Risk-Monotonicity in Statistical Learning

TL;DR

This paper introduces the first risk-monotonic algorithms for statistical learning that ensure non-increasing risk with more data, addressing a key challenge in understanding generalization and stability in machine learning.

Contribution

It provides the first consistent, risk-monotonic algorithms under weak assumptions and introduces new concentration inequalities for non-i.i.d. processes.

Findings

Algorithms achieve risk monotonicity with high probability.

Risk monotonicity does not worsen excess risk rates.

New concentration inequalities for Martingale Difference Sequences.

Abstract

Acquisition of data is a difficult task in many applications of machine learning, and it is only natural that one hopes and expects the population risk to decrease (better performance) monotonically with increasing data points. It turns out, somewhat surprisingly, that this is not the case even for the most standard algorithms that minimize the empirical risk. Non-monotonic behavior of the risk and instability in training have manifested and appeared in the popular deep learning paradigm under the description of double descent. These problems highlight the current lack of understanding of learning algorithms and generalization. It is, therefore, crucial to pursue this concern and provide a characterization of such behavior. In this paper, we derive the first consistent and risk-monotonic (in high probability) algorithms for a general statistical learning setting under weak assumptions, consequently answering some questions posed by Viering et al. 2019 on how to avoid non-monotonic behavior of risk curves. We further show that risk monotonicity need not necessarily come at the price of worse excess risk rates. To achieve this, we derive new empirical Bernstein-like concentration inequalities of independent interest that hold for certain non-i.i.d.~processes such as Martingale Difference Sequences.