Understanding transition events between metastable states is of great importance in the applied sciences. Wellknown examples of the transition events include nucleation events during phase transitions, conformational changes of bio-molecules, dislocation dynamics in crystalline solids, etc. The long time scale associated with these events is a consequence of the disparity between the effective thermal energy and typical energy barrier of the systems. The dynamics proceeds by long waiting periods around metastable states followed by sudden jumps from one state to another. For this reason, the transition event is called rare event. The main objective in the study of rare events is to understand the transition mechanism, such as the transition pathway and transition states.

Kernel ridge regression (KRR) is a well-known and popular nonparametric regression approach with many desirable properties, including minimax rate-optimality in estimating functions that belong to common reproducing kernel Hilbert spaces (RKHS). The approach, however, is computationally intensive for large data sets, due to the need to operate on a dense $n \times n$ kernel matrix, where $n$ is the sample size. Recently, various approximation schemes for solving KRR have been considered, and some analyzed. Some approaches such as Nystr\"{o}m approximation and sketching have been shown to preserve the rate optimality of KRR. In this paper, we consider the simplest approximation, namely, spectrally truncating the kernel matrix to its largest $r < n$ eigenvalues. We derive an exact expression for the maximum risk of this truncated KRR, over the unit ball of the RKHS. This result can be used to study the exact trade-off between the level of spectral truncation and the regularization parameter of the KRR. We show that, as long as the RKHS is infinite-dimensional, there is a threshold on $r$, above which, the spectrally-truncated KRR, surprisingly, outperforms the full KRR in terms of the minimax risk, where the minimum is taken over the regularization parameter. This strengthens the existing results on approximation schemes, by showing that not only one does not lose in terms of the rates, truncation can in fact improve the performance, for all finite samples (above the threshold). In other words, there is nothing to be gained by running the full KRR and one should always truncate. Our proof is elementary and distribution-free, only requiring the noise vector to be isotropic.

In classical federated learning a central server coordinates the training of a single model on a massively distributed network of devices. This setting can be naturally extended to a multi-task learning framework, to handle real-world federated datasets that typically show strong non-IID data distributions among devices. Even though federated multi-task learning has been shown to be an effective paradigm for real world datasets, it has been applied only to convex models. In this work we introduce VIRTUAL, an algorithm for federated multi-task learning with non-convex models. In VIRTUAL the federated network of the server and the clients is treated as a star-shaped Bayesian network, and learning is performed on the network using approximated variational inference. We show that this method is effective on real-world federated datasets, outperforming the current state-of-the-art for federated learning.

Active learning is an important technique to reduce the number of labeled examples in supervised learning. Active learning for binary classification has been well addressed. However, active learning of reject option classifier is still an unsolved problem. In this paper, we propose novel algorithms for active learning of reject option classifiers. We develop an active learning algorithm using double ramp loss function. We provide mistake bounds for this algorithm. We also propose a new loss function called double sigmoid loss function for reject option and corresponding active learning algorithm. We provide extensive experimental results to show the effectiveness of the proposed algorithms. The proposed algorithms efficiently reduce the number of label examples required.

This paper describes a new method, HMM gauge likelihood analysis, or GLA, of detecting anomalies in discrete time series using Hidden Markov Models and clustering. At the center of the method lies the comparison of subsequences. To achieve this, they first get assigned to their Hidden Markov Models using the Baum-Welch algorithm. Next, those models are described by an approximating representation of the probability distributions they define. Finally, this representation is then analyzed with the help of some clustering technique or other outlier detection tool and anomalies are detected. Clearly, HMMs could be substituted by some other appropriate model, e.g. some other dynamic Bayesian network. Our learning algorithm is unsupervised, so it does not require the labeling of large amounts of data. The usability of this method is demonstrated by applying it to synthetic and real-world syslog data.

