We study an online linear regression setting in which the observed feature vectors are corrupted by noise and the learner can pay to reduce the noise level. In practice, this may happen for several reasons: for example, because features can be measured more accurately using more expensive equipment, or because data providers can be incentivized to release less private features. Assuming feature vectors are drawn i.i.d. from a fixed but unknown distribution, we measure the learner's regret against the linear predictor minimizing a notion of loss that combines the prediction error and payment. We first study the case in which the mapping between payments and noise covariance is known and prove order-optimal regret bounds in the interaction length (up to log-factors). We then derive order-optimal bounds also when the noise covariance is unknown and prove that the regret rate is worse than the case of known covariances. Our analysis leverages matrix martingale concentration, showing that the empirical loss uniformly converges to the expected one for all payments and linear predictors.

Let $f(\theta, X_1),$ $ …,$ $ f(\theta, X_n)$ be a sequence of random elements, where $f$ is a fixed scalar function, $X_1, …, X_n$ are independent random variables (data), and $\theta$ is a random parameter distributed according to some data-dependent \emph{posterior} distribution $P_n$. In this paper, we consider the problem of proving concentration inequalities to estimate the mean of the sequence. An example of such a problem is the estimation of the generalization error of some predictor trained by a stochastic algorithm, such as a neural network, where $f$ is a loss function. Classically, this problem is approached through a \emph{PAC-Bayes} analysis where, in addition to the posterior, we choose a \emph{prior} distribution which captures our belief about the inductive bias of the learning problem. Then, the key quantity in PAC-Bayes concentration bounds is a divergence that captures the \emph{complexity} of the learning problem where the de facto standard choice is the Kullback-Leibler (KL) divergence. However, the tightness of this choice has rarely been questioned. In this paper, we challenge the tightness of the KL-divergence-based bounds by showing that it is possible to achieve a strictly tighter bound. In particular, we demonstrate new \emph{high-probability} PAC-Bayes bounds with a novel and \emph{better-than-KL} divergence that is inspired by Zhang et al. (2022). Our proof is inspired by recent advances in regret analysis of gambling algorithms, and its use to derive concentration inequalities. Our result is first-of-its-kind in that existing PAC-Bayes bounds with non-KL divergences are not known to be strictly better than KL. Thus, we believe our work marks the first step towards identifying optimal rates of PAC-Bayes bounds.

We study low-rank matrix trace regression and the related problem of low-rank matrix bandits. Assuming access to the distribution of the covariates, we propose a novel low-rank matrix estimation method called LowPopArt and provide its recovery guarantee that depends on a novel quantity denoted by $B(Q)$ that characterizes the hardness of the problem, where $Q$ is the covariance matrix of the measurement distribution. We show that our method can provide tighter recovery guarantees than classical nuclear norm penalized least squares (Koltchinskii et al., 2011) in several problems. To perform an efficient estimation with a limited number of measurements from an arbitrarily given measurement set $\mathcal{A}$, we also propose a novel experimental design criterion that minimizes $B(Q)$ with computational efficiency. We leverage our novel estimator and design of experiments to derive two low-rank linear bandit algorithms for general arm sets that enjoy improved regret upper bounds. This improves over previous works on low-rank bandits, which make somewhat restrictive assumptions that the arm set is the unit ball or that an efficient exploration distribution is given. To our knowledge, our experimental design criterion is the first one tailored to low-rank matrix estimation beyond the naive reduction to linear regression, which can be of independent interest.

We consider the fixed-confidence best arm identification (FC-BAI) problem in the Bayesian setting. This problem aims to find the arm of the largest mean with a fixed confidence level when the bandit model has been sampled from the known prior. Most studies on the FC-BAI problem have been conducted in the frequentist setting, where the bandit model is predetermined before the game starts. We show that the traditional FC-BAI algorithms studied in the frequentist setting, such as track-and-stop and top-two algorithms, result in arbitrarily suboptimal performances in the Bayesian setting. We also obtain a lower bound of the expected number of samples in the Bayesian setting and introduce a variant of successive elimination that has a matching performance with the lower bound up to a logarithmic factor. Simulations verify the theoretical results.

