Efficient and Near-Optimal Online Portfolio Selection

Abstract

In the problem of online portfolio selection as formulated by Cover (1991) [12], the trader repeatedly distributes her capital over d assets in each of T > 1 rounds, with the goal of maximizing the total return. Cover proposed an algorithm, termed Universal Portfolios, that performs nearly as well as the best (in hindsight) static assignment of a portfolio, with an O(d log(T )) logarithmic regret. Without imposing any restrictions on the market this guarantee is known to be worst-case optimal, and no other algorithm attaining it has been discovered so far. Unfortunately, Cover's algorithm crucially relies on computing certain d-dimensional integral, which must be approximated in any implementation; this results in a prohibitive Õ(d 4 (T + d) 14 ) per-round runtime for the fastest known implementation due to Kalai and Vempala (2002) [22]. We propose an algorithm for online portfolio selection that admits essentially the same regret guarantee as Universal Portfolios-up to a constant factor and replacement of log(T ) with log(T + d)-yet has a drastically reduced runtime of Õ(d 2 (T + d)) per round. The selected portfolio minimizes the observed logarithmic loss regularized with the log-determinant of its Hessian-equivalently, the hybrid logarithmic-volumetric barrier of the polytope specified by the asset return vectors. As such, our work reveals surprising connections of online portfolio selection with two classical topics in optimization theory: cutting-plane and interior-point algorithms.