Consider the task of matrix estimation in which a dataset $X \in \mathbb{R}^{n\times m}$ is observed with sparsity $p$, and we would like to estimate $\mathbb{E}[X]$, where $\mathbb{E}[X_{ui}] = f(\alpha_u, \beta_i)$ for some Holder smooth function $f$. We consider the setting where the row covariates $\alpha$ are unobserved yet the column covariates $\beta$ are observed. We provide an algorithm and accompanying analysis which shows that our algorithm improves upon naively estimating each row separately when the number of rows is not too small. Furthermore when the matrix is moderately proportioned, our algorithm achieves the minimax optimal nonparametric rate of an oracle algorithm that knows the row covariates. In simulated experiments we show our algorithm outperforms other baselines in low data regimes.

Recently, the information-theoretical framework has been proven to be able to obtain non-vacuous generalization bounds for large models trained by Stochastic Gradient Langevin Dynamics (SGLD) with isotropic noise. In this paper, we optimize the information-theoretical generalization bound by manipulating the noise structure in SGLD. We prove that with constraint to guarantee low empirical risk, the optimal noise covariance is the square root of the expected gradient covariance if both the prior and the posterior are jointly optimized. This validates that the optimal noise is quite close to the empirical gradient covariance. Technically, we develop a new information-theoretical bound that enables such an optimization analysis. We then apply matrix analysis to derive the form of optimal noise covariance. Presented constraint and results are validated by the empirical observations.

We present a variational method for online state estimation and parameter learning in state-space models (SSMs), a ubiquitous class of latent variable models for sequential data. As per standard batch variational techniques, we use stochastic gradients to simultaneously optimize a lower bound on the log evidence with respect to both model parameters and a variational approximation of the states' posterior distribution. However, unlike existing approaches, our method is able to operate in an entirely online manner, such that historic observations do not require revisitation after being incorporated and the cost of updates at each time step remains constant, despite the growing dimensionality of the joint posterior distribution of the states. This is achieved by utilizing backward decompositions of this joint posterior distribution and of its variational approximation, combined with Bellman-type recursions for the evidence lower bound and its gradients. We demonstrate the performance of this methodology across several examples, including high-dimensional SSMs and sequential Variational Auto-Encoders.

We present a framework for transfer learning based on modular variational Gaussian processes (GP). We develop a module-based method that having a dictionary of well fitted GPs, one could build ensemble GP models without revisiting any data. Each model is characterised by its hyperparameters, pseudo-inputs and their corresponding posterior densities. Our method avoids undesired data centralisation, reduces rising computational costs and allows the transfer of learned uncertainty metrics after training. We exploit the augmentation of high-dimensional integral operators based on the Kullback-Leibler divergence between stochastic processes to introduce an efficient lower bound under all the sparse variational GPs, with different complexity and even likelihood distribution. The method is also valid for multi-output GPs, learning correlations a posteriori between independent modules. Extensive results illustrate the usability of our framework in large-scale and multi-task experiments, also compared with the exact inference methods in the literature.

Generative models have been successfully used for generating realistic signals. Because the likelihood function is typically intractable in most of these models, the common practice is to use "implicit" models that avoid likelihood calculation. However, it is hard to obtain theoretical guarantees for such models. In particular, it is not understood when they can globally optimize their non-convex objectives. Here we provide such an analysis for the case of Maximum Mean Discrepancy (MMD) learning of generative models. We prove several optimality results, including for a Gaussian distribution with low rank covariance (where likelihood is inapplicable) and a mixture of Gaussians. Our analysis shows that that the MMD optimization landscape is benign in these cases, and therefore gradient based methods will globally minimize the MMD objective.

Variational Inference (VI) offers a method for approximating intractable likelihoods. In neural VI, inference of approximate posteriors is commonly done using an encoder. Alternatively, encoderless VI offers a framework for learning generative models from data without encountering suboptimalities caused by amortization via an encoder (e.g. in presence of missing or uncertain data). However, in absence of an encoder, such methods often suffer in convergence due to the slow nature of gradient steps required to learn the approximate posterior parameters. In this paper, we introduce Relay VI (RVI), a framework that dramatically improves both the convergence and performance of encoderless VI. In our experiments over multiple datasets, we study the effectiveness of RVI in terms of convergence speed, loss, representation power and missing data imputation. We find RVI to be a unique tool, often superior in both performance and convergence speed to previously proposed encoderless as well as amortized VI models (e.g.

