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### Regret Analysis for Continuous Dueling Bandit

The dueling bandit is a learning framework wherein the feedback information in the learning process is restricted to a noisy comparison between a pair of actions. In this research, we address a dueling bandit problem based on a cost function over a continuous space. We propose a stochastic mirror descent algorithm and show that the algorithm achieves an $O(\sqrt{T\log T})$-regret bound under strong convexity and smoothness assumptions for the cost function. Subsequently, we clarify the equivalence between regret minimization in dueling bandit and convex optimization for the cost function. Moreover, when considering a lower bound in convex optimization, our algorithm is shown to achieve the optimal convergence rate in convex optimization and the optimal regret in dueling bandit except for a logarithmic factor.

### Regret Analysis for Continuous Dueling Bandit

The dueling bandit is a learning framework wherein the feedback information in the learning process is restricted to a noisy comparison between a pair of actions. In this research, we address a dueling bandit problem based on a cost function over a continuous space. We propose a stochastic mirror descent algorithm and show that the algorithm achieves an O( T log T)-regret bound under strong convexity and smoothness assumptions for the cost function. Subsequently, we clarify the equivalence between regret minimization in dueling bandit and convex optimization forthe cost function. Moreover, when considering a lower bound in convex optimization, our algorithm is shown to achieve the optimal convergence rate in convex optimization and the optimal regret in dueling bandit except for a logarithmic factor.

### Taming the Monster: A Fast and Simple Algorithm for Contextualized Bandits • /r/MachineLearning

Abstract: We present a new algorithm for the contextual bandit learning problem, where the learner repeatedly takes one of K actions in response to the observed context, and observes the reward only for that chosen action. Our method assumes access to an oracle for solving fully supervised cost-sensitive classification problems and achieves the statistically optimal regret guarantee with only O ( p KT / log N) oracle calls across all T rounds, where N is the number of policies in the policy class we compete against. By doing so, we obtain the most practical contextual bandit learning algorithm amongst approaches that work for general policy classes. We further conduct a proof-of-concept experiment which demonstrates the excellent computational and prediction performance of (an online variant of) our algorithm relative to several baselines.

### Corralling a Band of Bandit Algorithms

We study the problem of combining multiple bandit algorithms (that is, online learning algorithms with partial feedback) with the goal of creating a master algorithm that performs almost as well as the best base algorithm if it were to be run on its own. The main challenge is that when run with a master, base algorithms unavoidably receive much less feedback and it is thus critical that the master not starve a base algorithm that might perform uncompetitively initially but would eventually outperform others if given enough feedback. We address this difficulty by devising a version of Online Mirror Descent with a special mirror map together with a sophisticated learning rate scheme. We show that this approach manages to achieve a more delicate balance between exploiting and exploring base algorithms than previous works yielding superior regret bounds. Our results are applicable to many settings, such as multi-armed bandits, contextual bandits, and convex bandits. As examples, we present two main applications. The first is to create an algorithm that enjoys worst-case robustness while at the same time performing much better when the environment is relatively easy. The second is to create an algorithm that works simultaneously under different assumptions of the environment, such as different priors or different loss structures.