Recently, several studies have proven the global convergence and generalization abilities of the gradient descent method for two-layer ReLU networks by making a positivity assumption of the Gram-matrix of the neural tangent kernel. However, the performance of gradient descent on classification problems has not been well studied, and further investigation of the problem structure is possible. In this work, we present a partially stronger but reasonable assumption for binary classification problems compared to the positivity assumption of the Gram-matrix, where a data distribution can be perfectly classifiable by a tangent model, and we provide a refined generalization analysis of the gradient descent method for two-layer networks with smooth activations. A remarkable point of this study is that our generalization bound has much better dependence on the network width compared to existing results. As a result, our theory significantly enlarges a class of over-parameterized networks having provable generalization ability, with respect to network width, while most studies require much higher over-parameterization.

Accurate estimation of the run time of computational codes has a number of significant advantages for scientific computing. It is required information for optimal resource allocation, improving turnaround times and utilization of science gateways. Furthermore, it allows users to better plan and schedule their research, streamlining workflows and improving the overall productivity of cyberinfrastructure. Predicting run time is challenging, however. The inputs to scientific codes can be complex and high dimensional. Their relationship to the run time may be highly non-linear, and, in the most general case is completely arbitrary and thus unpredictable (i.e., simply a random mapping from inputs to run time). Most codes are not so arbitrary, however, and there has been significant prior research on predicting the run time of applications and workloads. Such predictions are generally application-specific, however. In this paper, we focus on the Gaussian computational chemistry code. We characterize a data set of runs from the SEAGrid science gateway with a number of different studies. We also explore a number of different potential regression methods and present promising future directions.

We propose a scalable framework for inference in an inhomogeneous Poisson process modeled by a continuous sigmoidal Cox process that assumes the corresponding intensity function is given by a Gaussian process (GP) prior transformed with a scaled logistic sigmoid function. We present a tractable representation of the likelihood through augmentation with a superposition of Poisson processes. This view enables a structured variational approximation capturing dependencies across variables in the model. Our framework avoids discretization of the domain, does not require accurate numerical integration over the input space and is not limited to GPs with squared exponential kernels. We evaluate our approach on synthetic and real-world data showing that its benefits are particularly pronounced on multivariate input settings where it overcomes the limitations of mean-field methods and sampling schemes. We provide the state of-the-art in terms of speed, accuracy and uncertainty quantification trade-offs.

Optimal transport costs, which include the Wasserstein Distance and the Earth-Mover-Distance as special cases, have become useful tools in machine learning and statistics [16, 3, 1, 18, 8, 6]. The optimal transport cost between two distributions is computed (in its primal form) as a minimization problem, in which the cost of transporting one distribution to another is minimized over all possible joint distributions, leading to linear program (see for example,[26]). Optimal transport provides great flexibility when comparing (probability) measures and histograms. The transportation cost function (which we refer to as the cost function) can be used to capture key geometric characteristics [16]. It can be also used to compare discrete vs continuous distributions directly, without introducing smoothing, in contrast to alternatives such as the Kullback-Leibler divergence (see [3, 13] for more details). Also, by judiciously choosing the cost function, a Wasserstein distance can generate either the topology corresponding to weak convergence or the total variation distance. In data-driven applications, one needs to estimate the optimal transport cost by means of sampled data.

Human learners have the natural ability to use knowledge gained in one setting for learning in a different but related setting. This ability to transfer knowledge from one task to another is essential for effective learning. In this paper, we study transfer learning in the context of nonparametric classification based on observations from different distributions under the posterior drift model, which is a general framework and arises in many practical problems. We first establish the minimax rate of convergence and construct a rate-optimal two-sample weighted $K$-NN classifier. The results characterize precisely the contribution of the observations from the source distribution to the classification task under the target distribution. A data-driven adaptive classifier is then proposed and is shown to simultaneously attain within a logarithmic factor of the optimal rate over a large collection of parameter spaces. Simulation studies and real data applications are carried out where the numerical results further illustrate the theoretical analysis. Extensions to the case of multiple source distributions are also considered.

Gaussian Process Factor Analysis (GPFA) has been broadly applied to the problem of identifying smooth, low-dimensional temporal structure underlying large-scale neural recordings. However, spike trains are non-Gaussian, which motivates combining GPFA with discrete observation models for binned spike count data. The drawback to this approach is that GPFA priors are not conjugate to count model likelihoods, which makes inference challenging. Here we address this obstacle by introducing a fast, approximate inference method for non-conjugate GPFA models. Our approach uses orthogonal second-order polynomials to approximate the nonlinear terms in the non-conjugate log-likelihood, resulting in a method we refer to as polynomial approximate log-likelihood (PAL) estimators. This approximation allows for accurate closed-form evaluation of marginal likelihood and fast numerical optimization for parameters and hyperparameters. We derive PAL estimators for GPFA models with binomial, Poisson, and negative binomial observations, and additionally show that the parameters obtained can be used to initialize black-box variational inference, which significantly speeds up and stabilizes the inference procedure for these factor analytic models. We apply these methods to data from mouse visual cortex and monkey higher-order visual and parietal cortices, and compare GPFA under three different spike count observation models to traditional GPFA. We demonstrate that PAL estimators achieve fast and accurate extraction of latent structure from multi-neuron spike train data.

