From Bandits to Experts: A Tale of Domination and Independence – We consider the partial observability model for multi-armed bandits, introduced by Mannor and Shamir (2011). Our main result is a characterization of regret in the directed observability model in terms of the dominating and independence numbers of the observability graph. We also show that in the undirected case, the learner can achieve optimal regret without even accessing the observability graph before selecting an action. Both results are shown using variants of the Exp3 algorithm operating on the observability graph in a time-efficient manner.
Eluder Dimension and the Sample Complexity of Optimistic Exploration This paper considers the sample complexity of the multi-armed bandit with dependencies among the arms. Some of the most successful algorithms for this problem use the principle of optimism in the face of uncertainty to guide exploration. The clearest example of this is the class of upper confidence bound (UCB) algorithms, but recent work has shown that a simple posterior sampling algorithm, sometimes called Thompson sampling, also shares a close theoretical connection with optimistic approaches. In this paper, we develop a regret bound that holds for both classes of algorithms. This bound applies broadly and can be specialized to many model classes. It depends on a new notion we refer to as the eluder dimension, which measures the degree of dependence among action rewards. Compared to UCB algorithm regret bounds for specific model classes, our general bound matches the best available for linear models and is stronger than the best available for generalized linear models.
Adaptive Market Making via Online Learning – We consider the design of strategies for market making in a market like a stock, commodity, or currency exchange. In order to obtain profit guarantees for a market maker one typically requires very particular stochastic assumptions on the sequence of price fluctuations of the asset in question. We propose a class of spread-based market making strategies whose performance can be controlled even under worst-case (adversarial) settings. We prove structural properties of these strategies which allows us to design a master algorithm which obtains low regret relative to the best such strategy in hindsight. We run a set of experiments showing favorable performance on real-world price data.
Submodular Optimization with Submodular Cover and Submodular Knapsack Constraints – We investigate two new optimization problems – minimizing a submodular function subject to a submodular lower bound constraint (submodular cover) and maximizing a submodular function subject to a submodular upper bound constraint (submodular knapsack). We are motivated by a number of real-world applications in machine learning including sensor placement and data subset selection, which require maximizing a certain submodular function (like coverage or diversity) while simultaneously minimizing another (like cooperative cost). These problems are often posed as minimizing the difference between submodular functions [9, 23] which is in the worst case inapproximable. We show, however, that by phrasing these problems as constrained optimization, which is more natural for many applications, we achieve a number of bounded approximation guarantees. We also show that both these problems are closely related and, an approximation algorithm solving one can be used to obtain an approximation guarantee for the other. We provide hardness results for both problems thus showing that our approximation factors are tight up to log-factors. Finally, we empirically demonstrate the performance and good scalability properties of our algorithms.