Multiple linear regression provides a simple, but widely used, method to find associations between outcomes (responses) and a set of predictors (explanatory variables). It has been actively studied over more than a century, and there is a rich and vast literature on the subject [1]. In practical situations the number of predictor variables is often large, and it becomes desirable to induce sparsity in the regression coefficients to avoid overfitting [2, 3]. Sparse linear regression also serves as the foundation for non-linear techniques, such as trendfiltering [4, 5], which can estimate an underlying non-linear trend from time series data. Applications of sparse multiple linear regression and trendfiltering arise in a wide range of applications in modern science and engineering, including astronomy [6], atmospheric sciences [7], biology [8], economics [9, 10], genetics [11-15], geophysics [16], medical sciences [17, 18], social sciences [19] and text analysis [20]. Approaches to sparse linear regression can be broadly classified into two groups: (a) penalized linear regressions (PLR), which add a penalty term to the likelihood to penalize the magnitude of its parameters [21-23], and (b) Bayesian approaches [11-14, 24-29], which use a prior probability distribution on the model parameters to induce sparsity.

We report on the potential for using algorithms for non-negative matrix factorization (NMF) to improve parameter estimation in topic models. While several papers have studied connections between NMF and topic models, none have suggested leveraging these connections to develop new algorithms for fitting topic models. Importantly, NMF avoids the "sum-to-one" constraints on the topic model parameters, resulting in an optimization problem with simpler structure and more efficient computations. Building on recent advances in optimization algorithms for NMF, we show that first solving the NMF problem then recovering the topic model fit can produce remarkably better fits, and in less time, than standard algorithms for topic models. While we focus primarily on maximum likelihood estimation, we show that this approach also has the potential to improve variational inference for topic models. Our methods are implemented in the R package fastTopics.

The stochastic approximation method is behind the solution to many important, actively-studied problems in machine learning. Despite its far-reaching application, there is almost no work on applying stochastic approximation to learning problems with constraints. The reason for this, we hypothesize, is that no robust, widely-applicable stochastic approximation method exists for handling such problems. We propose that interior-point methods are a natural solution. We establish the stability of a stochastic interior-point approximation method both analytically and empirically, and demonstrate its utility by deriving an on-line learning algorithm that also performs feature selection via L1 regularization.

We describe a new algorithmic framework for inference in probabilistic models, and apply it to inference for latent Dirichlet allocation. Our framework adopts the methodology of variational inference, but unlike existing variational methods such as mean field and expectation propagation it is not restricted to tractable classes of approximating distributions. Our approach can also be viewed as a sequential Monte Carlo (SMC) method, but unlike existing SMC methods there is no need to design the artificial sequence of distributions. Notably, our framework offers a principled means to exchange the variance of an importance sampling estimate for the bias incurred through variational approximation. Experiments on a challenging inference problem in population genetics demonstrate improvements in stability and accuracy over existing methods, and at a comparable cost.

Despite all the attention paid to variational methods based on sum-product message passing (loopy belief propagation, tree-reweighted sum-product), these methods are still bound to inference on a small set of probabilistic models. Mean field approximations have been applied to a broader set of problems, but the solutions are often poor. We propose a new class of conditionally-specified variational approximations based on mean field theory. While not usable on their own, combined with sequential Monte Carlo they produce guaranteed improvements over conventional mean field. Moreover, experiments on a well-studied problem-- inferring the stable configurations of the Ising spin glass--show that the solutions can be significantly better than those obtained using sum-product-based methods.

Despite all the attention paid to variational methods based on sum-product message passing(loopy belief propagation, tree-reweighted sum-product), these methods are still bound to inference on a small set of probabilistic models. Mean field approximations have been applied to a broader set of problems, but the solutions are often poor. We propose a new class of conditionally-specified variational approximations basedon mean field theory. While not usable on their own, combined with sequential Monte Carlo they produce guaranteed improvements over conventional mean field. Moreover, experiments on a well-studied problem-- inferring the stable configurations of the Ising spin glass--show that the solutions can be significantly better than those obtained using sum-product-based methods.