Efficient Online Portfolio with Logarithmic Regret
This work addresses the decades-old challenge of efficient portfolio optimization for investors, offering a novel method that improves computational speed while maintaining near-optimal regret, though it is incremental relative to existing logarithmic regret algorithms.
The paper tackles the problem of online portfolio management by proposing the first algorithm with logarithmic regret that is not based on Cover's Universal Portfolio, achieving a regret of O(N^2(ln T)^4) and enabling faster implementation with O(TN^{2.5}) time per round.
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.