We consider the linear contextual multi-class multi-period packing problem (LMMP) where the goal is to pack items such that the total vector of consumption is below a given budget vector and the total value is as large as possible. We consider the setting where the reward and the consumption vector associated with each action is a class-dependent linear function of the context, and the decision-maker receives bandit feedback. LMMP includes linear contextual bandits with knapsacks and online revenue management as special cases. We establish a new estimator which guarantees a faster convergence rate, and consequently, a lower regret in such problems. We propose a bandit policy that is a closed-form function of said estimated parameters. When the contexts are non-degenerate, the regret of the proposed policy is sublinear in the context dimension, the number of classes, and the time horizon $T$ when the budget grows at least as $\sqrt{T}$. We also resolve an open problem posed by Agrawal & Devanur (2016) and extend the result to a multi-class setting. Our numerical experiments clearly demonstrate that the performance of our policy is superior to other benchmarks in the literature.

While Bayesian Optimization (BO) is a very popular method for optimizing expensive black-box functions, it fails to leverage the experience of domain experts. This causes BO to waste function evaluations on bad design choices (e.g., machine learning hyperparameters) that the expert already knows to work poorly. To address this issue, we introduce Prior-guided Bayesian Optimization (PrBO). PrBO allows users to inject their knowledge into the optimization process in the form of priors about which parts of the input space will yield the best performance, rather than BO's standard priors over functions (which are much less intuitive for users). PrBO then combines these priors with BO's standard probabilistic model to form a pseudo-posterior used to select which points to evaluate next. We show that PrBO is around 12x faster than state-of-the-art methods without user priors and 10,000x faster than random search on a common suite of benchmarks, and achieves a new state-of-the-art performance on a real-world hardware design application. We also show that PrBO converges faster even if the user priors are not entirely accurate and that it robustly recovers from misleading priors.

High-dimensional representations often have a lower dimensional underlying structure. This is particularly the case in many decision making settings. For example, when the representation of actions is generated from a deep neural network, it is reasonable to expect a low-rank structure whereas conventional structures like sparsity are not valid anymore. Subspace recovery methods, such as Principle Component Analysis (PCA) can find the underlying low-rank structures in the feature space and reduce the complexity of the learning tasks. In this work, we propose Projected Stochastic Linear Bandit (PSLB), an algorithm for high dimensional stochastic linear bandits (SLB) when the representation of actions has an underlying low-dimensional subspace structure. PSLB deploys PCA based projection to iteratively find the low rank structure in SLBs. We show that deploying projection methods assures dimensionality reduction and results in a tighter regret upper bound that is in terms of the dimensionality of the subspace and its properties, rather than the dimensionality of the ambient space. We modify the image classification task into the SLB setting and empirically show that, when a pre-trained DNN provides the high dimensional feature representations, deploying PSLB results in significant reduction of regret and faster convergence to an accurate model compared to state-of-art algorithm.

Abstract--We study stochastic multi-armed bandits with many players. The players do not know the number of players, cannot communicate with each other and if multiple players select a common arm they collide and none of them receive any reward. We consider the static scenario, where the number of players remains fixed, and the dynamic scenario, where the players enter and leave at any time. We provide algorithms based on a novel'trekking approach' that guarantees constant regret for the static case and sub-linear regret for the dynamic case with high probability. The trekking approach eliminates the need to estimate the number of players resulting in fewer collisions and improved regret performance compared to the state-of-theart algorithms. We also develop an epoch-less algorithm that eliminates any requirement of time synchronization across the players provided each player can detect the presence of other players on an arm. We validate our theoretical guarantees using simulation based and real test-bed based experiments. Multi-player multi-armed bandits (MPMAB) is a variant of the stochastic multi-armed bandits [1]-[3] where multiple players aim to maximize sum of their rewards playing the same set of arms. In this setting, the players do not communicate with each other and may not know number of other players in the game. If two or more players select the same arm simultaneously, they experience'collision' and none of them receive any reward. Our goal in this work is to develop distributed algorithms that aim to achieve high total rewards while keeping the number of collisions as low as possible. The study of MPMAB is mainly motivated from the ad-hoc cognitive radio networks (CRN) where multiple users transmit on a common set of channels (unlicensed spectrum) without any communication among them [4], [5]. Due to the ad hoc nature of such networks, a central controller, or a common control channels for coordination, may not be available and all channel selection decisions have to be done in a decentralized fashion [4]-[6]. Such models are being envisioned for futuristic ultra-dense wireless communication networks that can offer very high peak rates [7]. The quality of the channels are unknown to the users and their goal is to maximize number of successful transmissions (or sum rate/ throughput) in the network.