AISTATS 2019 Papers

We propose Learned Accept/Reject Sampling (LARS), a method for constructing richer priors using rejection sampling with a learned acceptance function. This work is motivated by recent analyses of the VAE objective, which pointed out that commonly used simple priors can lead to underfitting. As the distribution induced by LARS involves an intractable normalizing constant, we show how to estimate it and its gradients efficiently. We demonstrate that LARS priors improve VAE performance on several standard datasets both when they are learned jointly with the rest of the model and when they are fitted to a pretrained model. Finally, we show that LARS can be combined with existing methods for defining flexible priors for an additional boost in performance.

In this work, we study value function approximation in reinforcement learning (RL) problems with high dimensional state or action spaces via a generalized version of representation policy iteration (RPI). We consider the limitations of proto-value functions (PVFs) at accurately approximating the value function in low dimensions and we highlight the importance of features learning for an improved low-dimensional value function approximation. Then, we adopt different representation learning algorithms on graphs to learn the basis functions that best represent the value function. We empirically show that node2vec, an algorithm for scalable feature learning in networks, and Graph Auto-Encoder constantly outperform the commonly used smooth proto-value functions in low-dimensional feature space.

Reparameterizable densities are an important way to learn probability distributions in a deep learning setting. For many distributions it is possible to create low-variance gradient estimators by utilizing a ‘reparameterization trick’. Due to the absence of a general reparameterization trick, much research has recently been devoted to extend the number of reparameterizable distributional families. Unfortunately, this research has primarily focused on distributions defined in Euclidean space, ruling out the usage of one of the most influential class of spaces with non-trivial topologies: Lie groups. In this work we define a general framework to create reparameterizable densities on arbitrary Lie groups, and provide a detailed practitioners guide to further the ease of usage. We demonstrate how to create complex and multimodal distributions on the well known oriented group of 3D rotations, SO{3}, using normalizing flows. Our experiments on applying such distributions in a Bayesian setting for pose estimation on objects with discrete and continuous symmetries, showcase their necessity in achieving realistic uncertainty estimates.

In this paper, we present a Renyi Differentially Private stochastic gradient descent (SGD) algorithm for convex empirical risk minimization. The algorithm uses output perturbation and leverages randomness inside SGD, which creates a "randomized sensitivity", in order to reduce the amount of noise that is added. One of the benefits of output perturbation is that we can incorporate a periodic averaging step that serves to further reduce sensitivity while improving accuracy (reducing the well-known oscillating behavior of SGD near the optimum). Renyi Differential Privacy can be used to provide (epsilon, delta)-differential privacy guarantees and hence provide a comparison with prior work. An empirical evaluation demonstrates that the proposed method outperforms prior methods on differentially private ERM.

We consider the stochastic contextual bandit problem with additional regularization. The motivation comes from problems where the policy of the agent must be close to some baseline policy which is known to perform well on the task. To tackle this problem we use a nonparametric model and propose an algorithm splitting the context space into bins, and solving simultaneously — and independently — regularized multi-armed bandit instances on each bin. We derive slow and fast rates of convergence, depending on the unknown complexity of the problem. We also consider a new relevant margin condition to get problem-independent convergence rates, ending up in intermediate convergence rates interpolating between the aforementioned slow and fast rates.

We study a scenario of active learning where the input space is partitioned into different regions and where a distinct hypothesis is learned for each region. We first introduce a new active learning algorithm (EIWAL), which is an enhanced version of the IWAL algorithm, based on a finer analysis that results in more favorable learning guarantees. Then, we present a new learning algorithm for region-based active learning, ORIWAL, in which either IWAL or EIWAL serve as a subroutine. ORIWAL optimally allocates points to the subroutine algorithm for each region. We give a detailed theoretical analysis of ORIWAL, including generalization error guarantees and bounds on the number of points labeled, in terms of both the hypothesis set used in each region and the probability mass of that region. We also report the results of several experiments for our algorithm which demonstrate substantial benefits over existing non-region-based active learning algorithms, such as IWAL, and over passive learning.

