Algorithms with Logarithmic or Sublinear Regret for Constrained Contextual Bandits

We study contextual bandits with budget and time constraints, referred to as constrained contextual bandits.The time and budget constraints significantly complicate the exploration and exploitation tradeoff because they introduce complex coupling among contexts over time.Such coupling effects make it difficult to obtain oracle solutions that assume known statistics of bandits. To gain insight, we first study unit-cost systems with known context distribution. When the expected rewards are known, we develop an approximation of the oracle, referred to Adaptive-Linear-Programming (ALP), which achieves near-optimality and only requires the ordering of expected rewards. With these highly desirable features, we then combine ALP with the upper-confidence-bound (UCB) method in the general case where the expected rewards are unknown {\it a priori}. We show that the proposed UCB-ALP algorithm achieves logarithmic regret except for certain boundary cases. Further, we design algorithms and obtain similar regret analysis results for more general systems with unknown context distribution and heterogeneous costs. To the best of our knowledge, this is the first work that shows how to achieve logarithmic regret in constrained contextual bandits. Moreover, this work also sheds light on the study of computationally efficient algorithms for general constrained contextual bandits.Comment: 36 pages, 4 figures; accepted by the 29th Annual Conference on Neural Information Processing Systems (NIPS), Montr\'eal, Canada, Dec. 201

Linear Contextual Bandits with Knapsacks

We consider the linear contextual bandit problem with resource consumption, in addition to reward generation. In each round, the outcome of pulling an arm is a reward as well as a vector of resource consumptions. The expected values of these outcomes depend linearly on the context of that arm. The budget/capacity constraints require that the total consumption doesn't exceed the budget for each resource. The objective is once again to maximize the total reward. This problem turns out to be a common generalization of classic linear contextual bandits (linContextual), bandits with knapsacks (BwK), and the online stochastic packing problem (OSPP). We present algorithms with near-optimal regret bounds for this problem. Our bounds compare favorably to results on the unstructured version of the problem where the relation between the contexts and the outcomes could be arbitrary, but the algorithm only competes against a fixed set of policies accessible through an optimization oracle. We combine techniques from the work on linContextual, BwK, and OSPP in a nontrivial manner while also tackling new difficulties that are not present in any of these special cases

BISTRO: An Efficient Relaxation-Based Method for Contextual Bandits

We present efficient algorithms for the problem of contextual bandits with i.i.d. covariates, an arbitrary sequence of rewards, and an arbitrary class of policies. Our algorithm BISTRO requires d calls to the empirical risk minimization (ERM) oracle per round, where d is the number of actions. The method uses unlabeled data to make the problem computationally simple. When the ERM problem itself is computationally hard, we extend the approach by employing multiplicative approximation algorithms for the ERM. The integrality gap of the relaxation only enters in the regret bound rather than the benchmark. Finally, we show that the adversarial version of the contextual bandit problem is learnable (and efficient) whenever the full-information supervised online learning problem has a non-trivial regret guarantee (and efficient)

An efficient algorithm for contextual bandits with knapsacks, and an extension to concave objectives

We consider a contextual version of multi-armed bandit problem with global knapsack constraints. In each round, the outcome of pulling an arm is a scalar reward and a resource consumption vector, both dependent on the context, and the global knapsack constraints require the total consumption for each resource to be below some pre-fixed budget. The learning agent competes with an arbitrary set of context-dependent policies. This problem was introduced by Badanidiyuru et al. (2014), who gave a computationally inefficient algorithm with near-optimal regret bounds for it. We give a computationally efficient algorithm for this problem with slightly better regret bounds, by generalizing the approach of Agarwal et al. (2014) for the non-constrained version of the problem. The computational time of our algorithm scales logarithmically in the size of the policy space. This answers the main open question of Badanidiyuru et al. (2014). We also extend our results to a variant where there are no knapsack constraints but the objective is an arbitrary Lipschitz concave function of the sum of outcome vectors.Comment: Extended abstract appeared in COLT 201

Resourceful Contextual Bandits

We study contextual bandits with ancillary constraints on resources, which are common in real-world applications such as choosing ads or dynamic pricing of items. We design the first algorithm for solving these problems that handles constrained resources other than time, and improves over a trivial reduction to the non-contextual case. We consider very general settings for both contextual bandits (arbitrary policy sets, e.g. Dudik et al. (UAI'11)) and bandits with resource constraints (bandits with knapsacks, Badanidiyuru et al. (FOCS'13)), and prove a regret guarantee with near-optimal statistical properties.Comment: This is the full version of a paper in COLT 2014. Version history: (v2) Added some details to one of the proofs, (v3) a big revision following comments from COLT reviewers (but no new results), (v4) edits in related work, minor edits elsewhere. (v6) A correction for Theorem 3, corollary for contextual dynamic pricing with discretization; updated follow-up work & open question

