For smooth and strongly convex optimization, the optimal iteration complexity of the gradient-based algorithm is $O(\sqrt{\kappa}\log 1/\epsilon)$, where $\kappa$ is the conditional number. In the case that the optimization problem is ill-conditioned, we need to evaluate a larger number of full gradients, which could be computationally expensive. In this paper, we propose to reduce the number of full gradient required by allowing the algorithm to access the stochastic gradients of the objective function. To this end, we present a novel algorithm named Epoch Mixed Gradient Descent (EMGD) that is able to utilize two kinds of gradients. A distinctive step in EMGD is the mixed gradient descent, where we use an combination of the gradient and the stochastic gradient to update the intermediate solutions. By performing a fixed number of mixed gradient descents, we are able to improve the sub-optimality of the solution by a constant factor, and thus achieve a linear convergence rate. Theoretical analysis shows that EMGD is able to find an $\epsilon$-optimal solution by computing $O(\log 1/\epsilon)$ full gradients and $O(\kappa^2\log 1/\epsilon)$ stochastic gradients.

We establish theoretical results concerning all local optima of various regularized M-estimators, where both loss and penalty functions are allowed to be nonconvex. Our results show that as long as the loss function satisfies restricted strong convexity and the penalty function satisfies suitable regularity conditions, any local optimum of the composite objective function lies within statistical precision of the true parameter vector. Our theory covers a broad class of nonconvex objective functions, including corrected versions of the Lasso for errors-in-variables linear models; regression in generalized linear models using nonconvex regularizers such as SCAD and MCP; and graph and inverse covariance matrix estimation. On the optimization side, we show that a simple adaptation of composite gradient descent may be used to compute a global optimum up to the statistical precision epsilon in log(1/epsilon) iterations, which is the fastest possible rate of any first-order method. We provide a variety of simulations to illustrate the sharpness of our theoretical predictions.

Learning dynamic models from observed data has been a central issue in many scientific studies or engineering tasks. The usual setting is that data are collected sequentially from trajectories of some dynamical system operation. In quite a few modern scientific modeling tasks, however, it turns out that reliable sequential data are rather difficult to gather, whereas out-of-order snapshots are much easier to obtain. Examples include the modeling of galaxies, chronic diseases such Alzheimer's, or certain biological processes. Existing methods for learning dynamic model from non-sequence data are mostly based on Expectation-Maximization, which involves non-convex optimization and is thus hard to analyze. Inspired by recent advances in spectral learning methods, we propose to study this problem from a different perspective: moment matching and spectral decomposition. Under that framework, we identify reasonable assumptions on the generative process of non-sequence data, and propose learning algorithms based on the tensor decomposition method \cite{anandkumar2012tensor} to \textit{provably} recover first-order Markov models and hidden Markov models. To the best of our knowledge, this is the first formal guarantee on learning from non-sequence data. Preliminary simulation results confirm our theoretical findings.

With simultaneous measurements from ever increasing populations of neurons, there is a growing need for sophisticated tools to recover signals from individual neurons. In electrophysiology experiments, this classically proceeds in a two-step process: (i) threshold the waveforms to detect putative spikes and (ii) cluster the waveforms into single units (neurons). We extend previous Bayesian nonparamet- ric models of neural spiking to jointly detect and cluster neurons using a Gamma process model. Importantly, we develop an online approximate inference scheme enabling real-time analysis, with performance exceeding the previous state-of-the- art. Via exploratory data analysis—using data with partial ground truth as well as two novel data sets—we find several features of our model collectively contribute to our improved performance including: (i) accounting for colored noise, (ii) de- tecting overlapping spikes, (iii) tracking waveform dynamics, and (iv) using mul- tiple channels. We hope to enable novel experiments simultaneously measuring many thousands of neurons and possibly adapting stimuli dynamically to probe ever deeper into the mysteries of the brain.

