AISTATS 2019 Papers

This paper focuses on the problem of \emph{constrained} \emph{stochastic} optimization. A zeroth order Frank-Wolfe algorithm is proposed, which in addition to the projection-free nature of the vanilla Frank-Wolfe algorithm makes it gradient free. Under convexity and smoothness assumption, we show that the proposed algorithm converges to the optimal objective function at a rate $O\left(1/T^{1/3}\right)$, where $T$ denotes the iteration count. In particular, the primal sub-optimality gap is shown to have a dimension dependence of $O\left(d^{1/3}\right)$, which is the best known dimension dependence among all zeroth order optimization algorithms with one directional derivative per iteration. For non-convex functions, we obtain the \emph{Frank-Wolfe} gap to be $O\left(d^{1/3}T^{-1/4}\right)$. Experiments on black-box optimization setups demonstrate the efficacy of the proposed algorithm.

{\em “How much is my data worth?”} is an increasingly common question posed by organizations and individuals alike. An answer to this question could allow, for instance, fairly distributing profits among multiple data contributors and determining prospective compensation when data breaches happen. In this paper, we study the problem of \emph{data valuation} by utilizing the Shapley value, a popular notion of value which originated in coopoerative game theory. The Shapley value defines a unique payoff scheme that satisfies many desiderata for the notion of data value. However, the Shapley value often requires \emph{exponential} time to compute. To meet this challenge, we propose a repertoire of efficient algorithms for approximating the Shapley value. We also demonstrate the value of each training instance for various benchmark datasets.

Clustering mixtures of Gaussian distributions is a fundamental and challenging problem. State-of-the-art theoretical work on learning Gaussian mixture models has mostly focused on estimating the mixture parameters, where clustering is given as a byproduct. These methods have focused mostly on improving separation bounds for different mixture classes, and doing so in polynomial time and sample complexity. Less emphasis has been given to aligning these algorithms to the challenges of big data. In this paper, we focus on clustering $n$ samples from an arbitrary mixture of $c$-separated Gaussians in $\mathbb{R}^p$ in time that is linear in $p$ and $n$, and sample complexity that is independent of $p$. Our analysis suggests that for sufficiently separated Gaussians after $o(\log{p})$ random projections a good direction is found that yields a small clustering error. Specifically, for a user-specified error $e$, the expected number of such projections is small and bounded by $o(\ln p)$ when $\gamma\leq c\sqrt{\ln{\ln{p}}}$ and $\gamma=Q^{-1}(e)$ is the separation of the Gaussians with $Q$ as the tail distribution function of the normal distribution. Consequently, the expected overall running time of the algorithm is linear in $n$ and quasi-linear in $p$ at $o(\ln{p})O(np)$, and the sample complexity is independent of $p$. Unlike the methods that are based on $k$-means, our analysis is applicable to any mixture class (spherical or non-spherical). Finally, an extension to $k>2$ components is also provided.

This paper considers the following problem: we are given n coin tosses of coins with monotone increasing probability of getting heads (success). We study the performance of the monotone constrained likelihood estimate, which is equivalent to the estimate produced by isotonic regression. We derive adaptive and non-adaptive bounds on the performance of the isotonic estimate, i.e., we demonstrate that for some probability vectors the isotonic estimate converges much faster than in general. As an application of this framework we propose a two step procedure for the binary monotone single index model, which consists of running LASSO and consequently running an isotonic regression. We provide thorough numerical studies in support of our claims.

We propose a new variant of the top arm identification problem, \emph{top feasible arm identification}, where there are $K$ arms associated with $D$-dimensional distributions and the goal is to find $m$ arms that maximize some known linear function of their means subject to the constraint that their means belong to a given set $P \subset R^D$. This problem has many applications since in many settings, feedback is multi-dimensional and it is of interest to perform \emph{constrained maximization}. We present problem-dependent lower bounds for top feasible arm identification and upper bounds for several algorithms. Our most broadly applicable algorithm, TF-LUCB-B, has an upper bound that is loose by a factor of $O(D \log(K))$. Many problems of practical interest are two-dimensional and, for these, it is loose by a factor of $O(\log(K))$. Finally, we conduct experiments on synthetic and real-world datasets that demonstrate the effectiveness of our algorithms. Our algorithms are superior both in theory and in practice to a naive two-stage algorithm that first identifies the feasible arms and then applies a best arm identification algorithm to the feasible arms.

In this paper, we propose and analyze conservative value iteration, which unifies value iteration, soft value iteration, advantage learning, and dynamic policy programming. Our analysis shows that algorithms using a combination of gap-increasing and max operators are resilient to stochastic errors, but not to non-stochastic errors. In contrast, algorithms using a softmax operator without a gap-increasing operator are less susceptible to all types of errors, but may display poor asymptotic performance. Algorithms using a combination of gap-increasing and softmax operators are much more effective and may asymptotically outperform algorithms with the max operator. Not only do these theoretical results provide a deep understanding of various reinforcement learning algorithms, but they also highlight the effectiveness of gap-increasing operators, as well as the limitations of traditional greedy value updates by the max operator.

