[Paper Review] Majorizing Measures, Sequential Complexities, and Online Learning

Adam Block, Yuval Dagan+1 more|arXiv (Cornell University)|Feb 2, 2021

Computability, Logic, AI Algorithms22 references4 citations

TL;DR

This paper establishes a novel connection between majorizing measures and sequential Rademacher complexity, introducing a horizon-independent control of sequential complexity via fractional covering numbers and sequential scale-sensitive dimensions. It proves a tight contraction inequality for worst-case sequential Rademacher complexity, resolving long-standing open problems in online learning and extending classical empirical process theory to the sequential setting with sharp, integral-based bounds.

ABSTRACT

We introduce the technique of generic chaining and majorizing measures for controlling sequential Rademacher complexity. We relate majorizing measures to the notion of fractional covering numbers, which we show to be dominated in terms of sequential scale-sensitive dimensions in a horizon-independent way, and, under additional complexity assumptions establish a tight control on worst-case sequential Rademacher complexity in terms of the integral of sequential scale-sensitive dimension. Finally, we establish a tight contraction inequality for worst-case sequential Rademacher complexity. The above constitutes the resolution of a number of outstanding open problems in extending the classical theory of empirical processes to the sequential case, and, in turn, establishes sharp results for online learning.

Motivation & Objective

• To extend classical empirical process theory to the sequential setting by developing a framework for analyzing sequential Rademacher complexity.
• To resolve open problems in online learning by establishing tight control of worst-case sequential Rademacher complexity using majorizing measures.
• To relate sequential complexity to fractional covering numbers and sequential scale-sensitive dimensions in a horizon-independent manner.
• To prove a sharp contraction inequality for sequential Rademacher complexity, analogous to the classical Gaussian process result.
• To unify and strengthen existing results in online learning, private learning, and adversarial robustness through a unified complexity-theoretic framework.

Proposed method

• Introduces the use of generic chaining and majorizing measures to control sequential Rademacher complexity in the online learning setting.
• Defines and analyzes fractional covering numbers for sequential processes, showing they are dominated by sequential scale-sensitive dimensions in a horizon-independent way.
• Establishes that the integral of the sequential fat-shattering dimension controls worst-case sequential Rademacher complexity up to a constant factor.
• Derives a tight contraction inequality for sequential Rademacher complexity, proving it is sharp up to a universal constant.
• Applies the theory to bound uniform martingale laws of large numbers and minimax regret in online learning.
• Uses a recursive construction of measures and tree-based analysis to bound the integral of the logarithm of covering numbers over dyadic trees.

Experimental results

Research questions

• RQ1 Can majorizing measures be used to control sequential Rademacher complexity in the online learning setting?
• RQ2 Is there a horizon-independent relationship between fractional covering numbers and sequential scale-sensitive dimensions?
• RQ3 Can the worst-case sequential Rademacher complexity be tightly bounded using the integral of sequential fat-shattering dimension?
• RQ4 Does a sharp contraction inequality exist for sequential Rademacher complexity, analogous to the Gaussian case?
• RQ5 Can the theory unify and strengthen results in online learning, private learning, and adversarial robustness?

Key findings

• The worst-case sequential Rademacher complexity is bounded by a constant times the integral of the square root of the sequential fat-shattering dimension over the unit interval.
• Fractional covering numbers for sequential processes are dominated by the integral of the sequential scale-sensitive dimension, independent of the horizon length.
• A sharp contraction inequality for sequential Rademacher complexity is established, showing it is tight up to a universal constant.
• The theory resolves open problems in online learning by providing a horizon-independent complexity measure based on sequential fat-shattering dimension.
• The results imply tight bounds on minimax regret in online learning and uniform martingale laws of large numbers, with applications to private learning and adversarial robustness.
• The framework extends classical empirical process theory to the sequential case, providing a complete analogue of Dudley’s chaining and Fernique-Talagrand bounds for martingale processes.