Functional Bandits

We introduce the functional bandit problem, where the objective is to find an arm that optimises a known functional of the unknown arm-reward distributions. These problems arise in many settings such as maximum entropy methods in natural language processing, and risk-averse decision-making, but current best-arm identification techniques fail in these domains. We propose a new approach, that combines functional estimation and arm elimination, to tackle this problem. This method achieves provably efficient performance guarantees. In addition, we illustrate this method on a number of important functionals in risk management and information theory, and refine our generic theoretical results in those cases

Exploration vs Exploitation vs Safety: Risk-averse Multi-Armed Bandits

Motivated by applications in energy management, this paper presents the Multi-Armed Risk-Aware Bandit (MARAB) algorithm. With the goal of limiting the exploration of risky arms, MARAB takes as arm quality its conditional value at risk. When the user-supplied risk level goes to 0, the arm quality tends toward the essential infimum of the arm distribution density, and MARAB tends toward the MIN multi-armed bandit algorithm, aimed at the arm with maximal minimal value. As a first contribution, this paper presents a theoretical analysis of the MIN algorithm under mild assumptions, establishing its robustness comparatively to UCB. The analysis is supported by extensive experimental validation of MIN and MARAB compared to UCB and state-of-art risk-aware MAB algorithms on artificial and real-world problems.Comment: 16 page

Counterfactual Risk Minimization: Learning from Logged Bandit Feedback

We develop a learning principle and an efficient algorithm for batch learning from logged bandit feedback. This learning setting is ubiquitous in online systems (e.g., ad placement, web search, recommendation), where an algorithm makes a prediction (e.g., ad ranking) for a given input (e.g., query) and observes bandit feedback (e.g., user clicks on presented ads). We first address the counterfactual nature of the learning problem through propensity scoring. Next, we prove generalization error bounds that account for the variance of the propensity-weighted empirical risk estimator. These constructive bounds give rise to the Counterfactual Risk Minimization (CRM) principle. We show how CRM can be used to derive a new learning method -- called Policy Optimizer for Exponential Models (POEM) -- for learning stochastic linear rules for structured output prediction. We present a decomposition of the POEM objective that enables efficient stochastic gradient optimization. POEM is evaluated on several multi-label classification problems showing substantially improved robustness and generalization performance compared to the state-of-the-art.Comment: 10 page

Risk-averse multi-armed bandits and game theory

The multi-armed bandit (MAB) and game theory literature is mainly focused on the expected cumulative reward and the expected payoffs in a game, respectively. In contrast, the rewards and the payoffs are often random variables whose expected values only capture a vague idea of the overall distribution. The focus of this dissertation is to study the fundamental limits of the existing bandits and game theory problems in a risk-averse framework and propose new ideas that address the shortcomings. The author believes that human beings are mostly risk-averse, so studying multi-armed bandits and game theory from the point of view of risk aversion, rather than expected reward/payoff, better captures reality. In this manner, a specific class of multi-armed bandits, called explore-then-commit bandits, and stochastic games are studied in this dissertation, which are based on the notion of Risk-Averse Best Action Decision with Incomplete Information (R-ABADI, Abadi is the maiden name of the author's mother). The goal of the classical multi-armed bandits is to exploit the arm with the maximum score defined as the expected value of the arm reward. Instead, we propose a new definition of score that is derived from the joint distribution of all arm rewards and captures the reward of an arm relative to those of all other arms. We use a similar idea for games and propose a risk-averse R-ABADI equilibrium in game theory that is possibly different from the Nash equilibrium. The payoff distributions are taken into account to derive the risk-averse equilibrium, while the expected payoffs are used to find the Nash equilibrium. The fundamental properties of games, e.g. pure and mixed risk-averse R-ABADI equilibrium and strict dominance, are studied in the new framework and the results are expanded to finite-time games. Furthermore, the stochastic congestion games are studied from a risk-averse perspective and three classes of equilibria are proposed for such games. It is shown by examples that the risk-averse behavior of travelers in a stochastic congestion game can improve the price of anarchy in Pigou and Braess networks. Furthermore, the Braess paradox does not occur to the extent proposed originally when travelers are risk-averse. We also study an online affinity scheduling problem with no prior knowledge of the task arrival rates and processing rates of different task types on different servers. We propose the Blind GB-PANDAS algorithm that utilizes an exploration-exploitation scheme to load balance incoming tasks on servers in an online fashion. We prove that Blind GB-PANDAS is throughput optimal, i.e. it stabilizes the system as long as the task arrival rates are inside the capacity region. The Blind GB-PANDAS algorithm is compared to FCFS, Max-Weight, and c-mu-rule algorithms in terms of average task completion time through simulations, where the same exploration-exploitation approach as Blind GB-PANDAS is used for Max-Weight and c-$\mu$-rule. The extensive simulations show that the Blind GB-PANDAS algorithm conspicuously outperforms the three other algorithms at high loads

Conditionally Risk-Averse Contextual Bandits

Contextual bandits with average-case statistical guarantees are inadequate in risk-averse situations because they might trade off degraded worst-case behaviour for better average performance. Designing a risk-averse contextual bandit is challenging because exploration is necessary but risk-aversion is sensitive to the entire distribution of rewards; nonetheless we exhibit the first risk-averse contextual bandit algorithm with an online regret guarantee. We conduct experiments from diverse scenarios where worst-case outcomes should be avoided, from dynamic pricing, inventory management, and self-tuning software; including a production exascale data processing system