We study generalization properties of kernel regularized least squares regression based on a partitioning approach. We show that optimal rates of convergence are preserved if the number of local sets grows sufficiently slowly with the sample size. Moreover, the partitioning approach can be efficiently combined with local Nyström subsampling, improving computational cost twofold.

We consider the tensor completion problem of predicting the missing entries of a tensor. The commonly used CP model has a triple product form, but an alternate family of quadratic models which are the sum of pairwise products instead of a triple product have emerged from applications such as recommendation systems. Non-convex methods are the method of choice for learning quadratic models, and this work examines their sample complexity and error guarantee. Our main result is that with the number of samples being only linear in the dimension, all local minima of the mean squared error objective are global minima and recover the original tensor. We substantiate our theoretical results with experiments on synthetic and real-world data.

We revisit Rahimi and Recht (2007)’s kernel random Fourier features (RFF) method through the lens of the PAC-Bayesian theory. While the primary goal of RFF is to approximate a kernel, we look at the Fourier transform as a prior distribution over trigonometric hypotheses. It naturally suggests learning a posterior on these hypotheses. We derive generalization bounds that are optimized by learning a pseudo-posterior obtained from a closed-form expression. Based on this study, we consider two learning strategies: The first one finds a compact landmarks-based representation of the data where each landmark is given by a distribution-tailored similarity measure, while the second one provides a PAC-Bayesian justification to the kernel alignment method of Sinha and Duchi (2016).

Despite the raise to fame of stochastic variance reduced methods like SAGA and ProxSVRG, their use in non-smooth optimization is still limited to a few simple cases. Existing methods require to compute the proximal operator of the non-smooth term at each iteration, which, for complex penalties like the total variation, overlapping group lasso or trend filtering, is an iterative process that becomes unfeasible for moderately large problems. In this work we propose and analyze VRTOS, a variance-reduced method to solve problems with an arbitrary number of non-smooth terms. Like other variance reduced methods, it only requires to evaluate one gradient per iteration and converges with a constant step size, and so is ideally suited for large scale applications. Unlike existing variance reduced methods, it admits multiple non-smooth terms whose proximal operator only needs to be evaluated once per iteration. We provide a convergence rate analysis for the proposed methods that achieves the same asymptotic rate as their full gradient variants and illustrate its computational advantage on 4 different large scale datasets.

It has been shown that neural network classifiers are not robust. This raises concerns about their usage in safety-critical systems. We propose in this paper a regularization scheme for ReLU networks which provably improves the robustness of the classifier by maximizing the linear regions of the classifier as well as the distance to the decision boundary. Using our regularization we can even find the minimal adversarial perturbation for a certain fraction of test points for large networks. In the experiments we show that our approach improves upon pure adversarial training both in terms of lower and upper bounds on the robustness and is comparable or better than the state of the art in terms of test error and robustness.

In this paper, we propose the first computationally efficient projection-free algorithm for bandit convex optimization (BCO) with a general convex constraint. We show that our algorithm achieves a sublinear regret of $O(nT^{4/5})$ (where $T$ is the horizon and $n$ is the dimension) for any bounded convex functions with uniformly bounded gradients. We also evaluate the performance of our algorithm against baselines on both synthetic and real data sets for quadratic programming, portfolio selection and matrix completion problems.

The projection operation is a crucial step in applying Online Gradient Descent (OGD) and its stochastic version SGD. Unfortunately, in some cases, projection is computationally demanding and inhibits us from applying OGD. In this work we focus on the special case where the constraint set is smooth and we have an access to gradient and value oracles of the constraint function. Under these assumptions we design a new approximate projection operation that necessitates only logarithmically many calls to these oracles. We further show that combining OGD with this new approximate projection, results in a projection free variant which recovers the standard rates of the fully projected version. This applies to both convex and strongly-convex online settings.