Manifold regularization is a commonly used technique in semi-supervised learning. It guides the learning process by enforcing that the classification rule we find is smooth with respect to the data-manifold. In this paper we present sample and Rademacher complexity bounds for this method. We first derive distribution \emph{independent} sample complexity bounds by analyzing the general framework of adding a data dependent regularization term to a supervised learning process. We conclude that for these types of methods one can expect that the sample complexity improves at most by a constant, which depends on the hypothesis class. We then derive Rademacher complexities bounds which allow for a distribution \emph{dependent} complexity analysis. We illustrate how our bounds can be used for choosing an appropriate manifold regularization parameter. With our proposed procedure there is no need to use an additional labeled validation set.

We introduce Gluon Time Series (GluonTS, available at https://gluon-ts.mxnet.io), a library for deep-learning-based time series modeling. GluonTS simplifies the development of and experimentation with time series models for common tasks such as forecasting or anomaly detection. It provides all necessary components and tools that scientists need for quickly building new models, for efficiently running and analyzing experiments and for evaluating model accuracy.

In human perception and cognition, the fundamental operation that brains perform is interpretation: constructing coherent neural states from noisy, incomplete, and intrinsically ambiguous evidence. The problem of interpretation is well matched to an early and often overlooked architecture, the attractor network---a recurrent neural network that performs constraint satisfaction, imputation of missing features, and clean up of noisy data via energy minimization dynamics. We revisit attractor nets in light of modern deep learning methods, and propose a convolutional bipartite architecture with a novel training loss, activation function, and connectivity constraints. We tackle problems much larger than have been previously explored with attractor nets and demonstrate their potential for image denoising, completion, and super-resolution. We argue that this architecture is better motivated than ever-deeper feedforward models and is a viable alternative to more costly sampling-based methods on a range of supervised and unsupervised tasks.

Discovery of causal relations from observational data is essential for many disciplines of science and real-world applications. However, unlike traditional machine learning algorithms, whose developments have been greatly fostered by a large amount of available benchmark datasets, causal discovery algorithms are notoriously difficult to be systematically evaluated due to the fact that few datasets with known ground-truth causal relations are available. In this work, we handle the problem of evaluating causal discovery algorithms by building a flexible simulator in the medical setting. We develop a neuropathic pain simulator, inspired by the fact that the biological processes of neuropathic pathophysiology are well studied with well-understood causal influences. Our simulator exploits the causal graph of the neuropathic pain pathology, and its parameters in the generator are estimated from real-life patient cases. We show that data generated from our simulator have the same statistics as real-world data. As a clear advantage, the simulator can produce infinite samples without jeopardizing the privacy of real-world patients. Our simulator provides a natural tool for evaluating various types of causal discovery algorithms, including those to deal with practical issues in causal discovery, such as unknown confounders, selection bias, and missing data. Using our simulator, we have evaluated extensively causal discovery algorithms under various settings.

Variational autoencoders (VAEs) have become one of the most popular deep learning approaches to unsupervised learning and data generation. However, traditional VAEs suffer from the constraint that the latent space must distributionally match a simple prior (e.g. normal, uniform), independent of the initial data distribution. This leads to a number of issues around modeling manifold data, as there is no function with a bounded Jacobian that maps a normal distribution to certain manifolds (e.g. a hypersphere). Similarly, there are not many theoretical guarantees on the encoder and decoder created by the VAE. In this work, we propose a variational autoencoder that maps manifold valued data to its diffusion map coordinates in the latent space, resamples in a neighborhood around a given point in the latent space, and learns a decoder that maps the newly resampled points back to the manifold. The framework is built off of SpectralNet [Shaham et al., 2018a] and is capable of learning this data dependent latent space without computing the eigenfunction of the Laplacian explicitly. We prove that our method is capable of learning a locally bi-Lipschitz map between the manifold and the latent space, and that our resampling method around a point in the latent space $\psi(x)$ maps points back to the manifold around the point $x$, specifically into a neighborbood on the tangent space at the point $x$ on the manifold. We also provide empirical evidence of the benefits of using a diffusion map latent space on manifold data.