Constrained Best Arm Identification in Grouped Bandits

We study a grouped bandit setting where each arm comprises multiple independent sub-arms referred to as attributes. Each attribute of each arm has an independent stochastic reward. We impose the constraint that for an arm to be deemed feasible, the mean reward of all its attributes should exceed a specified threshold. The goal is to find the arm with the highest mean reward averaged across attributes among the set of feasible arms in the fixed confidence setting. We first characterize a fundamental limit on the performance of any policy. Following this, we propose a near-optimal confidence interval-based policy to solve this problem and provide analytical guarantees for the policy. We compare the performance of the proposed policy with that of two suitably modified versions of action elimination via simulations.

Influencing Bandits: Arm Selection for Preference Shaping

We consider a non stationary multi-armed bandit in which the population preferences are positively and negatively reinforced by the observed rewards. The objective of the algorithm is to shape the population preferences to maximize the fraction of the population favouring a predetermined arm. For the case of binary opinions, two types of opinion dynamics are considered -- decreasing elasticity (modeled as a Polya urn with increasing number of balls) and constant elasticity (using the voter model). For the first case, we describe an Explore-then-commit policy and a Thompson sampling policy and analyse the regret for each of these policies. We then show that these algorithms and their analyses carry over to the constant elasticity case. We also describe a Thompson sampling based algorithm for the case when more than two types of opinions are present. Finally, we discuss the case where presence of multiple recommendation systems gives rise to a trade-off between their popularity and opinion shaping objectives.

On the Regret of Online Edge Service Hosting

We consider the problem of service hosting where a service provider can dynamically rent edge resources via short term contracts to ensure better quality of service to its customers. The service can also be partially hosted at the edge, in which case, customers' requests can be partially served at the edge. The total cost incurred by the system is modeled as a combination of the rent cost, the service cost incurred due to latency in serving customers, and the fetch cost incurred as a result of the bandwidth used to fetch the code/databases of the service from the cloud servers to host the service at the edge. In this paper, we compare multiple hosting policies with regret as a metric, defined as the difference in the cost incurred by the policy and the optimal policy over some time horizon $T$. In particular we consider the Retro Renting (RR) and Follow The Perturbed Leader (FTPL) policies proposed in the literature and provide performance guarantees on the regret of these policies. We show that under i.i.d stochastic arrivals, RR policy has linear regret while FTPL policy has constant regret. Next, we propose a variant of FTPL, namely Wait then FTPL (W-FTPL), which also has constant regret while demonstrating much better dependence on the fetch cost. We also show that under adversarial arrivals, RR policy has linear regret while both FTPL and W-FTPL have regret $\mathrm{O}(\sqrt{T})$ which is order-optimal.

Multi-Model Federated Learning with Provable Guarantees

Federated Learning (FL) is a variant of distributed learning where edge devices collaborate to learn a model without sharing their data with the central server or each other. We refer to the process of training multiple independent models simultaneously in a federated setting using a common pool of clients as multi-model FL. In this work, we propose two variants of the popular FedAvg algorithm for multi-model FL, with provable convergence guarantees. We further show that for the same amount of computation, multi-model FL can have better performance than training each model separately. We supplement our theoretical results with experiments in strongly convex, convex, and non-convex settings.

Multi-Model Federated Learning

Federated learning is a form of distributed learning with the key challenge being the non-identically distributed nature of the data in the participating clients. In this paper, we extend federated learning to the setting where multiple unrelated models are trained simultaneously. Specifically, every client is able to train any one of M models at a time and the server maintains a model for each of the M models which is typically a suitably averaged version of the model computed by the clients. We propose multiple policies for assigning learning tasks to clients over time. In the first policy, we extend the widely studied FedAvg to multi-model learning by allotting models to clients in an i.i.d. stochastic manner. In addition, we propose two new policies for client selection in a multi-model federated setting which make decisions based on current local losses for each client-model pair. We compare the performance of the policies on tasks involving synthetic and real-world data and characterize the performance of the proposed policies. The key take-away from our work is that the proposed multi-model policies perform better or at least as good as single model training using FedAvg.

Greedy $k$-Center from Noisy Distance Samples

We study a variant of the canonical $k$-center problem over a set of vertices in a metric space, where the underlying distances are apriori unknown. Instead, we can query an oracle which provides noisy/incomplete estimates of the distance between any pair of vertices. We consider two oracle models: Dimension Sampling where each query to the oracle returns the distance between a pair of points in one dimension; and Noisy Distance Sampling where the oracle returns the true distance corrupted by noise. We propose active algorithms, based on ideas such as UCB and Thompson sampling developed in the closely related Multi-Armed Bandit problem, which adaptively decide which queries to send to the oracle and are able to solve the $k$-center problem within an approximation ratio of two with high probability. We analytically characterize instance-dependent query complexity of our algorithms and also demonstrate significant improvements over naive implementations via numerical evaluations on two real-world datasets (Tiny ImageNet and UT Zappos50K).

Dynamic social learning under graph constraints

We argue that graph-constrained dynamic choice with reinforcement can be viewed as a scaled version of a special instance of replicator dynamics. The latter also arises as the limiting differential equation for the empirical measures of a vertex reinforced random walk on a directed graph. We use this equivalence to show that for a class of positively $\alpha$-homogeneous rewards, $\alpha > 0$, the asymptotic outcome concentrates around the optimum in a certain limiting sense when `annealed' by letting $\alpha\uparrow\infty$ slowly. We also discuss connections with classical simulated annealing.

Contextual Bandits Evolving Over Finite Time

Contextual bandits have the same exploration-exploitation trade-off as standard multi-armed bandits. On adding positive externalities that decay with time, this problem becomes much more difficult as wrong decisions at the start are hard to recover from. We explore existing policies in this setting and highlight their biases towards the inherent reward matrix. We propose a rejection based policy that achieves a low regret irrespective of the structure of the reward probability matrix.