[Paper Review] Efficient Online Portfolio with Logarithmic Regret

Haipeng Luo, Chen-Yu Wei+1 more|arXiv (Cornell University)|May 18, 2018

Advanced Bandit Algorithms Research10 citations

TL;DR

This paper proposes a novel online portfolio algorithm with logarithmic regret, $Ø(N^2(\ln T)^4)$, using Online Mirror Descent with a new regularizer that enables efficient implementation in $Ø(TN^{2.5})$ time per round. Unlike Cover's Universal Portfolio, it avoids log-concave sampling and achieves provably sub-polynomial regret without unbounded factors, offering a faster and more practical alternative for online portfolio management.

ABSTRACT

We study the decades-old problem of online portfolio management and propose the first algorithm with logarithmic regret that is not based on Cover's Universal Portfolio algorithm and admits much faster implementation. Specifically Universal Portfolio enjoys optimal regret $\mathcal{O}(N\ln T)$ for $N$ financial instruments over $T$ rounds, but requires log-concave sampling and has a large polynomial running time. Our algorithm, on the other hand, ensures a slightly larger but still logarithmic regret of $\mathcal{O}(N^2(\ln T)^4)$, and is based on the well-studied Online Mirror Descent framework with a novel regularizer that can be implemented via standard optimization methods in time $\mathcal{O}(TN^{2.5})$ per round. The regret of all other existing works is either polynomial in $T$ or has a potentially unbounded factor such as the inverse of the smallest price relative.

Motivation & Objective

• To address the long-standing challenge of achieving low regret in online portfolio management with efficient computation.
• To develop an algorithm with provably logarithmic regret that avoids the computational bottlenecks of log-concave sampling in Cover's Universal Portfolio.
• To ensure the regret remains bounded and grows slowly with time, even in the presence of small price movements.
• To enable practical deployment by reducing running time from polynomial to near-linear in T per round.

Proposed method

• The algorithm is built on the Online Mirror Descent framework with a newly designed regularizer to control portfolio updates.
• The regularizer is constructed to be amenable to standard optimization techniques, enabling efficient computation.
• The method ensures regret scaling as $Ø(N^2(\ln T)^4)$, which is logarithmic in T and polynomial in N.
• The algorithm avoids reliance on log-concave sampling, a major bottleneck in previous optimal regret methods.
• The update rule is computed in $Ø(TN^{2.5})$ time per round, significantly faster than prior optimal regret algorithms.
• The framework ensures bounded regret even when the smallest asset price is very small, avoiding unbounded factors in the regret bound.

Experimental results

Research questions

• RQ1 Can an online portfolio algorithm achieve logarithmic regret without relying on log-concave sampling?
• RQ2 Is it possible to design a practical algorithm with sub-polynomial regret that runs in near-linear time per round?
• RQ3 Can the regret bound be made independent of the inverse of the smallest asset price?
• RQ4 How does the new regularizer in Online Mirror Descent affect the trade-off between regret and computational efficiency?

Key findings

• The proposed algorithm achieves a regret bound of $Ø(N^2(\ln T)^4)$, which is logarithmic in T and thus sub-polynomial.
• The algorithm avoids the need for log-concave sampling, a key computational bottleneck in Cover's Universal Portfolio.
• The running time per round is $Ø(TN^{2.5})$, making it significantly faster than prior optimal regret algorithms with polynomial time complexity.
• The regret bound remains bounded even when the smallest asset price approaches zero, eliminating unbounded factors present in other methods.
• The method is the first to achieve logarithmic regret via Online Mirror Descent without relying on Cover's algorithmic framework.