Efficient and Near-Optimal Online Portfolio Selection

In the problem of online portfolio selection as formulated by Cover [Cover TM (1991) Universal portfolios. Math. Finance 1(1):1-29], the trader repeatedly distributes the trader's 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 a certain d-dimensional integral, which must be approximated in any implementation; this results in a prohibitive O(sic)(d(4)(T + d)(14)) per-round runtime for the fastest known implementation. We propose an algorithm for online portfolio selection that satisfies 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 O(sic)(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 classic topics in optimization theory: cutting-plane and interior-point algorithms.