Combinatorial Semi-Bandits with Knapsacks

We unify two prominent lines of work on multi-armed bandits: bandits with knapsacks (BwK) and combinatorial semi-bandits. The former concerns limited "resources" consumed by the algorithm, e.g., limited supply in dynamic pricing. The latter allows a huge number of actions but assumes combinatorial structure and additional feedback to make the problem tractable. We define a common generalization, support it with several motivating examples, and design an algorithm for it. Our regret bounds are comparable with those for BwK and combinatorial semi- bandits

Exploration-Exploitation Trade-off in Reinforcement Learning on Online Markov Decision Processes with Global Concave Rewards

We consider an agent who is involved in a Markov decision process and receives a vector of outcomes every round. Her objective is to maximize a global concave reward function on the average vectorial outcome. The problem models applications such as multi-objective optimization, maximum entropy exploration, and constrained optimization in Markovian environments. In our general setting where a stationary policy could have multiple recurrent classes, the agent faces a subtle yet consequential trade-off in alternating among different actions for balancing the vectorial outcomes. In particular, stationary policies are in general sub-optimal. We propose a no-regret algorithm based on online convex optimization (OCO) tools (Agrawal and Devanur 2014) and UCRL2 (Jaksch et al. 2010). Importantly, we introduce a novel gradient threshold procedure, which carefully controls the switches among actions to handle the subtle trade-off. By delaying the gradient updates, our procedure produces a non-stationary policy that diversifies the outcomes for optimizing the objective. The procedure is compatible with a variety of OCO tools.Comment: 54 pages, 1 figur

Personalized Advertisement Recommendation: A Ranking Approach to Address the Ubiquitous Click Sparsity Problem

We study the problem of personalized advertisement recommendation (PAR), which consist of a user visiting a system (website) and the system displaying one of $K$ ads to the user. The system uses an internal ad recommendation policy to map the user's profile (context) to one of the ads. The user either clicks or ignores the ad and correspondingly, the system updates its recommendation policy. PAR problem is usually tackled by scalable \emph{contextual bandit} algorithms, where the policies are generally based on classifiers. A practical problem in PAR is extreme click sparsity, due to very few users actually clicking on ads. We systematically study the drawback of using contextual bandit algorithms based on classifier-based policies, in face of extreme click sparsity. We then suggest an alternate policy, based on rankers, learnt by optimizing the Area Under the Curve (AUC) ranking loss, which can significantly alleviate the problem of click sparsity. We conduct extensive experiments on public datasets, as well as three industry proprietary datasets, to illustrate the improvement in click-through-rate (CTR) obtained by using the ranker-based policy over classifier-based policies.Comment: Under revie

Deep Neural Linear Bandits: Overcoming Catastrophic Forgetting through Likelihood Matching

We study the neural-linear bandit model for solving sequential decision-making problems with high dimensional side information. Neural-linear bandits leverage the representation power of deep neural networks and combine it with efficient exploration mechanisms, designed for linear contextual bandits, on top of the last hidden layer. Since the representation is being optimized during learning, information regarding exploration with "old" features is lost. Here, we propose the first limited memory neural-linear bandit that is resilient to this phenomenon, which we term catastrophic forgetting. We evaluate our method on a variety of real-world data sets, including regression, classification, and sentiment analysis, and observe that our algorithm is resilient to catastrophic forgetting and achieves superior performance

ADARES: Adaptive Resource Management for Virtual Machines

Virtual execution environments allow for consolidation of multiple applications onto the same physical server, thereby enabling more efficient use of server resources. However, users often statically configure the resources of virtual machines through guesswork, resulting in either insufficient resource allocations that hinder VM performance, or excessive allocations that waste precious data center resources. In this paper, we first characterize real-world resource allocation and utilization of VMs through the analysis of an extensive dataset, consisting of more than 250k VMs from over 3.6k private enterprise clusters. Our large-scale analysis confirms that VMs are often misconfigured, either overprovisioned or underprovisioned, and that this problem is pervasive across a wide range of private clusters. We then propose ADARES, an adaptive system that dynamically adjusts VM resources using machine learning techniques. In particular, ADARES leverages the contextual bandits framework to effectively manage the adaptations. Our system exploits easily collectible data, at the cluster, node, and VM levels, to make more sensible allocation decisions, and uses transfer learning to safely explore the configurations space and speed up training. Our empirical evaluation shows that ADARES can significantly improve system utilization without sacrificing performance. For instance, when compared to threshold and prediction-based baselines, it achieves more predictable VM-level performance and also reduces the amount of virtual CPUs and memory provisioned by up to 35% and 60% respectively for synthetic workloads on real clusters