Sparse Principal Component Analysis (SPCA) and Sparse Linear Regression (SLR) have a wide range of applications and have attracted a tremendous amount of attention in the last two decades as canonical examples of statistical problems in high dimension. A variety of algorithms have been proposed for both SPCA and SLR, but an explicit connection between the two had not been made. We show how to efficiently transform a black-box solver for SLR into an algorithm for SPCA: assuming the SLR solver satisfies prediction error guarantees achieved by existing efficient algorithms such as those based on the Lasso, the SPCA algorithm derived from it achieves near state of the art guarantees for testing and for support recovery for the single spiked covariance model as obtained by the current best polynomial-time algorithms. Our reduction not only highlights the inherent similarity between the two problems, but also, from a practical standpoint, allows one to obtain a collection of algorithms for SPCA directly from known algorithms for SLR. We provide experimental results on simulated data comparing our proposed framework to other algorithms for SPCA.

Subsampling is a common and often effective method to deal with the computational challenges of large datasets. However, for most statistical models, there is no well-motivated approach for drawing a non-uniform subsample. We show that the concept of an asymptotically linear estimator and the associated influence function leads to asymptotically optimal sampling probabilities for a wide class of popular models. This is the only tight optimality result for subsampling we are aware of as other methods only provide probabilistic error bounds or optimal rates. We also show that these optimal weights can differ depending on whether the task is parameter estimation or prediction. Furthermore, for linear regression models, which have well-studied procedures for non-uniform subsampling, we empirically show our optimal influence function based method outperforms previous approaches even when using approximations to the optimal probabilities.

We use surrogate losses to obtain several new regret bounds and new algorithms for contextual bandit learning. Using the ramp loss, we derive a new margin-based regret bound in terms of standard sequential complexity measures of a benchmark class of real-valued regression functions. Using the hinge loss, we derive an efficient algorithm with a $\sqrt{dT}$-type mistake bound against benchmark policies induced by $d$-dimensional regressors. Under realizability assumptions, our results also yield classical regret bounds.

We consider a high dimensional linear regression problem where the goal is to efficiently recover an unknown vector \beta^* from n noisy linear observations Y=X \beta^*+W in R^n, for known X in R^{n \times p} and unknown W in R^n. Unlike most of the literature on this model we make no sparsity assumption on \beta^*. Instead we adopt a regularization based on assuming that the underlying vectors \beta^* have rational entries with the same denominator Q. We call this Q-rationality assumption. We propose a new polynomial-time algorithm for this task which is based on the seminal Lenstra-Lenstra-Lovasz (LLL) lattice basis reduction algorithm. We establish that under the Q-rationality assumption, our algorithm recovers exactly the vector \beta^* for a large class of distributions for the iid entries of X and non-zero noise W. We prove that it is successful under small noise, even when the learner has access to only one observation (n=1). Furthermore, we prove that in the case of the Gaussian white noise for W, n=o(p/\log p) and Q sufficiently large, our algorithm tolerates a nearly optimal information-theoretic level of the noise.

This paper studies the problem of sparse regression where the goal is to learn a sparse vector that best optimizes a given objective function. Under the assumption that the objective function satisfies restricted strong convexity (RSC), we analyze orthogonal matching pursuit (OMP), a greedy algorithm that is used heavily in applications, and obtain support recovery result as well as a tight generalization error bound for OMP. Furthermore, we obtain lower bounds for OMP, showing that both our results on support recovery and generalization error are tight up to logarithmic factors. To the best of our knowledge, these support recovery and generalization bounds are the first such matching upper and lower bounds (up to logarithmic factors) for {\em any} sparse regression algorithm under the RSC assumption.

The process of learning new behaviors over time is a problem of great interest in both neuroscience and artificial intelligence. However, most standard analyses of animal training data either treat behavior as fixed or track only coarse performance statistics (e.g., accuracy, bias), providing limited insight into the evolution of the policies governing behavior. To overcome these limitations, we propose a dynamic psychophysical model that efficiently tracks trial-to-trial changes in behavior over the course of training. Our model consists of a dynamic logistic regression model, parametrized by a set of time-varying weights that express dependence on sensory stimuli as well as task-irrelevant covariates, such as stimulus, choice, and answer history. Our implementation scales to large behavioral datasets, allowing us to infer 500K parameters (e.g. 10 weights over 50K trials) in minutes on a desktop computer. We optimize hyperparameters governing how rapidly each weight evolves over time using the decoupled Laplace approximation, an efficient method for maximizing marginal likelihood in non-conjugate models. To illustrate performance, we apply our method to psychophysical data from both rats and human subjects learning a delayed sensory discrimination task. The model successfully tracks the psychophysical weights of rats over the course of training, capturing day-to-day and trial-to-trial fluctuations that underlie changes in performance, choice bias, and dependencies on task history. Finally, we investigate why rats frequently make mistakes on easy trials, and suggest that apparent lapses can be explained by sub-optimal weighting of known task covariates.

