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Register a new userGeneralized Linear Bandits with Local Differential Privacy
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Contextual bandit algorithms are useful in personalized online decision-making. However, many applications such as personalized medicine and online advertising require the utilization of individual-specific information for effective learning, while users data should remain private from the server due to privacy concerns. This motivates the introduction of local differential privacy (LDP), a stringent notion in privacy, to contextual bandits. In this paper, we design LDP algorithms for stochastic generalized linear bandits to achieve the same regret bound as in non-privacy settings. Our main idea is to develop a stochastic gradient-based estimator and update mechanism to ensure LDP. We then exploit the flexibility of stochastic gradient descent (SGD), whose theoretical guarantee for bandit problems is rarely explored, in dealing with generalized linear bandits. We also develop an estimator and update mechanism based on Ordinary Least Square (OLS) for linear bandits. Finally, we conduct experiments with both simulation and real-world datasets to demonstrate the consistently superb performance of our algorithms under LDP constraints with reasonably small parameters $(varepsilon, delta)$ to ensure strong privacy protection.
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Standard approaches to decision-making under uncertainty focus on sequential exploration of the space of decisions. However, textit{simultaneously} proposing a batch of decisions, which leverages available resources for parallel experimentation, has the potential to rapidly accelerate exploration. We present a family of (parallel) contextual linear bandit algorithms, whose regret is nearly identical to their perfectly sequential counterparts -- given access to the same total number of oracle queries -- up to a lower-order burn-in term that is dependent on the context-set geometry. We provide matching information-theoretic lower bounds on parallel regret performance to establish our algorithms are asymptotically optimal in the time horizon. Finally, we also present an empirical evaluation of these parallel algorithms in several domains, including materials discovery and biological sequence design problems, to demonstrate the utility of parallelized bandits in practical settings.
Stochastic linear bandits with high-dimensional sparse features are a practical model for a variety of domains, including personalized medicine and online advertising. We derive a novel $Omega(n^{2/3})$ dimension-free minimax regret lower bound for sparse linear bandits in the data-poor regime where the horizon is smaller than the ambient dimension and where the feature vectors admit a well-conditioned exploration distribution. This is complemented by a nearly matching upper bound for an explore-then-commit algorithm showing that that $Theta(n^{2/3})$ is the optimal rate in the data-poor regime. The results complement existing bounds for the data-rich regime and provide another example where carefully balancing the trade-off between information and regret is necessary. Finally, we prove a dimension-free $O(sqrt{n})$ regret upper bound under an additional assumption on the magnitude of the signal for relevant features.
In this paper we study the adversarial combinatorial bandit with a known non-linear reward function, extending existing work on adversarial linear combinatorial bandit. {The adversarial combinatorial bandit with general non-linear reward is an important open problem in bandit literature, and it is still unclear whether there is a significant gap from the case of linear reward, stochastic bandit, or semi-bandit feedback.} We show that, with $N$ arms and subsets of $K$ arms being chosen at each of $T$ time periods, the minimax optimal regret is $widetildeTheta_{d}(sqrt{N^d T})$ if the reward function is a $d$-degree polynomial with $d< K$, and $Theta_K(sqrt{N^K T})$ if the reward function is not a low-degree polynomial. {Both bounds are significantly different from the bound $O(sqrt{mathrm{poly}(N,K)T})$ for the linear case, which suggests that there is a fundamental gap between the linear and non-linear reward structures.} Our result also finds applications to adversarial assortment optimization problem in online recommendation. We show that in the worst-case of adversarial assortment problem, the optimal algorithm must treat each individual $binom{N}{K}$ assortment as independent.
In this paper we introduce the transductive linear bandit problem: given a set of measurement vectors $mathcal{X}subset mathbb{R}^d$, a set of items $mathcal{Z}subset mathbb{R}^d$, a fixed confidence $delta$, and an unknown vector $theta^{ast}in mathbb{R}^d$, the goal is to infer $text{argmax}_{zin mathcal{Z}} z^toptheta^ast$ with probability $1-delta$ by making as few sequentially chosen noisy measurements of the form $x^toptheta^{ast}$ as possible. When $mathcal{X}=mathcal{Z}$, this setting generalizes linear bandits, and when $mathcal{X}$ is the standard basis vectors and $mathcal{Z}subset {0,1}^d$, combinatorial bandits. Such a transductive setting naturally arises when the set of measurement vectors is limited due to factors such as availability or cost. As an example, in drug discovery the compounds and dosages $mathcal{X}$ a practitioner may be willing to evaluate in the lab in vitro due to cost or safety reasons may differ vastly from those compounds and dosages $mathcal{Z}$ that can be safely administered to patients in vivo. Alternatively, in recommender systems for books, the set of books $mathcal{X}$ a user is queried about may be restricted to well known best-sellers even though the goal might be to recommend more esoteric titles $mathcal{Z}$. In this paper, we provide instance-dependent lower bounds for the transductive setting, an algorithm that matches these up to logarithmic factors, and an evaluation. In particular, we provide the first non-asymptotic algorithm for linear bandits that nearly achieves the information theoretic lower bound.
Stochastic sparse linear bandits offer a practical model for high-dimensional online decision-making problems and have a rich information-regret structure. In this work we explore the use of information-directed sampling (IDS), which naturally balances the information-regret trade-off. We develop a class of information-theoretic Bayesian regret bounds that nearly match existing lower bounds on a variety of problem instances, demonstrating the adaptivity of IDS. To efficiently implement sparse IDS, we propose an empirical Bayesian approach for sparse posterior sampling using a spike-and-slab Gaussian-Laplace prior. Numerical results demonstrate significant regret reductions by sparse IDS relative to several baselines.
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