Dynamic Batch Learning in High-Dimensional Sparse Linear Contextual Bandits

We study the problem of dynamic batch learning in high-dimensional sparse linear contextual bandits, where a decision maker, under a given maximum-number-of-batch constraint and only able to observe rewards at the end of each batch, can dynamically decide how many individuals to include in the next batch (at the end of the current batch) and what personalized action-selection scheme to adopt within each batch. Such batch constraints are ubiquitous in a variety of practical contexts, including personalized product offerings in marketing and medical treatment selection in clinical trials. We characterize the fundamental learning limit in this problem via a regret lower bound and provide a matching upper bound (up to log factors), thus prescribing an optimal scheme for this problem. To the best of our knowledge, our work provides the first inroad into a theoretical understanding of dynamic batch learning in high-dimensional sparse linear contextual bandits. Notably, even a special case of our result (when no batch constraint is present) yields the first minimax optimal $\tilde{O}(\sqrt{s_0T})$ regret bound for standard online learning in high-dimensional linear contextual bandits (for the no-margin case), where $s_0$ is the sparsity parameter (or an upper bound thereof) and $T$ is the learning horizon. This result (both that $\tilde{O}(\sqrt{s_0 T})$ is achievable and that $\Omega(s_0 T)$ is a lower bound) appears to be unknown in the emerging literature of high-dimensional contextual bandits.

Federated Learning's Blessing: FedAvg has Linear Speedup

Federated learning (FL) learns a model jointly from a set of participating devices without sharing each other's privately held data. The characteristics of non-iid data across the network, low device participation, and the mandate that data remain private bring challenges in understanding the convergence of FL algorithms, particularly in regards to how convergence scales with the number of participating devices. In this paper, we focus on Federated Averaging (FedAvg)--the most widely used and effective FL algorithm in use today--and provide a comprehensive study of its convergence rate. Although FedAvg has recently been studied by an emerging line of literature, it remains open as to how FedAvg's convergence scales with the number of participating devices in the FL setting--a crucial question whose answer would shed light on the performance of FedAvg in large FL systems. We fill this gap by establishing convergence guarantees for FedAvg under three classes of problems: strongly convex smooth, convex smooth, and overparameterized strongly convex smooth problems. We show that FedAvg enjoys linear speedup in each case, although with different convergence rates. For each class, we also characterize the corresponding convergence rates for the Nesterov accelerated FedAvg algorithm in the FL setting: to the best of our knowledge, these are the first linear speedup guarantees for FedAvg when Nesterov acceleration is used. To accelerate FedAvg, we also design a new momentum-based FL algorithm that further improves the convergence rate in overparameterized linear regression problems. Empirical studies of the algorithms in various settings have supported our theoretical results.

Learning to Bid Optimally and Efficiently in Adversarial First-price Auctions

First-price auctions have very recently swept the online advertising industry, replacing second-price auctions as the predominant auction mechanism on many platforms. This shift has brought forth important challenges for a bidder: how should one bid in a first-price auction, where unlike in second-price auctions, it is no longer optimal to bid one's private value truthfully and hard to know the others' bidding behaviors? In this paper, we take an online learning angle and address the fundamental problem of learning to bid in repeated first-price auctions, where both the bidder's private valuations and other bidders' bids can be arbitrary. We develop the first minimax optimal online bidding algorithm that achieves an $\widetilde{O}(\sqrt{T})$ regret when competing with the set of all Lipschitz bidding policies, a strong oracle that contains a rich set of bidding strategies. This novel algorithm is built on the insight that the presence of a good expert can be leveraged to improve performance, as well as an original hierarchical expert-chaining structure, both of which could be of independent interest in online learning. Further, by exploiting the product structure that exists in the problem, we modify this algorithm--in its vanilla form statistically optimal but computationally infeasible--to a computationally efficient and space efficient algorithm that also retains the same $\widetilde{O}(\sqrt{T})$ minimax optimal regret guarantee. Additionally, through an impossibility result, we highlight that one is unlikely to compete this favorably with a stronger oracle (than the considered Lipschitz bidding policies). Finally, we test our algorithm on three real-world first-price auction datasets obtained from Verizon Media and demonstrate our algorithm's superior performance compared to several existing bidding algorithms.

Distributional Robust Batch Contextual Bandits

Policy learning using historical observational data is an important problem that has found widespread applications. Examples include selecting offers, prices, advertisements to send to customers, as well as selecting which medication to prescribe to a patient. However, existing literature rests on the crucial assumption that the future environment where the learned policy will be deployed is the same as the past environment that has generated the data--an assumption that is often false or too coarse an approximation. In this paper, we lift this assumption and aim to learn a distributional robust policy with incomplete (bandit) observational data. We propose a novel learning algorithm that is able to learn a robust policy to adversarial perturbations and unknown covariate shifts. We first present a policy evaluation procedure in the ambiguous environment and then give a performance guarantee based on the theory of uniform convergence. Additionally, we also give a heuristic algorithm to solve the distributional robust policy learning problems efficiently.