Natural actor-critics are a popular class of policy search algorithms for finding locally optimal policies for Markov decision processes. In this paper we address a drawback of natural actor-critics that limits their real-world applicability - their lack of safety guarantees. We present a principled algorithm for performing natural gradient descent over a constrained domain. In the context of reinforcement learning, this allows for natural actor-critic algorithms that are guaranteed to remain within a known safe region of policy space. While deriving our class of constrained natural actor-critic algorithms, which we call Projected Natural Actor-Critics (PNACs), we also elucidate the relationship between natural gradient descent and mirror descent.

Research on distributed machine learning algorithms has focused primarily on one of two extremes---algorithms that obey strict concurrency constraints or algorithms that obey few or no such constraints. We consider an intermediate alternative in which algorithms optimistically assume that conflicts are unlikely and if conflicts do arise a conflict-resolution protocol is invoked. We view this optimistic concurrency control'' paradigm as particularly appropriate for large-scale machine learning algorithms, particularly in the unsupervised setting. We demonstrate our approach in three problem areas: clustering, feature learning and online facility location. We evaluate our methods via large-scale experiments in a cluster computing environment. "

A common problem in knowledge representation and related fields is reasoning over a large joint knowledge graph, represented as triples of a relation between two entities. The goal of this paper is to develop a more powerful neural network model suitable for inference over these relationships. Previous models suffer from weak interaction between entities or simple linear projection of the vector space. We address these problems by introducing a neural tensor network (NTN) model which allow the entities and relations to interact multiplicatively. Additionally, we observe that such knowledge base models can be further improved by representing each entity as the average of vectors for the words in the entity name, giving an additional dimension of similarity by which entities can share statistical strength. We assess the model by considering the problem of predicting additional true relations between entities given a partial knowledge base. Our model outperforms previous models and can classify unseen relationships in WordNet and FreeBase with an accuracy of 86.2% and 90.0%, respectively.

It is a common practice to approximate complicated'' functions with more friendly ones. In large-scale machine learning applications, nonsmooth losses/regularizers that entail great computational challenges are usually approximated by smooth functions. We re-examine this powerful methodology and point out a nonsmooth approximation which simply pretends the linearity of the proximal map. The new approximation is justified using a recent convex analysis tool---proximal average, and yields a novel proximal gradient algorithm that is strictly better than the one based on smoothing, without incurring any extra overhead. Numerical experiments conducted on two important applications, overlapping group lasso and graph-guided fused lasso, corroborate the theoretical claims."

Expectation Propagation (EP) is a popular approximate posterior inference algorithm that often provides a fast and accurate alternative to sampling-based methods. However, while the EP framework in theory allows for complex non-Gaussian factors, there is still a significant practical barrier to using them within EP, because doing so requires the implementation of message update operators, which can be difficult and require hand-crafted approximations. In this work, we study the question of whether it is possible to automatically derive fast and accurate EP updates by learning a discriminative model e.g., a neural network or random forest) to map EP message inputs to EP message outputs. We address the practical concerns that arise in the process, and we provide empirical analysis on several challenging and diverse factors, indicating that there is a space of factors where this approach appears promising.

The l1-regularized Gaussian maximum likelihood estimator (MLE) has been shown to have strong statistical guarantees in recovering a sparse inverse covariance matrix even under high-dimensional settings. However, it requires solving a difficult non-smooth log-determinant program with number of parameters scaling quadratically with the number of Gaussian variables. State-of-the-art methods thus do not scale to problems with more than 20,000 variables. In this paper, we develop an algorithm BigQUIC, which can solve 1 million dimensional l1-regularized Gaussian MLE problems (which would thus have 1000 billion parameters) using a single machine, with bounded memory. In order to do so, we carefully exploit the underlying structure of the problem. Our innovations include a novel block-coordinate descent method with the blocks chosen via a clustering scheme to minimize repeated computations; and allowing for inexact computation of specific components. In spite of these modifications, we are able to theoretically analyze our procedure and show that BigQUIC can achieve super-linear or even quadratic convergence rates.