[Paper Review] Combining Online Learning Guarantees

Ashok Cutkosky|arXiv (Cornell University)|Feb 24, 2019

Advanced Bandit Algorithms Research25 references3 citations

TL;DR

This paper introduces a simple yet powerful meta-algorithm that combines any two parameter-free online learning algorithms by adding their iterates, resulting in a new algorithm whose regret is the minimum of the two base algorithms' regrets. It further extends this idea to create a generic, safe optimistic online learning framework that adapts to multiple hint sequences and achieves dimension-free, parameter-free regret guarantees even in unconstrained settings.

ABSTRACT

We show how to take any two parameter-free online learning algorithms with different regret guarantees and obtain a single algorithm whose regret is the minimum of the two base algorithms. Our method is embarrassingly simple: just add the iterates. This trick can generate efficient algorithms that adapt to many norms simultaneously, as well as providing diagonal-style algorithms that still maintain dimension-free guarantees. We then proceed to show how a variant on this idea yields a black-box procedure for generating optimistic online learning algorithms. This yields the first optimistic regret guarantees in the unconstrained setting and generically increases adaptivity. Further, our optimistic algorithms are guaranteed to do no worse than their non-optimistic counterparts regardless of the quality of the optimistic estimates provided to the algorithm.

Motivation & Objective

• To develop a black-box method for combining online learning algorithms with different regret guarantees to achieve the best of both in terms of regret.
• To enable parameter-free online learning that adapts to multiple norms simultaneously without prior knowledge of the comparison point or gradient norms.
• To introduce a generic, safe reduction for generating optimistic online learning algorithms that maintain worst-case performance even when hints are poor.
• To extend optimistic online learning to unconstrained domains, where prior methods were limited to bounded domains.
• To provide a unified framework that competes with the best fixed hint sequence in hindsight, improving adaptivity.

Proposed method

• The core technique is to combine two online learning algorithms by summing their iterates at each round, which results in a new algorithm whose regret is pointwise bounded by the minimum of the two individual regrets.
• The method applies to any parameter-free algorithm with bounded regret at zero, enabling dimension-free and norm-adaptive guarantees.
• For optimistic learning, the paper introduces a reduction that takes any adaptive algorithm and transforms it into an optimistic one using a dual optimization scheme over hints.
• The optimistic algorithm uses a surrogate loss function to minimize regret with respect to the best hint sequence, leveraging online convex optimization to generate adaptive hints.
• A variant of the method uses a modified loss function in the hint selection process to tighten bounds in unconstrained settings, replacing standard sum-of-squares with a more refined expression involving inner products.
• The framework is robust: even when hints are arbitrarily poor, the regret remains no worse than that of the original non-optimistic algorithm.

Experimental results

Research questions

• RQ1 Can we combine two online learning algorithms with different regret guarantees into a single algorithm that achieves the minimum of the two regrets?
• RQ2 Can we design a generic, black-box method to generate optimistic online learning algorithms that are safe against poor hints and adapt to multiple hint sequences?
• RQ3 Can we extend optimistic online learning to unconstrained domains while maintaining dimension-free, parameter-free regret guarantees?
• RQ4 Can we construct a hint generation strategy that competes with the best fixed hint in hindsight, improving adaptivity?
• RQ5 Can this framework be used to derive new regret bounds, such as an empirical Bernstein bound, in Hilbert spaces?

Key findings

• The iterate-summing method achieves regret bounded by the pointwise minimum of two base algorithms' regrets, enabling 'best-of-all-worlds' performance without prior knowledge of the optimal algorithm.
• The method produces an efficient algorithm that adapts to multiple norms simultaneously, maintaining dimension-free regret guarantees.
• The optimistic variant of the algorithm achieves regret bounds of the form $ R_T(u) \leq O\left[B(u) \min\left(\sqrt{\sum \|g_t\|^2}, \sqrt{\sum \|g_t - h_t\|^2}\right)\right] $, improving on prior results.
• In the unconstrained setting, the bound is further refined to $ \sum \|g_t - h_t\|^2 - \|h_t\|^2 $, which allows tighter performance when gradients are correlated.
• The algorithm is provably safe: even with arbitrarily poor hints, the regret never exceeds that of the original non-optimistic algorithm.
• A simple hint generation scheme using online learning over the loss $ \ell_t(h) = \|g_t - h\|^2 $ achieves regret comparable to the best fixed hint, with $ \tilde{O}(\|u\|\sqrt{\sum \|g_t - \bar{g}\|^2}) $, generalizing prior results to unconstrained domains.