Georgia Institute of Technology College of Sciences

In the problem of online portfolio selection as formulated by Cover (1991), 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 called Universal Portfolios, that performs nearly as well as the best (in hindsight) static assignment of a portfolio, with 

an $ O(d\log(T)) $ regret in terms of the logarithmic return. 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 the expectation over certain log-concave density in R^d, so in a practical implementation this expectation has to be approximated via sampling, which is computationally challenging. In particular, the fastest known implementation, proposed by Kalai and Vempala in 2002, runs in $ O( d^4 (T+d)^{14} ) $ per round, which rules out any practical application scenario. Proposing a practical algorithm with a near-optimal regret is a long-standing open problem. We propose an algorithm for online portfolio selection with a near-optimal regret guarantee of $ O( d \log(T+d) ) $ and the runtime of only $ O( d^2 (T+d) ) $ per round. In a nutshell, our algorithm is a variant of the follow-the-regularized-leader scheme, with a time-dependent regularizer given by the volumetric barrier for the sum of observed losses. Thus, our result gives a fresh perspective on the concept of volumetric barrier, initially proposed in the context of cutting-plane methods and interior-point methods, correspondingly by Vaidya (1989) and Nesterov and Nemirovski (1994). Our side contribution, of independent interest, is deriving the volumetrically regularized portfolio as a variational approximation of the universal portfolio: namely, we show that it minimizes Gibbs's free energy functional, with accuracy of order $ O( d \log(T+d) ) $. This is a joint work with Remi Jezequel and Pierre Gaillard.