Evaluated CMI Bounds for Meta Learning: Tightness and Expressiveness

TL;DR

This paper develops new generalization bounds for meta learning using the evaluated conditional mutual information (e-CMI), bridging information-theoretic and classical learning theory, and demonstrating the bounds' tightness and expressiveness in representation learning.

Contribution

It introduces novel e-CMI-based generalization bounds for meta learning, connecting information-theoretic and classical approaches, and applies them to representation learning with matching known complexity scalings.

Findings

e-CMI bounds scale as rac{5C(\u1ef9H)}{n\u001hat n} + rac{5C()}{n}

Bounds match Gaussian complexity results in representation learning

Bridges the gap between information-theoretic and classical generalization bounds in meta learning

Abstract

Recent work has established that the conditional mutual information (CMI) framework of Steinke and Zakynthinou (2020) is expressive enough to capture generalization guarantees in terms of algorithmic stability, VC dimension, and related complexity measures for conventional learning (Harutyunyan et al., 2021, Haghifam et al., 2021). Hence, it provides a unified method for establishing generalization bounds. In meta learning, there has so far been a divide between information-theoretic results and results from classical learning theory. In this work, we take a first step toward bridging this divide. Specifically, we present novel generalization bounds for meta learning in terms of the evaluated CMI (e-CMI). To demonstrate the expressiveness of the e-CMI framework, we apply our bounds to a representation learning setting, with $n$ samples from $\hat n$ tasks parameterized by functions of the form $f_i \circ h$. Here, each $f_i \in \mathcal F$ is a task-specific function, and $h \in \mathcal H$ is the shared representation. For this setup, we show that the e-CMI framework yields a bound that scales as $\sqrt{ \mathcal C(\mathcal H)/(n\hat n) + \mathcal C(\mathcal F)/n} $, where $\mathcal C(\cdot)$ denotes a complexity measure of the hypothesis class. This scaling behavior coincides with the one reported in Tripuraneni et al. (2020) using Gaussian complexity.