We propose a fast method with statistical guarantees for learning an exponential family density model where the natural parameter is in a reproducing kernel Hilbert space, and may be infinite-dimensional. The model is learned by fitting the derivative of the log density, the score, thus avoiding the need to compute a normalization constant. Our approach improves the computational efficiency of an earlier solution by using a low-rank, Nystr\"om-like solution. The new solution retains the consistency and convergence rates of the full-rank solution (exactly in Fisher distance, and nearly in other distances), with guarantees on the degree of cost and storage reduction. We evaluate the method in experiments on density estimation and in the construction of an adaptive Hamiltonian Monte Carlo sampler. Compared to an existing score learning approach using a denoising autoencoder, our estimator is empirically more data-efficient when estimating the score, runs faster, and has fewer parameters (which can be tuned in a principled and interpretable way), in addition to providing statistical guarantees.

The goal of two-sample tests is to assess whether two samples, $S_P \sim P^n$ and $S_Q \sim Q^m$, are drawn from the same distribution. Perhaps intriguingly, one relatively unexplored method to build two-sample tests is the use of binary classifiers. In particular, construct a dataset by pairing the $n$ examples in $S_P$ with a positive label, and by pairing the $m$ examples in $S_Q$ with a negative label. If the null hypothesis "$P = Q$" is true, then the classification accuracy of a binary classifier on a held-out subset of this dataset should remain near chance-level. As we will show, such Classifier Two-Sample Tests (C2ST) learn a suitable representation of the data on the fly, return test statistics in interpretable units, have a simple null distribution, and their predictive uncertainty allow to interpret where $P$ and $Q$ differ. The goal of this paper is to establish the properties, performance, and uses of C2ST. First, we analyze their main theoretical properties. Second, we compare their performance against a variety of state-of-the-art alternatives. Third, we propose their use to evaluate the sample quality of generative models with intractable likelihoods, such as Generative Adversarial Networks (GANs). Fourth, we showcase the novel application of GANs together with C2ST for causal discovery.

The abundance of high-dimensional data in the modern sciences has generated tremendous interest in penalized estimators such as the lasso, scaled lasso, square-root lasso, elastic net, and many others. In this paper, we establish a general oracle inequality for prediction in high-dimensional linear regression with such methods. Since the proof relies only on convexity and continuity arguments, the result holds irrespective of the design matrix and applies to a wide range of penalized estimators. Overall, the bound demonstrates that generic estimators can provide consistent prediction with any design matrix. From a practical point of view, the bound can help to identify the potential of specific estimators, and they can help to get a sense of the prediction accuracy in a given application.

I have a question regarding the optimization technique used for updating the weights. People generally use gradient descent for the optimization whether its SGD or adaptive. Why can't we use other techniques like Newton Raphson.

We develop a multi-kernel based regression method for graph signal processing where the target signal is assumed to be smooth over a graph. In multi-kernel regression, an effective kernel function is expressed as a linear combination of many basis kernel functions. We estimate the linear weights to learn the effective kernel function by appropriate regularization based on graph smoothness. We show that the resulting optimization problem is shown to be convex and pro- pose an accelerated projected gradient descent based solution. Simulation results using real-world graph signals show efficiency of the multi-kernel based approach over a standard kernel based approach.

We investigate regularized algorithms combining with projection for least-squares regression problem over a Hilbert space, covering nonparametric regression over a reproducing kernel Hilbert space. We prove convergence results with respect to variants of norms, under a capacity assumption on the hypothesis space and a regularity condition on the target function. As a result, we obtain optimal rates for regularized algorithms with randomized sketches, provided that the sketch dimension is proportional to the effective dimension up to a logarithmic factor. As a byproduct, we obtain similar results for Nystr\"{o}m regularized algorithms. Our results are the first ones with optimal, distribution-dependent rates that do not have any saturation effect for sketched/Nystr\"{o}m regularized algorithms, considering both the attainable and non-attainable cases.

In conventional ODE modelling coefficients of an equation driving the system state forward in time are estimated. However, for many complex systems it is practically impossible to determine the equations or interactions governing the underlying dynamics. In these settings, parametric ODE model cannot be formulated. Here, we overcome this issue by introducing a novel paradigm of nonparametric ODE modelling that can learn the underlying dynamics of arbitrary continuous-time systems without prior knowledge. We propose to learn non-linear, unknown differential functions from state observations using Gaussian process vector fields within the exact ODE formalism. We demonstrate the model's capabilities to infer dynamics from sparse data and to simulate the system forward into future.

The move from hand-designed to learned optimizers in machine learning has been quite successful for gradient-based and -free optimizers. When facing a constrained problem, however, maintaining feasibility typically requires a projection step, which might be computationally expensive and not differentiable. We show how the design of projection-free convex optimization algorithms can be cast as a learning problem based on Frank-Wolfe Networks: recurrent networks implementing the Frank-Wolfe algorithm aka. conditional gradients. This allows them to learn to exploit structure when, e.g., optimizing over rank-1 matrices. Our LSTM-learned optimizers outperform hand-designed as well learned but unconstrained ones. We demonstrate this for training support vector machines and softmax classifiers.

We present the first framework for Gaussian-process-modulated Poisson processes when the temporal data appear in the form of panel counts. Panel count data frequently arise when experimental subjects are observed only at discrete time points and only the numbers of occurrences of the events between subsequent observation times are available. The exact occurrence timestamps of the events are unknown. The method of conducting the efficient variational inference is presented, based on the assumption of a Gaussian-process-modulated intensity function. We derive a tractable lower bound to alleviate the problems of the intractable evidence lower bound inherent in the variational inference framework. Our algorithm outperforms classical methods on both synthetic and three real panel count sets.

In this paper, we introduce the use of a personalized Gaussian Process model (pGP) to predict the key metrics of Alzheimer's Disease progression (MMSE, ADAS-Cog13, CDRSB and CS) based on each patient's previous visits. We start by learning a population-level model using multi-modal data from previously seen patients using the base Gaussian Process (GP) regression. Then, this model is adapted sequentially over time to a new patient using domain adaptive GPs to form the patient's pGP. We show that this new approach, together with an auto-regressive formulation, leads to significant improvements in forecasting future clinical status and cognitive scores for target patients when compared to modeling the population with traditional GPs.