[Paper Review] Online Improper Learning with an Approximation Oracle

Elad Hazan, Wei Hu+2 more|arXiv (Cornell University)|Jan 1, 2018

Advanced Bandit Algorithms Research10 citations

TL;DR

This paper presents a novel reduction from online learning to offline approximate optimization, enabling efficient online learning using only logarithmically many calls to an approximation oracle per iteration. It resolves an open problem by Kalai and Vempala and Garber, achieving near-optimal regret in improper learning settings with efficient oracle usage.

ABSTRACT

We study the following question: given an efficient approximation algorithm for an optimization problem, can we learn efficiently in the same setting? We give a formal affirmative answer to this question in the form of a reduction from online learning to offline approximate optimization using an efficient algorithm that guarantees near optimal regret. The algorithm is efficient in terms of the number of oracle calls to a given approximation oracle – it makes only logarithmically many such calls per iteration. This resolves an open question by Kalai and Vempala, and by Garber. Furthermore, our result applies to the more general improper learning problems.

Motivation & Objective

• To address the open question of whether efficient approximation algorithms can enable efficient online learning.
• To design a reduction from online learning to offline approximate optimization that maintains low regret.
• To ensure the proposed algorithm is efficient in terms of oracle queries, making only logarithmic calls per iteration.
• To extend the applicability to improper learning problems, where predictions may not lie in the original hypothesis class.
• To provide a formal affirmative answer to the question of whether approximation oracles can support efficient online learning.

Proposed method

• The method employs a reduction from online learning to offline approximate optimization using a carefully designed iterative update rule.
• It leverages a separation oracle and a dual averaging framework to maintain regret bounds while using only logarithmic oracle calls per round.
• The algorithm maintains a distribution over hypotheses and updates it based on feedback, using the approximation oracle to compute near-optimal solutions efficiently.
• The approach ensures that the regret grows sublinearly with time, achieving near-optimal performance despite using approximate solutions.
• It introduces a novel analysis technique that bounds the regret in terms of the approximation ratio of the oracle and the number of iterations.
• The method is general and applies to a wide range of improper learning problems, not restricted to specific hypothesis classes.

Experimental results

Research questions

• RQ1 Can an efficient approximation algorithm for an optimization problem be used to achieve efficient online learning?
• RQ2 What is the minimal number of oracle calls required per iteration to maintain near-optimal regret in online learning?
• RQ3 Can the reduction from online learning to approximate offline optimization be made efficient in terms of oracle usage?
• RQ4 Does the approach extend to improper learning, where the hypothesis class is not closed under convex combinations?
• RQ5 Can the regret be bounded using only logarithmic oracle calls per iteration while maintaining near-optimality?

Key findings

• The proposed algorithm achieves near-optimal regret with only logarithmic number of oracle calls per iteration, significantly improving upon prior methods.
• The reduction is general and applies to improper learning problems, extending the scope of online learning with approximation oracles.
• The method resolves an open question posed by Kalai and Vempala and by Garber regarding the feasibility of efficient online learning using approximation oracles.
• The regret bound is sublinear and depends on the approximation ratio of the oracle, ensuring scalability and efficiency.
• The algorithm maintains computational efficiency by avoiding repeated expensive optimization, relying instead on efficient oracle queries.
• The theoretical analysis confirms that the approach achieves regret close to the optimal benchmark, even when using approximate solutions.