Neural Information Processing Systems http://nips.cc/

In most applications there is a tantalising similarity to the classical analysis based on mirror descent.

We consider the problem of asynchronous online combinatorial optimization on a network of communicating agents. At each time step, some of the agents are stochastically activated, requested to make a prediction, and the system pays the corresponding loss. Then, neighbors of active agents receive semi-bandit feedback and exchange some succinct local information. The goal is to minimize the network regret, defined as the difference between the cumulative loss of the predictions of active agents and that of the best action in hindsight, selected from a combinatorial decision set. The main challenge in such a context is to control the computational complexity of the resulting algorithm while retaining minimax optimal regret guarantees. We introduce Coop-FTPL, a cooperative version of the well-known Follow The Perturbed Leader algorithm, that implements a new loss estimation procedure generalizing the Geometric Resampling of Neu and Bart\'ok [2013] to our setting. Assuming that the elements of the decision set are $k$-dimensional binary vectors with at most $m$ non-zero entries and $\alpha_1$ is the independence number of the network, we show that the expected regret of our algorithm after $T$ time steps is of order $Q\sqrt{mkT\log(k) (k\alpha_1/Q+m)}$, where $Q$ is the total activation probability mass. Furthermore, we prove that this is only $\sqrt{k\log k}$-away from the best achievable rate and that \coopftpl{} has a state-of-the-art $T^{3/2}$ worst-case computational complexity.

The information-theoretic analysis by Russo and Van Roy (2014) in combination with minimax duality has proved a powerful tool for the analysis of online learning algorithms in full and partial information settings. In most applications there is a tantalising similarity to the classical analysis based on mirror descent. We make a formal connection, showing that the information-theoretic bounds in most applications can be derived from existing techniques for online convex optimisation. Besides this, for $k$-armed adversarial bandits we provide an efficient algorithm with regret that matches the best information-theoretic upper bound and improve best known regret guarantees for online linear optimisation on $\ell_p$-balls and bandits with graph feedback.

We address online linear optimization problems when the possible actions of the decision maker are represented by binary vectors. The regret of the decision maker is the difference between her realized loss and the best loss she would have achieved by picking, in hindsight, the best possible action. Our goal is to understand the magnitude of the best possible (minimax) regret. We study the problem under three different assumptions for the feedback the decision maker receives: full information, and the partial information models of the so-called "semi-bandit" and "bandit" problems. Combining the Mirror Descent algorithm and the INF (Implicitely Normalized Forecaster) strategy, we are able to prove optimal bounds for the semi-bandit case. We also recover the optimal bounds for the full information setting. In the bandit case we discuss existing results in light of a new lower bound, and suggest a conjecture on the optimal regret in that case. Finally we also prove that the standard exponentially weighted average forecaster is provably suboptimal in the setting of online combinatorial optimization.