[Paper Review] Compression, Generalization and Learning

Marco C. Campi, Simone Garatti|arXiv (Cornell University)|Jan 30, 2023

Machine Learning and Algorithms5 citations

TL;DR

This paper establishes a novel theoretical framework linking compression set size to the probability of change in compression, offering finite-sample bounds for the risk of misclassification without distributional assumptions. It shows that the cardinality of the compressed set consistently estimates the probability of change, enabling tight, agnostic bounds via a preference condition.

ABSTRACT

A compression function is a map that slims down an observational set into a subset of reduced size, while preserving its informational content. In multiple applications, the condition that one new observation makes the compressed set change is interpreted that this observation brings in extra information and, in learning theory, this corresponds to misclassification, or misprediction. In this paper, we lay the foundations of a new theory that allows one to keep control on the probability of change of compression (which maps into the statistical "risk" in learning applications). Under suitable conditions, the cardinality of the compressed set is shown to be a consistent estimator of the probability of change of compression (without any upper limit on the size of the compressed set); moreover, unprecedentedly tight finite-sample bounds to evaluate the probability of change of compression are obtained under a generally applicable condition of preference. All results are usable in a fully agnostic setup, i.e., without requiring any a priori knowledge on the probability distribution of the observations. Not only these results offer a valid support to develop trust in observation-driven methodologies, they also play a fundamental role in learning techniques as a tool for hyper-parameter tuning.

Motivation & Objective

• To develop a theory that controls the probability of change in compression functions without prior knowledge of data distribution.
• To establish the cardinality of the compressed set as a consistent estimator of the probability of change, even with unbounded set sizes.
• To derive unprecedentedly tight finite-sample bounds for the risk of misclassification using a general preference condition.
• To enable application of compression theory in supervised and unsupervised learning, as well as other domains, in a fully agnostic manner.
• To support trust in observation-driven methodologies and hyper-parameter tuning through rigorous statistical control.

Proposed method

• Introduces a compression function that maps observation sets into reduced subsets while preserving informational content.
• Defines the probability of change of compression as the likelihood that a new observation alters the compressed set.
• Uses a preference condition to derive finite-sample bounds on the probability of change, independent of data distribution.
• Applies measure-theoretic constructions involving multisets and probabilistic measures to model the evolution of compressed sets.
• Employs a limiting argument with approximating sequences of measures to prove tight bounds, leveraging non-decreasing functions and mass redistribution.
• Establishes that the cardinality of the compressed set consistently estimates the risk of change, even without an upper bound on size.

Experimental results

Research questions

• RQ1 How can the probability of change in a compression function be bounded in finite samples without assuming a data distribution?
• RQ2 Can the size of the compressed set serve as a consistent estimator for the probability of change in compression?
• RQ3 What conditions enable the derivation of tight, agnostic finite-sample bounds on the risk of misclassification via compression?
• RQ4 How does the preference condition facilitate tighter bounds compared to existing approaches in statistical learning theory?
• RQ5 In what ways can the proposed framework support trust and hyper-parameter tuning in learning systems without distributional assumptions?

Key findings

• The cardinality of the compressed set is a consistent estimator of the probability of change of compression, even when the set size is unbounded.
• Finite-sample bounds on the probability of change are derived under a general preference condition, providing tighter guarantees than previous methods.
• The bounds are agnostic, requiring no prior knowledge of the underlying data distribution, making them applicable across diverse learning settings.
• The framework enables risk evaluation in both supervised and unsupervised learning contexts through compression-only analysis.
• The theoretical results support the use of compression size as a proxy for statistical risk, facilitating hyper-parameter tuning in learning systems.
• The proof technique involves constructing approximating measure sequences that preserve constraints while converging to optimal bounds, ensuring tightness.