First-order regret bounds for combinatorial semi-bandits

We consider the problem of online combinatorial optimization under semi-bandit feedback, where a learner has to repeatedly pick actions from a combinatorial decision set in order to minimize the total losses associated with its decisions. After making each decision, the learner observes the losses associated with its action, but not other losses. For this problem, there are several learning algorithms that guarantee that the learner's expected regret grows as $\widetilde{O}(\sqrt{T})$ with the number of rounds $T$. In this paper, we propose an algorithm that improves this scaling to $\widetilde{O}(\sqrt{{L_T^*}})$, where $L_T^*$ is the total loss of the best action. Our algorithm is among the first to achieve such guarantees in a partial-feedback scheme, and the first one to do so in a combinatorial setting.Comment: To appear at COLT 201

Adaptive Online Prediction by Following the Perturbed Leader

When applying aggregating strategies to Prediction with Expert Advice, the learning rate must be adaptively tuned. The natural choice of sqrt(complexity/current loss) renders the analysis of Weighted Majority derivatives quite complicated. In particular, for arbitrary weights there have been no results proven so far. The analysis of the alternative "Follow the Perturbed Leader" (FPL) algorithm from Kalai & Vempala (2003) (based on Hannan's algorithm) is easier. We derive loss bounds for adaptive learning rate and both finite expert classes with uniform weights and countable expert classes with arbitrary weights. For the former setup, our loss bounds match the best known results so far, while for the latter our results are new.Comment: 25 page

An efficient algorithm for learning with semi-bandit feedback

We consider the problem of online combinatorial optimization under semi-bandit feedback. The goal of the learner is to sequentially select its actions from a combinatorial decision set so as to minimize its cumulative loss. We propose a learning algorithm for this problem based on combining the Follow-the-Perturbed-Leader (FPL) prediction method with a novel loss estimation procedure called Geometric Resampling (GR). Contrary to previous solutions, the resulting algorithm can be efficiently implemented for any decision set where efficient offline combinatorial optimization is possible at all. Assuming that the elements of the decision set can be described with d-dimensional binary vectors with at most m non-zero entries, we show that the expected regret of our algorithm after T rounds is O(m sqrt(dT log d)). As a side result, we also improve the best known regret bounds for FPL in the full information setting to O(m^(3/2) sqrt(T log d)), gaining a factor of sqrt(d/m) over previous bounds for this algorithm.Comment: submitted to ALT 201

Joint Channel Selection and Power Control in Infrastructureless Wireless Networks: A Multi-Player Multi-Armed Bandit Framework

This paper deals with the problem of efficient resource allocation in dynamic infrastructureless wireless networks. Assuming a reactive interference-limited scenario, each transmitter is allowed to select one frequency channel (from a common pool) together with a power level at each transmission trial; hence, for all transmitters, not only the fading gain, but also the number of interfering transmissions and their transmit powers are varying over time. Due to the absence of a central controller and time-varying network characteristics, it is highly inefficient for transmitters to acquire global channel and network knowledge. Therefore a reasonable assumption is that transmitters have no knowledge of fading gains, interference, and network topology. Each transmitting node selfishly aims at maximizing its average reward (or minimizing its average cost), which is a function of the action of that specific transmitter as well as those of all other transmitters. This scenario is modeled as a multi-player multi-armed adversarial bandit game, in which multiple players receive an a priori unknown reward with an arbitrarily time-varying distribution by sequentially pulling an arm, selected from a known and finite set of arms. Since players do not know the arm with the highest average reward in advance, they attempt to minimize their so-called regret, determined by the set of players' actions, while attempting to achieve equilibrium in some sense. To this end, we design in this paper two joint power level and channel selection strategies. We prove that the gap between the average reward achieved by our approaches and that based on the best fixed strategy converges to zero asymptotically. Moreover, the empirical joint frequencies of the game converge to the set of correlated equilibria. We further characterize this set for two special cases of our designed game