Exponential Weights on the Hypercube in Polynomial Time

We study a general online linear optimization problem(OLO). At each round, a subset of objects from a fixed universe of $n$ objects is chosen, and a linear cost associated with the chosen subset is incurred. We use regret as a measure of performance of our algorithms. Regret is the difference between the total cost incurred over all iterations and the cost of the best fixed subset in hindsight. We consider Full Information, Semi-Bandit and Bandit feedback for this problem. Using characteristic vectors of the subsets, this problem reduces to OLO on the $\{0,1\}^n$ hypercube. The Exp2 algorithm and its bandit variants are commonly used strategies for this problem. It was previously unknown if it is possible to run Exp2 on the hypercube in polynomial time. In this paper, we present a polynomial time algorithm called PolyExp for OLO on the hypercube. We show that our algorithm is equivalent to both Exp2 on $\{0,1\}^n$ as well as Online Mirror Descent(OMD) with Entropic regularization on $[0,1]^n$ and Bernoulli Sampling. We consider $L_\infty$ adversarial losses. We show PolyExp achieves expected regret bounds that are a factor of $\sqrt{n}$ better than Exp2 in all the three settings. Because of the equivalence of these algorithms, this implies an improvement on Exp2's regret bounds. Moreover, PolyExp's regret bounds match the $L_\infty$ lowerbounds in Audibert et al. (2011). Finally, we show how to use PolyExp on the $\{-1,+1\}^n$ hypercube, solving an open problem in Bubeck et al. (2012).