Minimax linkage was first introduced by Ao et al. [3] in 2004, as an alternative to standard linkage methods used in hierarchical clustering. Minimax linkage relies on distances to a prototype for each cluster; this prototype can be thought of as a representative object in the cluster, hence improving the interpretability of clustering results. Bien and Tibshirani analyzed properties of this method in 2011 [2], popularizing the method within the statistics community. Additionally, they performed comparisons of minimax linkage to standard linkage methods, making use of five data sets and two different evaluation metrics (distance to prototype and misclassification rate). In an effort to expand upon their work and evaluate minimax linkage more comprehensively, our benchmark study focuses on thorough method evaluation via multiple performance metrics on several well-described data sets. We also make all code and data publicly available through an R package, for full reproducibility. Similarly to [2], we find that minimax linkage often produces the smallest maximum minimax radius of all linkage methods, meaning that minimax linkage produces clusters where objects in a cluster are tightly clustered around their prototype. This is true across a range of values for the total number of clusters (k). However, this is not always the case, and special attention should be paid to the case when k is the true known value. For true k, minimax linkage does not always perform the best in terms of all the evaluation metrics studied, including maximum minimax radius. This paper was motivated by the IFCS Cluster Benchmarking Task Force's call for clustering benchmark studies and the white paper [5], which put forth guidelines and principles for comprehensive benchmarking in clustering. Our work is designed to be a neutral benchmark study of minimax linkage.

In this work we show that Evolution Strategies (ES) are a viable method for learning non-differentiable parameters of large supervised models. ES are black-box optimization algorithms that estimate distributions of model parameters; however they have only been used for relatively small problems so far. We show that it is possible to scale ES to more complex tasks and models with millions of parameters. While using ES for differentiable parameters is computationally impractical (although possible), we show that a hybrid approach is practically feasible in the case where the model has both differentiable and non-differentiable parameters. In this approach we use standard gradient-based methods for learning differentiable weights, while using ES for learning non-differentiable parameters - in our case sparsity masks of the weights. This proposed method is surprisingly competitive, and when parallelized over multiple devices has only negligible training time overhead compared to training with gradient descent. Additionally, this method allows to train sparse models from the first training step, so they can be much larger than when using methods that require training dense models first.

Compared to in-clinic balance training, in-home training is not as effective. This is, in part, due to the lack of feedback from physical therapists (PTs). Here, we analyze the feasibility of using trunk sway data and machine learning (ML) techniques to automatically evaluate balance, providing accurate assessments outside of the clinic. We recruited sixteen participants to perform standing balance exercises. For each exercise, we recorded trunk sway data and had a PT rate balance performance on a scale of 1 to 5. The rating scale was adapted from the Functional Independence Measure. From the trunk sway data, we extracted a 61-dimensional feature vector representing performance of each exercise. Given these labeled data, we trained a multi-class support vector machine (SVM) to map trunk sway features to PT ratings. Evaluated in a leave-one-participant-out scheme, the model achieved a classification accuracy of 82%. Compared to participant self-assessment ratings, the SVM outputs were significantly closer to PT ratings. The results of this pilot study suggest that in the absence of PTs, ML techniques can provide accurate assessments during standing balance exercises. Such automated assessments could reduce PT consultation time and increase user compliance outside of the clinic.

Ensemble models refer to methods that combine a typically large number of classifiers into a compound prediction. The output of an ensemble method is the result of fitting a base-learning algorithm to a given data set, and obtaining diverse answers by reweighting the observations or by resampling them using a given probabilistic selection. A key challenge of using ensembles in large-scale multidimensional data lies in the complexity and the computational burden associated with them. The models created by ensembles are often difficult, if not impossible, to interpret and their implementation requires more computational power than single classifiers. Recent research effort in the field has concentrated in reducing ensemble size, while maintaining their predictive accuracy. We propose a method to prune an ensemble solution by optimizing its margin distribution, while increasing its diversity. The proposed algorithm results in an ensemble that uses only a fraction of the original classifiers, with improved or similar generalization performance. We analyze and test our method on both synthetic and real data sets. The simulations show that the proposed method compares favorably to the original ensemble solutions and to other existing ensemble pruning methodologies.