We consider a generalization of mixed regression where the response is an additive combination of several mixture components. Standard mixed regression is a special case where each response is generated from exactly one component. Typical approaches to the mixture regression problem employ local search methods such as Expectation Maximization (EM) that are prone to spurious local optima. On the other hand, a number of recent theoretically-motivated \emph{Tensor-based methods} either have high sample complexity, or require the knowledge of the input distribution, which is not available in most of practical situations. In this work, we study a novel convex estimator \emph{MixLasso} for the estimation of generalized mixed regression, based on an atomic norm specifically constructed to regularize the number of mixture components. Our algorithm gives a risk bound that trades off between prediction accuracy and model sparsity without imposing stringent assumptions on the input/output distribution, and can be easily adapted to the case of non-linear functions. In our numerical experiments on mixtures of linear as well as nonlinear regressions, the proposed method yields high-quality solutions in a wider range of settings than existing approaches.

Bayesian optimization (BO) is a model-based approach for gradient-free black-box function optimization, such as hyperparameter optimization. Typically, BO relies on conventional Gaussian process (GP) regression, whose algorithmic complexity is cubic in the number of evaluations. As a result, GP-based BO cannot leverage large numbers of past function evaluations, for example, to warm-start related BO runs. We propose a multi-task adaptive Bayesian linear regression model for transfer learning in BO, whose complexity is linear in the function evaluations: one Bayesian linear regression model is associated to each black-box function optimization problem (or task), while transfer learning is achieved by coupling the models through a shared deep neural net. Experiments show that the neural net learns a representation suitable for warm-starting the black-box optimization problems and that BO runs can be accelerated when the target black-box function (e.g., validation loss) is learned together with other related signals (e.g., training loss). The proposed method was found to be at least one order of magnitude faster than competing methods recently published in the literature.

Summarizing high-dimensional data using a small number of parameters is a ubiquitous first step in the analysis of neuronal population activity. Recently developed methods use "targeted" approaches that work by identifying multiple, distinct low-dimensional subspaces of activity that capture the population response to individual experimental task variables, such as the value of a presented stimulus or the behavior of the animal. These methods have gained attention because they decompose total neural activity into what are ostensibly different parts of a neuronal computation. However, existing targeted methods have been developed outside of the confines of probabilistic modeling, making some aspects of the procedures ad hoc, or limited in flexibility or interpretability. Here we propose a new model-based method for targeted dimensionality reduction based on a probabilistic generative model of the population response data. The low-dimensional structure of our model is expressed as a low-rank factorization of a linear regression model. We perform efficient inference using a combination of expectation maximization and direct maximization of the marginal likelihood. We also develop an efficient method for estimating the dimensionality of each subspace. We show that our approach outperforms alternative methods in both mean squared error of the parameter estimates, and in identifying the correct dimensionality of encoding using simulated data. We also show that our method provides more accurate inference of low-dimensional subspaces of activity than a competing algorithm, demixed PCA.

Hamiltonian Monte Carlo (HMC) is a widely deployed method to sample from high-dimensional distributions in Statistics and Machine learning. HMC is known to run very efficiently in practice and its popular second-order ``leapfrog" implementation has long been conjectured to run in $d^{1/4}$ gradient evaluations. Here we show that this conjecture is true when sampling from strongly log-concave target distributions that satisfy a weak third-order regularity property associated with the input data. Our regularity condition is weaker than the Lipschitz Hessian property and allows us to show faster convergence bounds for a much larger class of distributions than would be possible with the usual Lipschitz Hessian constant alone. Important distributions that satisfy our regularity condition include posterior distributions used in Bayesian logistic regression for which the data satisfies an ``incoherence" property. Our result compares favorably with the best available bounds for the class of strongly log-concave distributions, which grow like $d^{{1}/{2}}$ gradient evaluations with the dimension. Moreover, our simulations on synthetic data suggest that, when our regularity condition is satisfied, leapfrog HMC performs better than its competitors -- both in terms of accuracy and in terms of the number of gradient evaluations it requires.