We consider the properties of the bootstrap as a tool for inference concerning the eigenvalues of a sample covariance matrix computed from an n x p data matrix X. We focus on the modern framework where p/n is not close to 0 but remains bounded as n and p tend to infinity. Through a mix of numerical and theoretical considerations, we show that the non-parametric bootstrap is not in general a reliable inferential tool in the setting we consider. However, in the case where the population covariance matrix is well-approximated by a finite rank matrix, the non-parametric bootstrap performs as it does in finite dimension.

In this work, we consider the problem of autonomously discovering behavioral abstractions, or options, for reinforcement learning agents. We propose an algorithm that focuses on the termination function, as opposed to - as is common - the policy. The termination function is usually trained to optimize a control objective: an option ought to terminate if another has better value. We offer a different, information-theoretic perspective, and propose that terminations should focus instead on the compressibility of the option’s encoding - arguably a key reason for using abstractions. To achieve this algorithmically, we leverage the classical options framework, and learn the option transition model as a "critic" for the termination function. Using this model, we derive gradients that optimize the desired criteria. We show that the resulting options are non-trivial, intuitively meaningful, and useful for learning.

In variational autoencoders, the prior on the latent codes $z$ is often treated as an afterthought, but the prior shapes the kind of latent representation that the model learns. If the goal is to learn a representation that is interpretable and useful, then the prior should reflect the ways in which the high-level factors that describe the data vary. The “default” prior is a standard normal, but if the natural factors of variation in the dataset exhibit discrete structure or are not independent, then the isotropic-normal prior will actually encourage learning representations that \emph{mask} this structure. To alleviate this problem, we propose using a flexible Bayesian nonparametric hierarchical clustering prior based on the time-marginalized coalescent (TMC). To scale learning to large datasets, we develop a new inducing-point approximation and inference algorithm. We then apply the method without supervision to several datasets and examine the interpretability and practical performance of the inferred hierarchies and learned latent space.

Multi-output regression models must exploit dependencies between outputs to maximise predictive performance. The application of Gaussian processes (GPs) to this setting typically yields models that are computationally demanding and have limited representational power. We present the Gaussian Process Autoregressive Regression (GPAR) model, a scalable multi-output GP model that is able to capture nonlinear, possibly input-varying, dependencies between outputs in a simple and tractable way: the product rule is used to decompose the joint distribution over the outputs into a set of conditionals, each of which is modelled by a standard GP. GPAR’s efficacy is demonstrated on a variety of synthetic and real-world problems, outperforming existing GP models and achieving state-of-the-art performance on established benchmarks.

Testing for conditional independence is a core aspect of constraint-based causal discovery. Although commonly used tests are perfect in theory, they often fail to reject independence in practice—especially when conditioning on multiple variables. We focus on discrete data and propose a new test based on the notion of algorithmic independence that we instantiate using stochastic complexity. Amongst others, we show that our proposed test, SCI, is an asymptotically unbiased as well as L2 consistent estimator for conditional mutual information (CMI). Further, we show that SCI can be reformulated to find a sensible threshold for CMI that works well on limited samples. Empirical evaluation shows that SCI has a lower type II error than commonly used tests. As a result, we obtain a higher recall when we use SCI in causal discovery algorithms, without compromising the precision.

We study the problem of distribution testing when the samples can only be accessed using a locally differentially private mechanism and focus on two representative testing questions of identity (goodness-of-fit) and independence testing for discrete distributions. First, we construct tests that use existing, general-purpose locally differentially private mechanisms such as the popular RAPPOR or the recently introduced Hadamard Response for collecting data and show that our proposed tests are sample optimal, when we insist on using these mechanisms. Next, we allow bespoke mechanisms designed specifically for testing and introduce the Randomized Aggregated Private Testing Optimal Response (RAPTOR) mechanism which is remarkably simple and requires only one bit of communication per sample. We show that our proposed mechanism yields sample-optimal tests, and in particular outperforms any test based on RAPPOR or Hadamard response. A distinguishing feature of our optimal mechanism is that, in contrast to existing mechanisms, it uses public randomness.

The importance of survival analysis in many disciplines (especially in medicine) has led to the development of a variety of approaches to modeling the survival function. Models constructed via various approaches offer different strengths and weaknesses in terms of discriminative performance and calibration, but no one model is best across all datasets or even across all time horizons within a single dataset. Because we require both good calibration and good discriminative performance over different time horizons, conventional model selection and ensemble approaches are not applicable. This paper develops a novel approach that combines the collective intelligence of different underlying survival models to produce a valid survival function that is well-calibrated and offers superior discriminative performance at different time horizons. Empirical results show that our approach provides significant gains over the benchmarks on a variety of real-world datasets.