We study stochastic linear bandits where, in each round, the learner receives a set of actions (i.e., feature vectors), from which it chooses an element and obtains a stochastic reward. The expected reward is a fixed but unknown linear function of the chosen action. We study \emph{sparse} regret bounds, that depend on the number $S$ of non-zero coefficients in the linear reward function. Previous works focused on the case where $S$ is known, or the action sets satisfy additional assumptions. In this work, we obtain the first sparse regret bounds that hold when $S$ is unknown and the action sets are adversarially generated. Our techniques combine online to confidence set conversions with a novel randomized model selection approach over a hierarchy of nested confidence sets. When $S$ is known, our analysis recovers state-of-the-art bounds for adversarial action sets. We also show that a variant of our approach, using Exp3 to dynamically select the confidence sets, can be used to improve the empirical performance of stochastic linear bandits while enjoying a regret bound with optimal dependence on the time horizon.

We consider the problem of estimating the mean of a sequence of random elements $f(\theta, X_1)$ $, \ldots, $ $f(\theta, X_n)$ where $f$ is a fixed scalar function, $S=(X_1, \ldots, X_n)$ are independent random variables, and $\theta$ is a possibly $S$-dependent parameter. An example of such a problem would be to estimate the generalization error of a neural network trained on $n$ examples where $f$ is a loss function. Classically, this problem is approached through concentration inequalities holding uniformly over compact parameter sets of functions $f$, for example as in Rademacher or VC type analysis. However, in many problems, such inequalities often yield numerically vacuous estimates. Recently, the \emph{PAC-Bayes} framework has been proposed as a better alternative for this class of problems for its ability to often give numerically non-vacuous bounds. In this paper, we show that we can do even better: we show how to refine the proof strategy of the PAC-Bayes bounds and achieve \emph{even tighter} guarantees. Our approach is based on the \emph{coin-betting} framework that derives the numerically tightest known time-uniform concentration inequalities from the regret guarantees of online gambling algorithms. In particular, we derive the first PAC-Bayes concentration inequality based on the coin-betting approach that holds simultaneously for all sample sizes. We demonstrate its tightness showing that by \emph{relaxing} it we obtain a number of previous results in a closed form including Bernoulli-KL and empirical Bernstein inequalities. Finally, we propose an efficient algorithm to numerically calculate confidence sequences from our bound, which often generates nonvacuous confidence bounds even with one sample, unlike the state-of-the-art PAC-Bayes bounds.

In sparse linear bandits, a learning agent sequentially selects an action from a fixed action set and receives reward feedback, and the reward function depends linearly on a few coordinates of the covariates of the actions. This has applications in many real-world sequential decision making problems. In this paper, we devise a simple, novel sparse linear estimation method called $\textrm{PopArt}$ that enjoys a tighter $\ell_1$ recovery guarantee compared to Lasso (Tibshirani, 1996). Our bound naturally motivates an experimental design criterion that is convex and thus computationally efficient to solve. Based on our novel estimator and design criterion, we derive sparse linear bandit algorithms that enjoy improved regret upper bounds upon the state of the art (Hao et al., 2020), especially in terms of the geometry of the given action set. Finally, we prove a matching lower bound for sparse linear bandits in the data-poor regime, which closes the gap between upper and lower bounds in prior work.

We consider the bilinear bandit problem where the learner chooses a pair of arms, each from two different action spaces of dimension $d_1$ and $d_2$, respectively. The learner then receives a reward whose expectation is a bilinear function of the two chosen arms with an unknown matrix parameter $\Theta^*\in\mathbb{R}^{d_1 \times d_2}$ with rank $r$. Despite abundant applications such as drug discovery, the optimal regret rate is unknown for this problem, though it was conjectured to be $\tilde O(\sqrt{d_1d_2(d_1+d_2)r T})$ by Jun et al. (2019) where $\tilde O$ ignores polylogarithmic factors in $T$. In this paper, we make progress towards closing the gap between the upper and lower bound on the optimal regret. First, we reject the conjecture above by proposing algorithms that achieve the regret $\tilde O(\sqrt{d_1 d_2 (d_1+d_2) T})$ using the fact that the action space dimension $O(d_1+d_2)$ is significantly lower than the matrix parameter dimension $O(d_1 d_2)$. Second, we additionally devise an algorithm with better empirical performance than previous algorithms.