Pushing the Efficiency-Regret Pareto Frontier for Online Learning of Portfolios and Quantum States

Abstract

We revisit the classical online portfolio selection problem. It is widely assumed that a trade-off between computational complexity and regret is unavoidable, with Cover's Universal Portfolios algorithm, SOFT-BAYES and ADA-BARRONS currently constituting its state-of-the-art Pareto frontier. In this paper, we present the first efficient algorithm, BISONS, that obtains polylogarithmic regret with memory and per-step running time requirements that are polynomial in the dimension, displacing ADA-BARRONS from the Pareto frontier. Additionally, we resolve a COLT 2020 open problem by showing that a certain Follow-The-Regularized-Leader algorithm with log-barrier regularization suffers an exponentially larger dependence on the dimension than previously conjectured. Thus, we rule out this algorithm as a candidate for the Pareto frontier. We also extend our algorithm and analysis to a more general problem than online portfolio selection, viz. online learning of quantum states with log loss. This algorithm, called SCHR ÖDINGER'S-BISONS, is the first efficient algorithm with polylogarithmic regret for this more general problem.

Efficient and Near-Optimal Online Portfolio Selection

“…Finally, putting µ = 0 in (VB-FTRL) we recover Follow-The-Leader regularized with a logarithmic barrier (LB-FTRL), the algorithm conjectured to have an O(d log(T )) regret in [46]. As we have already observed in Section 1.1, the recent refutation of this conjecture in [49] indicates that timedependent regularization is instrumental in achieving regret of the right order with an FTRL-type strategy. The regret analysis of VB-FTRL in Section 3 demonstrates this on a technical level.…”

“…In addition, the per-round runtime deteriorated to O(d 2.5 T ), thus becoming T -dependent. Very recently, two competing works [49,34] achieved a T -independent per-round runtime while preserving the O(d 2 ) regret guarantee of Ada-BARRONS. In both cases, the crucial step was to combine Ada-BARRONS with an appropriate scheme of adaptive restarts.…”

“…Since such an algorithm could be implemented with O(d 2 T ) per-round runtime via Newton's method, this would result in a regretoptimal and computationally feasible procedure. Yet, [49] recently disproved this conjecture: as it turns out, the regret of LB-FTRL can be of order 2 d log(T ) log(log(T )) whenever T > poly (2 d ).…”

On Confidence Sequences for Bounded Random Processes via Universal Gambling Strategies

“…An interested reader is referred to [3] and the references therein for a detailed treatment of these topics and their interconnections: Chapter 6 explores the horse-race formalism as well as establishes connections between betting on horse races and data compression; Chapters 11 and 13 discuss universal compression and connections to prediction and hypothesis testing; Chapter 14 investigates the closely allied concept of Kolmogorov complexity and the minimum description length (MDL) principle (see also [9] and [7] for a detailed treatise on the MDL principle); and Chapter 16 delves into portfolio selection and the UP method. Universal portfolio selection still remains a widely studied topic because a method that simultaneously achieves low regret and low time complexity has been elusive-see [16,17,26,31] and the references within for recent work in this direction.…”