We consider online content recommendation with implicit feedback through pairwise comparisons, formalized as the so-called dueling bandit problem. We study the dueling bandit problem in the Condorcet winner setting, and consider two notions of regret: the more well-studied strong regret, which is 0 only when both arms pulled are the Condorcet winner; and the less well-studied weak regret, which is 0 if either arm pulled is the Condorcet winner. We propose a new algorithm for this problem, Winner Stays (WS), with variations for each kind of regret: WS for weak regret (WS-W) has expected cumulative weak regret that is @math , and @math if arms have a total order; WS for strong regret (WS-S) has expected cumulative strong regret of @math , and @math if arms have a total order. WS-W is the first dueling bandit algorithm with weak regret that is constant in time. WS is simple to compute, even for problems with many arms, and we demonstrate through numerical experiments on simulated and real data that WS has significantly smaller regret than existing algorithms in both the weak- and strong-regret settings.
Most work on dueling bandits focuses on strong regret. shows that the worst-case expected cumulative strong regret up to time T for any algorithm is @math . Algorithms have been proposed that reach this lower bound under the Condorcet winner assumption in the finite-horizon setting: Interleaved Filter (IF) @cite_8 and Beat the Mean (BTM) @cite_13 . Relative Upper Confidence Bound (RUCB) @cite_0 also reaches this lower bound in the horizonless setting. Relative Minimum Empirical Divergence (RMED) @cite_12 is the first algorithm to have a regret bound that matches this lower bound. proposed two algorithms, Copeland Confidence Bound (CCB) and Scalable Copeland Bandits (SCB), which achieve an optimal regret bound without assuming existence of a Condorcet winner.
@cite_0 @cite_13 @cite_12 @cite_8
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We study the K-armed dueling bandit problem, a variation of the standard stochastic bandit problem where the feedback is limited to relative comparisons of a pair of arms. We introduce a tight asymptotic regret lower bound that is based on the information divergence. An algorithm that is inspired by the Deterministic Minimum Empirical Divergence algorithm (Honda and Takemura, 2010) is proposed, and its regret is analyzed. The proposed algorithm is found to be the first one with a regret upper bound that matches the lower bound. Experimental comparisons of dueling bandit algorithms show that the proposed algorithm significantly outperforms existing ones. We present algorithms for reducing the Dueling Bandits problem to the conventional (stochastic) Multi-Armed Bandits problem. The Dueling Bandits problem is an online model of learning with ordinal feedback of the form "A is preferred to B" (as opposed to cardinal feedback like "A has value 2.5"), giving it wide applicability in learning from implicit user feedback and revealed and stated preferences. In contrast to existing algorithms for the Dueling Bandits problem, our reductions -- named @math , @math and @math -- provide a generic schema for translating the extensive body of known results about conventional Multi-Armed Bandit algorithms to the Dueling Bandits setting. For @math and @math we prove regret upper bounds in both finite and infinite settings, and conjecture about the performance of @math which empirically outperforms the other two as well as previous algorithms in our experiments. In addition, we provide the first almost optimal regret bound in terms of second order terms, such as the differences between the values of the arms. This paper proposes a new method for the K-armed dueling bandit problem, a variation on the regular K-armed bandit problem that offers only relative feedback about pairs of arms. Our approach extends the Upper Confidence Bound algorithm to the relative setting by using estimates of the pairwise probabilities to select a promising arm and applying Upper Confidence Bound with the winner as a benchmark. We prove a sharp finite-time regret bound of order O(K log T) on a very general class of dueling bandit problems that matches a lower bound proven in (, 2012). In addition, our empirical results using real data from an information retrieval application show that it greatly outperforms the state of the art. This work deals with four classical prediction settings, namely full information, bandit, label efficient and bandit label efficient as well as four different notions of regret: pseudo-regret, expected regret, high probability regret and tracking the best expert regret. We introduce a new forecaster, INF (Implicitly Normalized Forecaster) based on an arbitrary function ψ for which we propose a unified analysis of its pseudo-regret in the four games we consider. In particular, for ψ(x)=exp(η x) + γ K, INF reduces to the classical exponentially weighted average forecaster and our analysis of the pseudo-regret recovers known results while for the expected regret we slightly tighten the bounds. On the other hand with ψ(x)=(η -x)q + γ K, which defines a new forecaster, we are able to remove the extraneous logarithmic factor in the pseudo-regret bounds for bandits games, and thus fill in a long open gap in the characterization of the minimax rate for the pseudo-regret in the bandit game. We also provide high probability bounds depending on the cumulative reward of the optimal action. Finally, we consider the stochastic bandit game, and prove that an appropriate modification of the upper confidence bound policy UCB1 (, 2002a) achieves the distribution-free optimal rate while still having a distribution-dependent rate logarithmic in the number of plays.