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Estimators computed from adaptively collected data do not behave like their non-adaptive brethren. Rather, the sequential dependence of the collection policy can lead to severe distributional biases that persist even in the infinite data limit. We develop a general method -- $mathbf{W}$-decorrelation -- for transforming the bias of adaptive linear regression estimators into variance. The method uses only coarse-grained information about the data collection policy and does not need access to propensity scores or exact knowledge of the policy. We bound the finite-sample bias and variance of the $mathbf{W}$-estimator and develop asymptotically correct confidence intervals based on a novel martingale central limit theorem. We then demonstrate the empirical benefits of the generic $mathbf{W}$-decorrelation procedure in two different adaptive data settings: the multi-armed bandit and the autoregressive time series.
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We develop a sequential low-complexity inference procedure for Dirichlet process mixtures of Gaussians for online clustering and parameter estimation when the number of clusters are unknown a-priori. We present an easily computable, closed form parametric expression for the conditional likelihood, in which hyperparameters are recursively updated as a function of the streaming data assuming conjugate priors. Motivated by large-sample asymptotics, we propose a novel adaptive low-complexity design for the Dirichlet process concentration parameter and show that the number of classes grow at most at a logarithmic rate. We further prove that in the large-sample limit, the conditional likelihood and data predictive distribution become asymptotically Gaussian. We demonstrate through experiments on synthetic and real data sets that our approach is superior to other online state-of-the-art methods.
We consider the problem of model selection for two popular stochastic linear bandit settings, and propose algorithms that adapts to the unknown problem complexity. In the first setting, we consider the $K$ armed mixture bandits, where the mean reward of arm $i in [K]$, is $mu_i+ langle alpha_{i,t},theta^* rangle $, with $alpha_{i,t} in mathbb{R}^d$ being the known context vector and $mu_i in [-1,1]$ and $theta^*$ are unknown parameters. We define $|theta^*|$ as the problem complexity and consider a sequence of nested hypothesis classes, each positing a different upper bound on $|theta^*|$. Exploiting this, we propose Adaptive Linear Bandit (ALB), a novel phase based algorithm that adapts to the true problem complexity, $|theta^*|$. We show that ALB achieves regret scaling of $O(|theta^*|sqrt{T})$, where $|theta^*|$ is apriori unknown. As a corollary, when $theta^*=0$, ALB recovers the minimax regret for the simple bandit algorithm without such knowledge of $theta^*$. ALB is the first algorithm that uses parameter norm as model section criteria for linear bandits. Prior state of art algorithms cite{osom} achieve a regret of $O(Lsqrt{T})$, where $L$ is the upper bound on $|theta^*|$, fed as an input to the problem. In the second setting, we consider the standard linear bandit problem (with possibly an infinite number of arms) where the sparsity of $theta^*$, denoted by $d^* leq d$, is unknown to the algorithm. Defining $d^*$ as the problem complexity, we show that ALB achieves $O(d^*sqrt{T})$ regret, matching that of an oracle who knew the true sparsity level. This methodology is then extended to the case of finitely many arms and similar results are proven. This is the first algorithm that achieves such model selection guarantees. We further verify our results via synthetic and real-data experiments.
Stochastic gradient descent (SGD) and projected stochastic gradient descent (PSGD) are scalable algorithms to compute model parameters in unconstrained and constrained optimization problems. In comparison with stochastic gradient descent (SGD), PSGD forces its iterative values into the constrained parameter space via projection. The convergence rate of PSGD-type estimates has been exhaustedly studied, while statistical properties such as asymptotic distribution remain less explored. From a purely statistical point of view, this paper studies the limiting distribution of PSGD-based estimate when the true parameters satisfying some linear-equality constraints. Our theoretical findings reveal the role of projection played in the uncertainty of the PSGD estimate. As a byproduct, we propose an online hypothesis testing procedure to test the linear-equality constraints. Simulation studies on synthetic data and an application to a real-world dataset confirm our theory.
Over-parameterization and adaptive methods have played a crucial role in the success of deep learning in the last decade. The widespread use of over-parameterization has forced us to rethink generalization by bringing forth new phenomena, such as implicit regularization of optimization algorithms and double descent with training progression. A series of recent works have started to shed light on these areas in the quest to understand -- why do neural networks generalize well? The setting of over-parameterized linear regression has provided key insights into understanding this mysterious behavior of neural networks. In this paper, we aim to characterize the performance of adaptive methods in the over-parameterized linear regression setting. First, we focus on two sub-classes of adaptive methods depending on their generalization performance. For the first class of adaptive methods, the parameter vector remains in the span of the data and converges to the minimum norm solution like gradient descent (GD). On the other hand, for the second class of adaptive methods, the gradient rotation caused by the pre-conditioner matrix results in an in-span component of the parameter vector that converges to the minimum norm solution and the out-of-span component that saturates. Our experiments on over-parameterized linear regression and deep neural networks support this theory.
When data is collected in an adaptive manner, even simple methods like ordinary least squares can exhibit non-normal asymptotic behavior. As an undesirable consequence, hypothesis tests and confidence intervals based on asymptotic normality can lead to erroneous results. We propose an online debiasing estimator to correct these distributional anomalies in least squares estimation. Our proposed method takes advantage of the covariance structure present in the dataset and provides sharper estimates in directions for which more information has accrued. We establish an asymptotic normality property for our proposed online debiasing estimator under mild conditions on the data collection process, and provide asymptotically exact confidence intervals. We additionally prove a minimax lower bound for the adaptive linear regression problem, thereby providing a baseline by which to compare estimators. There are various conditions under which our proposed estimator achieves the minimax lower bound up to logarithmic factors. We demonstrate the usefulness of our theory via applications to multi-armed bandit, autoregressive time series estimation, and active learning with exploration.
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