Classification as a supervised learning concept is an important content in machine learning. It aims at categorizing a set of data into classes. There are several commonly-used classification methods nowadays such as k-nearest neighbors, random forest, and support vector machine. Each of them has its own pros and cons, and none of them is invincible for all kinds of problems. In this thesis, we focus on Quadratic Multiform Separation (QMS), a classification method recently proposed by Michael Fan et al. (2019). Its fresh concept, rich mathematical structure, and innovative definition of loss function set it apart from the existing classification methods. Inspired by QMS, we propose utilizing a gradient-based optimization method, Adam, to obtain a classifier that minimizes the QMS-specific loss function. In addition, we provide suggestions regarding model tuning through explorations of the relationships between hyperparameters and accuracies. Our empirical result shows that QMS performs as good as most classification methods in terms of accuracy. Its superior performance is almost comparable to those of gradient boosting algorithms that win massive machine learning competitions.

In this community review report, we discuss applications and techniques for fast machine learning (ML) in science -- the concept of integrating power ML methods into the real-time experimental data processing loop to accelerate scientific discovery. The material for the report builds on two workshops held by the Fast ML for Science community and covers three main areas: applications for fast ML across a number of scientific domains; techniques for training and implementing performant and resource-efficient ML algorithms; and computing architectures, platforms, and technologies for deploying these algorithms. We also present overlapping challenges across the multiple scientific domains where common solutions can be found. This community report is intended to give plenty of examples and inspiration for scientific discovery through integrated and accelerated ML solutions. This is followed by a high-level overview and organization of technical advances, including an abundance of pointers to source material, which can enable these breakthroughs.

Meta-learning methods aim to build learning algorithms capable of quickly adapting to new tasks in low-data regime. One of the main benchmarks of such an algorithms is a few-shot learning problem. In this paper we investigate the modification of standard meta-learning pipeline that takes a multi-task approach during training. The proposed method simultaneously utilizes information from several meta-training tasks in a common loss function. The impact of each of these tasks in the loss function is controlled by the corresponding weight. Proper optimization of these weights can have a big influence on training of the entire model and might improve the quality on test time tasks. In this work we propose and investigate the use of methods from the family of simultaneous perturbation stochastic approximation (SPSA) approaches for meta-train tasks weights optimization. We have also compared the proposed algorithms with gradient-based methods and found that stochastic approximation demonstrates the largest quality boost in test time. Proposed multi-task modification can be applied to almost all methods that use meta-learning pipeline. In this paper we study applications of this modification on Prototypical Networks and Model-Agnostic Meta-Learning algorithms on CIFAR-FS, FC100, tieredImageNet and miniImageNet few-shot learning benchmarks. During these experiments, multi-task modification has demonstrated improvement over original methods. The proposed SPSA-Tracking algorithm shows the largest accuracy boost. Our code is available online.

Spatial and spatiotemporal prediction tasks are common in applications ranging from environmental sciences to archaeology and epidemiology. While sophisticated mathematical frameworks have long been developed in spatial statistics to characterize predictive uncertainties under well-defined mathematical assumptions such as intrinsic stationarity (e.g., Cressie 1993), computational estimation procedures have only been proposed more recently to assess predictive performances of spatial and spatiotemporal prediction models (Brenning 2005, 2012; Pohjankukka, Pahikkala, Nevalainen, and Heikkonen 2017; Roberts, Bahn, Ciuti, Boyce, Elith, Guillera-Arroita, Hauenstein, Lahoz-Monfort, Schröder, Thuiller, Warton, Wintle, Hartig, and Dormann 2017). Although alternatives such as the bootstrap exist since some decades (Efron and Gong 1983; Hand 1997), cross-validation (CV) is a particularly well-established, easy-to-implement algorithm for model assessment of supervised machine-learning models (Efron and Gong 1983, and next section) and model selection (Arlot and Celisse 2010). In its basic form, CV is based on resampling the data without paying attention to any possible dependence structure, which may arise from, e.g., grouped or structured data, or underlying environmental processes inducing some sort of spatial coherence at the landscape scale. In treating dependent observations as independent, or ignoring autocorrelation, CV test samples may in fact be heavily correlated with, or even pseudo-replicates of, the data used for training the model, which introduces a potentially severe bias in assessing the transferability of flexible machine-learning (ML) models.