In this paper we consider the dynamic assortment selection problem under an uncapacitated multinomial-logit (MNL) model. By carefully analyzing a revenue potential function, we show that a trisection based algorithm achieves an item-independent regret bound of @math , which matches information theoretical lower bounds up to iterated logarithmic terms. Our proof technique draws tools from the unimodal convex bandit literature as well as adaptive confidence parameters in minimax multi-armed bandit problems.
Another relevant line of research is @cite_2 @cite_21 @cite_6 @cite_5 , in which discrete or continuous multi-armed bandit problems are considered with additional unimodality constraints on the means of the arms. Apart from unimodality, additional structures such as inverse Lipschitz continuity'' (e.g., @math for some constant @math , where @math denotes the mean reward of the @math -th arm) or convexity are imposed to ensure smaller regret compared to unstructured multi-armed bandits . However, both conditions fail to hold for the revenue potential function arising from uncapacitated MNL-based assortment planning problems. In addition, under the gap-free setting where an @math regret is to be expected, most previous works have additional @math terms in their regret upper bounds (except for the work of @cite_5 which introduces additional strong regularity conditions on the underlying functions). In @cite_14 , a more general problem of optimizing piecewise-constant function is considered, without assuming a unimodal structure of the function. Consequently, a weaker @math regret is derived.
@cite_14 @cite_21 @cite_6 @cite_2 @cite_5
2757755218 2953359952 2567882221 2096382242
This work deals with the local rapid exponential stabilization for a Boussinesq system of KdV-KdV type introduced by J. Bona, M. Chen and J.-C. Saut. This is a model for the motion of small amplitude long waves on the surface of an ideal fluid. Here, we will consider the Boussinesq system of KdV-KdV type posed on a finite domain, with homogeneous Dirichlet--Neumann boundary controls acting at the right end point of the interval. Our goal is to build suitable integral transformations to get a feedback control law that leads to the stabilization of the system. More precisely, we will prove that the solution of the closed-loop system decays exponentially to zero in the @math --norm and the decay rate can be tuned to be as large as desired if the initial data is small enough. We consider stochastic multi-armed bandits where the expected reward is a unimodal function over partially ordered arms. This important class of problems has been recently investigated in (Cope 2009, Yu 2011). The set of arms is either discrete, in which case arms correspond to the vertices of a finite graph whose structure represents similarity in rewards, or continuous, in which case arms belong to a bounded interval. For discrete unimodal bandits, we derive asymptotic lower bounds for the regret achieved under any algorithm, and propose OSUB, an algorithm whose regret matches this lower bound. Our algorithm optimally exploits the unimodal structure of the problem, and surprisingly, its asymptotic regret does not depend on the number of arms. We also provide a regret upper bound for OSUB in non-stationary environments where the expected rewards smoothly evolve over time. The analytical results are supported by numerical experiments showing that OSUB performs significantly better than the state-of-the-art algorithms. For continuous sets of arms, we provide a brief discussion. We show that combining an appropriate discretization of the set of arms with the UCB algorithm yields an order-optimal regret, and in practice, outperforms recently proposed algorithms designed to exploit the unimodal structure. This paper addresses the problem of minimizing a convex, Lipschitz function @math over a convex, compact set @math under a stochastic bandit (i.e., noisy zeroth-order) feedback model. In this model, the algorithm is allowed to observe noisy realizations of the function value @math at any query point @math . The quantity of interest is the regret of the algorithm, which is the sum of the function values at algorithm's query points minus the optimal function value. We demonstrate a generalization of the ellipsoid algorithm that incurs @math regret. Since any algorithm has regret at least @math on this problem, our algorithm is optimal in terms of the scaling with @math . We consider a class of multi-armed bandit problems where the set of available actions can be mapped to a convex, compact region of Ropfd, sometimes denoted as the ldquocontinuum-armed banditrdquo problem. The paper establishes bounds on the efficiency of any arm-selection procedure under certain conditions on the class of possible underlying reward functions. Both finite-time lower bounds on the growth rate of the regret, as well as asymptotic upper bounds on the rates of convergence of the selected control values to the optimum are derived. We explicitly characterize the dependence of these convergence rates on the minimal rate of variation of the mean reward function in a neighborhood of the optimal control. The bounds can be used to demonstrate the asymptotic optimality of the Kiefer-Wolfowitz method of stochastic approximation with regard to a large class of possible mean reward functions.