Active learning has shown to reduce the number of experiments needed to obtain high-confidence drug-target predictions. However, in order to actually save experiments using active learning, it is crucial to have a method to evaluate the quality of the current prediction and decide when to stop the experimentation process. Only by applying reliable stoping criteria to active learning, time and costs in the experimental process can be actually saved. We compute active learning traces on simulated drug-target matrices in order to learn a regression model for the accuracy of the active learner. By analyzing the performance of the regression model on simulated data, we design stopping criteria for previously unseen experimental matrices. We demonstrate on four previously characterized drug effect data sets that applying the stopping criteria can result in upto 40% savings of the total experiments for highly accurate predictions.

Many modern tools in machine learning and signal processing, such as sparse dictionary learning, principal component analysis (PCA), non-negative matrix factorization (NMF), $K$-means clustering, etc., rely on the factorization of a matrix obtained by concatenating high-dimensional vectors from a training collection. While the idealized task would be to optimize the expected quality of the factors over the underlying distribution of training vectors, it is achieved in practice by minimizing an empirical average over the considered collection. The focus of this paper is to provide sample complexity estimates to uniformly control how much the empirical average deviates from the expected cost function. Standard arguments imply that the performance of the empirical predictor also exhibit such guarantees. The level of genericity of the approach encompasses several possible constraints on the factors (tensor product structure, shift-invariance, sparsity \ldots), thus providing a unified perspective on the sample complexity of several widely used matrix factorization schemes. The derived generalization bounds behave proportional to $\sqrt{\log(n)/n}$ w.r.t.\ the number of samples $n$ for the considered matrix factorization techniques.

Matrix completion has attracted significant recent attention in many fields including statistics, applied mathematics and electrical engineering. Current literature on matrix completion focuses primarily on independent sampling models under which the individual observed entries are sampled independently. Motivated by applications in genomic data integration, we propose a new framework of structured matrix completion (SMC) to treat structured missingness by design. Specifically, our proposed method aims at efficient matrix recovery when a subset of the rows and columns of an approximately low-rank matrix are observed. We provide theoretical justification for the proposed SMC method and derive lower bound for the estimation errors, which together establish the optimal rate of recovery over certain classes of approximately low-rank matrices. Simulation studies show that the method performs well in finite sample under a variety of configurations. The method is applied to integrate several ovarian cancer genomic studies with different extent of genomic measurements, which enables us to construct more accurate prediction rules for ovarian cancer survival.

Protein contacts contain important information for protein structure and functional study, but contact prediction from sequence remains very challenging. Both evolutionary coupling (EC) analysis and supervised machine learning methods are developed to predict contacts, making use of different types of information, respectively. This paper presents a group graphical lasso (GGL) method for contact prediction that integrates joint multi-family EC analysis and supervised learning. Different from existing single-family EC analysis that uses residue co-evolution information in only the target protein family, our joint EC analysis uses residue co-evolution in both the target family and its related families, which may have divergent sequences but similar folds. To implement joint EC analysis, we model a set of related protein families using Gaussian graphical models (GGM) and then co-estimate their precision matrices by maximum-likelihood, subject to the constraint that the precision matrices shall share similar residue co-evolution patterns. To further improve the accuracy of the estimated precision matrices, we employ a supervised learning method to predict contact probability from a variety of evolutionary and non-evolutionary information and then incorporate the predicted probability as prior into our GGL framework. Experiments show that our method can predict contacts much more accurately than existing methods, and that our method performs better on both conserved and family-specific contacts.

Recent years have demonstrated that using random feature maps can significantly decrease the training and testing times of kernel-based algorithms without significantly lowering their accuracy. Regrettably, because random features are target-agnostic, typically thousands of such features are necessary to achieve acceptable accuracies. In this work, we consider the problem of learning a small number of explicit polynomial features. Our approach, named Tensor Machines, finds a parsimonious set of features by optimizing over the hypothesis class introduced by Kar and Karnick for random feature maps in a target-specific manner. Exploiting a natural connection between polynomials and tensors, we provide bounds on the generalization error of Tensor Machines. Empirically, Tensor Machines behave favorably on several real-world datasets compared to other state-of-the-art techniques for learning polynomial features, and deliver significantly more parsimonious models.