Learning domain-invariant representations has become a popular approach to unsupervised domain adaptation and is often justified by invoking a particular suite of theoretical results. We argue that there are two significant flaws in such arguments. First, the results in question hold only for a fixed representation and do not account for information lost in non-invertible transformations. Second, domain invariance is often a far too strict requirement and does not always lead to consistent estimation, even under strong and favorable assumptions. In this work, we give generalization bounds for unsupervised domain adaptation that hold for any representation function by acknowledging the cost of non-invertibility. In addition, we show that penalizing distance between densities is often wasteful and propose a bound based on measuring the extent to which the support of the source domain covers the target domain. We perform experiments on well-known benchmarks that illustrate the short-comings of current standard practice.

Sparse regularization is a central technique for both machine learning (to achieve supervised features selection or unsupervised mixture learning) and imaging sciences (to achieve super-resolution). Existing performance guaranties assume a separation of the spikes based on an ad-hoc (usually Euclidean) minimum distance condition, which ignores the geometry of the problem. In this article, we study the BLASSO (i.e. the off-the-grid version of l1 LASSO regularization) and show that the Fisher-Rao distance is the natural way to ensure and quantify support recovery, since it preserves the invariance of the problem under reparameterization. We prove that under mild regularity and curvature conditions, stable support identification is achieved even in the presence of randomized sub-sampled observations (which is the case in compressed sensing or learning scenario). On deconvolution problems, which are translation invariant, this generalizes to the multi-dimensional setting existing results of the literature. For more complex translation-varying problems, such as Laplace transform inversion, this gives the first geometry-aware guarantees for sparse recovery.

We study the problem of subsampling in differential privacy (DP), a question that is the centerpiece behind many successful differentially private machine learning algorithms. Specifically, we provide a tight upper bound on the Renyi Differential Privacy (RDP) [Mironov 2017] parameters for algorithms that: (1) subsample the dataset, and then (2) applies a randomized mechanism M to the subsample, in terms of the RDP parameters of M and the subsampling probability parameter. Our results generalize the moments accounting technique, developed by [Abadi et al. 2016] for the Gaussian mechanism, to any subsampled RDP mechanism.

Constrained submodular function maximization has been used in subset selection problems such as selection of most informative sensor locations. While these models have been quite popular, the solutions obtained via this approach are unstable to perturbations in data defining the submodular functions. Robust submodular maximization has been proposed as a richer model that aims to overcome this discrepancy as well as increase the modeling scope of submodular optimization. In this work, we consider robust submodular maximization with structured combinatorial constraints and give efficient algorithms with provable guarantees. Our approach is applicable to constraints defined by single or multiple matroids, knapsack as well as distributionally robust criteria. We consider both the offline setting where the data defining the problem is known in advance as well as the online setting where the input data is revealed over time. For the offline setting, we give a nearly optimal bi-criteria approximation algorithm that relies on new extensions of the classical greedy algorithm. For the online version of the problem, we give an algorithm that returns a bi-criteria solution with sub-linear regret.

We present Variational Aspect-based Latent Topic Allocation (VALTA), a family of autoencoding topic models that learn aspect-based representations of reviews. VALTA defines a user-item encoder that maps bag-of-words vectors for combined reviews associated with each paired user and item onto structured embeddings, which in turn define per-aspect topic weights. We model individual reviews in a structured manner by inferring an aspect assignment for each sentence in a given review, where the per-aspect topic weights obtained by the user-item encoder serve to define a mixture over topics, conditioned on the aspect. The result is an autoencoding neural topic model for reviews, which can be trained in a fully unsupervised manner to learn topics that are structured into aspects. Experimental evaluation on large number of datasets demonstrates that aspects are interpretable, yield higher coherence scores than non-structured autoencoding topic model variants, and can be utilized to perform aspect-based comparison and genre discovery.

Deep latent-variable models learn representations of high-dimensional data in an unsupervised manner. A number of recent efforts have focused on learning representations that disentangle statistically independent axes of variation by introducing modifications to the standard objective function. These approaches generally assume a simple diagonal Gaussian prior and as a result are not able to reliably disentangle discrete factors of variation. We propose a two-level hierarchical objective to control relative degree of statistical independence between blocks of variables and individual variables within blocks. We derive this objective as a generalization of the evidence lower bound, which allows us to explicitly represent the trade-offs between mutual information between data and representation, KL divergence between representation and prior, and coverage of the support of the empirical data distribution. Experiments on a variety of datasets demonstrate that our objective can not only disentangle discrete variables, but that doing so also improves disentanglement of other variables and, importantly, generalization even to unseen combinations of factors.