Distributional Soft Actor Critic for Risk Sensitive Learning

Most of reinforcement learning (RL) algorithms aim at maximizing the expectation of accumulated discounted returns. Since the accumulated discounted return is a random variable, its distribution includes more information than its expectation. Meanwhile, entropy of policy indicates its diversity and it can help improve the exploration capability of algorithms. In this paper, we present a new RL algorithm named Distributional Soft Actor Critic (DSAC), combining distributional RL and maximum entropy RL together. Taking the randomness both in action and discounted return into consideration, DSAC over performs the state-of-the-art baselines with more stability in several continuous control benchmarks. Moreover, distributional information of returns can also be used to measure metrics other than expectation, such as risk-related metrics. With a fully parameterized quantile function, DSAC is easily adopted to optimize policy under different risk preferences. Our experiments demonstrate that with distribution modeling in RL the agent performs better both for risk-averse and risk-seeking control tasks.

Sequential Batch Learning in Finite-Action Linear Contextual Bandits

We study the sequential batch learning problem in linear contextual bandits with finite action sets, where the decision maker is constrained to split incoming individuals into (at most) a fixed number of batches and can only observe outcomes for the individuals within a batch at the batch's end. Compared to both standard online contextual bandits learning or offline policy learning in contexutal bandits, this sequential batch learning problem provides a finer-grained formulation of many personalized sequential decision making problems in practical applications, including medical treatment in clinical trials, product recommendation in e-commerce and adaptive experiment design in crowdsourcing. We study two settings of the problem: one where the contexts are arbitrarily generated and the other where the contexts are \textit{iid} drawn from some distribution. In each setting, we establish a regret lower bound and provide an algorithm, whose regret upper bound nearly matches the lower bound. As an important insight revealed therefrom, in the former setting, we show that the number of batches required to achieve the fully online performance is polynomial in the time horizon, while for the latter setting, a pure-exploitation algorithm with a judicious batch partition scheme achieves the fully online performance even when the number of batches is less than logarithmic in the time horizon. Together, our results provide a near-complete characterization of sequential decision making in linear contextual bandits when batch constraints are present.

Optimal No-regret Learning in Repeated First-price Auctions

We study online learning in repeated first-price auctions with censored feedback, where a bidder, only observing the winning bid at the end of each auction, learns to adaptively bid in order to maximize her cumulative payoff. To achieve this goal, the bidder faces a challenging dilemma: if she wins the bid--the only way to achieve positive payoffs--then she is not able to observe the highest bid of the other bidders, which we assume is iid drawn from an unknown distribution. This dilemma, despite being reminiscent of the exploration-exploitation trade-off in contextual bandits, cannot directly be addressed by the existing UCB or Thompson sampling algorithms in that literature, mainly because contrary to the standard bandits setting, when a positive reward is obtained here, nothing about the environment can be learned. In this paper, by exploiting the structural properties of first-price auctions, we develop the first learning algorithm that achieves $O(\sqrt{T}\log^2 T)$ regret bound when the bidder's private values are stochastically generated. We do so by providing an algorithm on a general class of problems, which we call monotone group contextual bandits, where the same regret bound is established under stochastically generated contexts. Further, by a novel lower bound argument, we characterize an $\Omega(T^{2/3})$ lower bound for the case where the contexts are adversarially generated, thus highlighting the impact of the contexts generation mechanism on the fundamental learning limit. Despite this, we further exploit the structure of first-price auctions and develop a learning algorithm that operates sample-efficiently (and computationally efficiently) in the presence of adversarially generated private values. We establish an $O(\sqrt{T}\log^3 T)$ regret bound for this algorithm, hence providing a complete characterization of optimal learning guarantees for this problem.

Diagonal Preconditioning: Theory and Algorithms

Diagonal preconditioning has been a staple technique in optimization and machine learning. It often reduces the condition number of the design or Hessian matrix it is applied to, thereby speeding up convergence. However, rigorous analyses of how well various diagonal preconditioning procedures improve the condition number of the preconditioned matrix and how that translates into improvements in optimization are rare. In this paper, we first provide an analysis of a popular diagonal preconditioning technique based on column standard deviation and its effect on the condition number using random matrix theory. Then we identify a class of design matrices whose condition numbers can be reduced significantly by this procedure. We then study the problem of optimal diagonal preconditioning to improve the condition number of any full-rank matrix and provide a bisection algorithm and a potential reduction algorithm with $O(\log(\frac{1}{\epsilon}))$ iteration complexity, where each iteration consists of an SDP feasibility problem and a Newton update using the Nesterov-Todd direction, respectively. Finally, we extend the optimal diagonal preconditioning algorithm to an adaptive setting and compare its empirical performance at reducing the condition number and speeding up convergence for regression and classification problems with that of another adaptive preconditioning technique, namely batch normalization, that is essential in training machine learning models.