Semantic inpainting is the task of inferring missing pixels in an image given surrounding pixels and high level image semantics. Most semantic inpainting algorithms are deterministic: given an image with missing regions, a single inpainted image is generated. However, there are often several plausible inpaintings for a given missing region. In this paper, we propose a method to perform probabilistic semantic inpainting by building a model, based on PixelCNNs, that learns a distribution of images conditioned on a subset of visible pixels. Experiments on the MNIST and CelebA datasets show that our method produces diverse and realistic inpaintings.

Latent variable models (LVMs) learn probabilistic models of data manifolds lying in an ambient Euclidean space. In a number of applications, a priori known spatial constraints can shrink the ambient space into a considerably smaller manifold. Additionally, in these applications the Euclidean geometry might induce a suboptimal similarity measure, which could be improved by choosing a different metric. Euclidean models ignore such information and assign probability mass to data points that can never appear as data, and vastly different likelihoods to points that are similar under the desired metric.We propose the wrapped Gaussian process latent variable model (WGPLVM), that extends Gaussian process latent variable models to take values strictly on a given Riemannian manifold, making the model blind to impossible data points. This allows non-linear, probabilistic inference of low-dimensional Riemannian submanifolds from data. Our evaluation on diverse datasets show that we improve performance on several tasks, including encoding, visualization and uncertainty quantification.

We propose a novel probabilistic approach to multilevel clustering problems based on composite transportation distance, which is a variant of transportation distance where the underlying metric is Kullback-Leibler divergence. Our method involves solving a joint optimization problem over spaces of probability measures to simultaneously discover grouping structures within groups and among groups. By exploiting the connection of our method to the problem of finding composite transportation barycenters, we develop fast and efficient optimization algorithms even for potentially large-scale multilevel datasets. Finally, we present experimental results with both synthetic and real data to demonstrate the efficiency and scalability of the proposed approach.

In this paper, we propose a flexible method for probabilistic modeling with conditional quantile functions using monotonic regression splines. The shape of the spline is parameterized by a neural network whose parameters are learned by minimizing the continuous ranked probability score. Within this framework, we propose a method for probabilistic time series forecasting, which combines the modeling capacity of recurrent neural networks with the flexibility of a spline-based representation of the output distribution. Unlike methods based on parametric probability density functions and maximum likelihood estimation, the proposed method can flexibly adapt to different output distributions without manual intervention. We empirically demonstrate the effectiveness of the approach on synthetic and real-world data sets.

Classical supervised learning produces unreliable models when training and target distributions differ, with most existing solutions requiring samples from the target domain. We propose a proactive approach which learns a relationship in the training domain that will generalize to the target domain by incorporating prior knowledge of aspects of the data generating process that are expected to differ as expressed in a causal selection diagram. Specifically, we remove variables generated by unstable mechanisms from the joint factorization to yield the Surgery Estimator—an interventional distribution that is invariant to the differences across environments. We prove that the surgery estimator finds stable relationships in strictly more scenarios than previous approaches which only consider conditional relationships, and demonstrate this in simulated experiments. We also evaluate on real world data for which the true causal diagram is unknown, performing competitively against entirely data-driven approaches.

Estimating conditional dependence graphs and precision matrices are some of the most common problems in modern statistics and machine learning. When data are fully observed, penalized maximum likelihood-type estimators have become standard tools for estimating graphical models under sparsity conditions. Extensions of these methods to more complex settings where data are contaminated with additive or multiplicative noise have been developed in recent years. In these settings, however, the relative performance of different methods is not well understood and algorithmic gaps still exist. In particular, in high-dimensional settings these methods require using non-positive semidefinite matrices as inputs, presenting novel optimization challenges. We develop an alternating direction method of multipliers (ADMM) algorithm for these problems, providing a feasible algorithm to estimate precision matrices with indefinite input and potentially nonconvex penalties. We compare this method with existing alternative solutions and empirically characterize the tradeoffs between them. Finally, we use this method to explore the networks among US senators estimated from voting records data.