Meta-Learning for Simple Regret Minimization

We develop a meta-learning framework for simple regret minimization in bandits. In this framework, a learning agent interacts with a sequence of bandit tasks, which are sampled i.i.d.\ from an unknown prior distribution, and learns its meta-parameters to perform better on future tasks. We propose the first Bayesian and frequentist algorithms for this meta-learning problem. The Bayesian algorithm has access to a prior distribution over the meta-parameters and its meta simple regret over $m$ bandit tasks with horizon $n$ is mere $\tilde{O}(m / \sqrt{n})$. This is while we show that the meta simple regret of the frequentist algorithm is $\tilde{O}(\sqrt{m} n + m/ \sqrt{n})$, and thus, worse. However, the algorithm is more general, because it does not need a prior distribution over the meta-parameters, and is easier to implement for various distributions. We instantiate our algorithms for several classes of bandit problems. Our algorithms are general and we complement our theory by evaluating them empirically in several environments.

Deep Hierarchy in Bandits

Mean rewards of actions are often correlated. The form of these correlations may be complex and unknown a priori, such as the preferences of a user for recommended products and their categories. To maximize statistical efficiency, it is important to leverage these correlations when learning. We formulate a bandit variant of this problem where the correlations of mean action rewards are represented by a hierarchical Bayesian model with latent variables. Since the hierarchy can have multiple layers, we call it deep. We propose a hierarchical Thompson sampling algorithm (HierTS) for this problem, and show how to implement it efficiently for Gaussian hierarchies. The efficient implementation is possible due to a novel exact hierarchical representation of the posterior, which itself is of independent interest. We use this exact posterior to analyze the Bayes regret of HierTS in Gaussian bandits. Our analysis reflects the structure of the problem, that the regret decreases with the prior width, and also shows that hierarchies reduce the regret by non-constant factors in the number of actions. We confirm these theoretical findings empirically, in both synthetic and real-world experiments.

IMO$^3$: Interactive Multi-Objective Off-Policy Optimization

Most real-world optimization problems have multiple objectives. A system designer needs to find a policy that trades off these objectives to reach a desired operating point. This problem has been studied extensively in the setting of known objective functions. We consider a more practical but challenging setting of unknown objective functions. In industry, this problem is mostly approached with online A/B testing, which is often costly and inefficient. As an alternative, we propose interactive multi-objective off-policy optimization (IMO$^3$). The key idea in our approach is to interact with a system designer using policies evaluated in an off-policy fashion to uncover which policy maximizes her unknown utility function. We theoretically show that IMO$^3$ identifies a near-optimal policy with high probability, depending on the amount of feedback from the designer and training data for off-policy estimation. We demonstrate its effectiveness empirically on multiple multi-objective optimization problems.

Hierarchical Bayesian Bandits

Meta-, multi-task, and federated learning can be all viewed as solving similar tasks, drawn from an unknown distribution that reflects task similarities. In this work, we provide a unified view of all these problems, as learning to act in a hierarchical Bayesian bandit. We analyze a natural hierarchical Thompson sampling algorithm (hierTS) that can be applied to any problem in this class. Our regret bounds hold under many instances of such problems, including when the tasks are solved sequentially or in parallel; and capture the structure of the problems, such that the regret decreases with the width of the task prior. Our proofs rely on novel total variance decompositions, which can be applied to other graphical model structures. Finally, our theory is complemented by experiments, which show that the hierarchical structure helps with knowledge sharing among the tasks. This confirms that hierarchical Bayesian bandits are a universal and statistically-efficient tool for learning to act with similar bandit tasks.

Safe Optimal Design with Applications in Policy Learning

Motivated by practical needs in online experimentation and off-policy learning, we study the problem of safe optimal design, where we develop a data logging policy that efficiently explores while achieving competitive rewards with a baseline production policy. We first show, perhaps surprisingly, that a common practice of mixing the production policy with uniform exploration, despite being safe, is sub-optimal in maximizing information gain. Then we propose a safe optimal logging policy for the case when no side information about the actions' expected rewards is available. We improve upon this design by considering side information and also extend both approaches to a large number of actions with a linear reward model. We analyze how our data logging policies impact errors in off-policy learning. Finally, we empirically validate the benefit of our designs by conducting extensive experiments.

Optimal Probing with Statistical Guarantees for Network Monitoring at Scale

Cloud networks are difficult to monitor because they grow rapidly and the budgets for monitoring them are limited. We propose a framework for estimating network metrics, such as latency and packet loss, with guarantees on estimation errors for a fixed monitoring budget. Our proposed algorithms produce a distribution of probes across network paths, which we then monitor; and are based on A- and E-optimal experimental designs in statistics. Unfortunately, these designs are too computationally costly to use at production scale. We propose their scalable and near-optimal approximations based on the Frank-Wolfe algorithm. We validate our approaches in simulation on real network topologies, and also using a production probing system in a real cloud network. We show major gains in reducing the probing budget compared to both production and academic baselines, while maintaining low estimation errors, even with very low probing budgets.

No Regrets for Learning the Prior in Bandits

We propose ${\tt AdaTS}$, a Thompson sampling algorithm that adapts sequentially to bandit tasks that it interacts with. The key idea in ${\tt AdaTS}$ is to adapt to an unknown task prior distribution by maintaining a distribution over its parameters. When solving a bandit task, that uncertainty is marginalized out and properly accounted for. ${\tt AdaTS}$ is a fully-Bayesian algorithm that can be implemented efficiently in several classes of bandit problems. We derive upper bounds on its Bayes regret that quantify the loss due to not knowing the task prior, and show that it is small. Our theory is supported by experiments, where ${\tt AdaTS}$ outperforms prior algorithms and works well even in challenging real-world problems.

Random Effect Bandits

This paper studies regret minimization in multi-armed bandits, a classical online learning problem. To develop more statistically-efficient algorithms, we propose to use the assumption of a random-effect model. In this model, the mean rewards of arms are drawn independently from an unknown distribution, whose parameters we estimate. We provide an estimator of the arm means in this model and also analyze its uncertainty. Based on these results, we design a UCB algorithm, which we call ReUCB. We analyze ReUCB and prove a Bayes regret bound on its $n$-round regret, which matches an existing lower bound. Our experiments show that ReUCB can outperform Thompson sampling in various scenarios, without assuming that the prior distribution of arm means is known.

Fixed-Budget Best-Arm Identification in Contextual Bandits: A Static-Adaptive Algorithm

We study the problem of best-arm identification (BAI) in contextual bandits in the fixed-budget setting. We propose a general successive elimination algorithm that proceeds in stages and eliminates a fixed fraction of suboptimal arms in each stage. This design takes advantage of the strengths of static and adaptive allocations. We analyze the algorithm in linear models and obtain a better error bound than prior work. We also apply it to generalized linear models (GLMs) and bound its error. This is the first BAI algorithm for GLMs in the fixed-budget setting. Our extensive numerical experiments show that our algorithm outperforms the state of art.

Thompson Sampling with a Mixture Prior

We study Thompson sampling (TS) in online decision-making problems where the uncertain environment is sampled from a mixture distribution. This is relevant to multi-task settings, where a learning agent is faced with different classes of problems. We incorporate this structure in a natural way by initializing TS with a mixture prior -- dubbed MixTS -- and develop a novel, general technique for analyzing the regret of TS with such priors. We apply this technique to derive Bayes regret bounds for MixTS in both linear bandits and tabular Markov decision processes (MDPs). Our regret bounds reflect the structure of the problem and depend on the number of components and confidence width of each component of the prior. Finally, we demonstrate the empirical effectiveness of MixTS in both synthetic and real-world experiments.