Clustering of data sets is a standard problem in many areas of science and engineering. The method of spectral clustering is based on embedding the data set using a kernel function, and using the top eigenvectors of the normalized Laplacian to recover the connected components. We study the performance of spectral clustering in recovering the latent labels of i.i.d. samples from a finite mixture of nonparametric distributions. The difficulty of this label recovery problem depends on the overlap between mixture components and how easily a mixture component is divided into two nonoverlapping components. When the overlap is small compared to the indivisibility of the mixture components, the principal eigenspace of the population-level normalized Laplacian operator is approximately spanned by the square-root kernelized component densities. In the finite sample setting, and under the same assumption, embedded samples from different components are approximately orthogonal with high probability when the sample size is large. As a corollary we control the fraction of samples mislabeled by spectral clustering under finite mixtures with nonparametric components.

We show that accelerated gradient descent, averaged gradient descent and the heavy-ball method for non-strongly-convex problems may be reformulated as constant parameter second-order difference equation algorithms, where stability of the system is equivalent to convergence at rate O(1/n 2), where n is the number of iterations. We provide a detailed analysis of the eigenvalues of the corresponding linear dynamical system , showing various oscillatory and non-oscillatory behaviors, together with a sharp stability result with explicit constants. We also consider the situation where noisy gradients are available, where we extend our general convergence result, which suggests an alternative algorithm (i.e., with different step sizes) that exhibits the good aspects of both averaging and acceleration.

Conditional Random Fields (CRF) have been widely used in a variety of computer vision tasks. Conventional CRFs typically define edges on neighboring image pixels, resulting in a sparse graph such that efficient inference can be performed. However, these CRFs fail to model long-range contextual relationships. Fully-connected CRFs have thus been proposed. While there are efficient approximate inference methods for such CRFs, usually they are sensitive to initialization and make strong assumptions. In this work, we develop an efficient, yet general algorithm for inference on fully-connected CRFs. The algorithm is based on a scalable SDP algorithm and the low- rank approximation of the similarity/kernel matrix. The core of the proposed algorithm is a tailored quasi-Newton method that takes advantage of the low-rank matrix approximation when solving the specialized SDP dual problem. Experiments demonstrate that our method can be applied on fully-connected CRFs that cannot be solved previously, such as pixel-level image co-segmentation.

We show that unconverged stochastic gradient descent can be interpreted as a procedure that samples from a nonparametric variational approximate posterior distribution. This distribution is implicitly defined as the transformation of an initial distribution by a sequence of optimization updates. By tracking the change in entropy over this sequence of transformations during optimization, we form a scalable, unbiased estimate of the variational lower bound on the log marginal likelihood. We can use this bound to optimize hyperparameters instead of using cross-validation. This Bayesian interpretation of SGD suggests improved, overfitting-resistant optimization procedures, and gives a theoretical foundation for popular tricks such as early stopping and ensembling. We investigate the properties of this marginal likelihood estimator on neural network models.

Finding minima of a real valued non-convex function over a high dimensional space is a major challenge in science. We provide evidence that some such functions that are defined on high dimensional domains have a narrow band of values whose pre-image contains the bulk of its critical points. This is in contrast with the low dimensional picture in which this band is wide. Our simulations agree with the previous theoretical work on spin glasses that proves the existence of such a band when the dimension of the domain tends to infinity. Furthermore our experiments on teacher-student networks with the MNIST dataset establish a similar phenomenon in deep networks. We finally observe that both the gradient descent and the stochastic gradient descent methods can reach this level within the same number of steps.