushin's condition. Indeed, we show that the standard complexity measures of Gaussian and Rademacher complexities and VC dimension are sufficient measures of complexity for the purposes of bounding the generalization error and learning rates of hypothesis classes in our setting. Moreover, our generalization bounds only degrade by constant factors compared to their i.i.d. analogs, and our learnability bounds degrade by log factors in the size of the training set.