AISTATS 2021 Papers

The dominant term in PAC-Bayes bounds is often the Kullback-Leibler divergence between the posterior and prior. For so-called linear PAC-Bayes risk bounds based on the empirical risk of a fixed posterior kernel, it is possible to minimize the expected value of the bound by choosing the prior to be the expected posterior, which we call the

The privacy implications of generative adversarial networks (GANs) are a topic of great interest, leading to several recent algorithms for training GANs with privacy guarantees. By drawing connections to the generalization properties of GANs, we prove that under some assumptions, GAN-generated samples inherently satisfy some (weak) privacy guarantees. First, we show that if a GAN is trained on m samples and used to generate n samples, the generated samples are (epsilon, delta)-differentially-private for (epsilon, delta) pairs where delta scales as O(n/m). We show that under some special conditions, this upper bound is tight. Next, we study the robustness of GAN-generated samples to membership inference attacks. We model membership inference as a hypothesis test in which the adversary must determine whether a given sample was drawn from the training dataset or from the underlying data distribution. We show that this adversary can achieve an area under the ROC curve that scales no better than O(m^{-1/4}).

We study the convergence rates of the EM algorithm for learning two-component mixed linear regression under all regimes of signal-to-noise ratio (SNR). We resolve a long-standing question that many recent results have attempted to tackle: we completely characterize the convergence behavior of EM, and show that the EM algorithm achieves minimax optimal sample complexity under all SNR regimes. In particular, when the SNR is sufficiently large, the EM updates converge to the true parameter $\theta^{*}$ at the standard parametric convergence rate $\calo((d/n)^{1/2})$ after $\calo(\log(n/d))$ iterations. In the regime where the SNR is above $\calo((d/n)^{1/4})$ and below some constant, the EM iterates converge to a $\calo({\rm SNR}^{-1} (d/n)^{1/2})$ neighborhood of the true parameter, when the number of iterations is of the order $\calo({\rm SNR}^{-2} \log(n/d))$. In the low SNR regime where the SNR is below $\calo((d/n)^{1/4})$, we show that EM converges to a $\calo((d/n)^{1/4})$ neighborhood of the true parameters, after $\calo((n/d)^{1/2})$ iterations. Notably, these results are achieved under mild conditions of either random initialization or an efficiently computable local initialization. By providing tight convergence guarantees of the EM algorithm in middle-to-low SNR regimes, we fill the remaining gap in the literature, and significantly, reveal that in low SNR, EM changes rate, matching the $n^{-1/4}$ rate of the MLE, a behavior that previous work had been unable to show.

Tensor-power (TP) recurrent model is a family of non-linear dynamical systems, of which the recurrence relation consists of a p-fold (a.k.a., degree-p) tensor product. Despite such the model frequently appears in the advanced recurrent neural networks (RNNs), to this date there is limited study on its memory property, a critical characteristic in sequence tasks. In this work, we conduct a thorough investigation of the memory mechanism of TP recurrent models. Theoretically, we prove that a large degree p is an essential condition to achieve the long memory effect, yet it would lead to unstable dynamical behaviors. Empirically, we tackle this issue by extending the degree p from discrete to a differentiable domain, such that it is efficiently learnable from a variety of datasets. Taken together, the new model is expected to benefit from the long memory effect in a stable manner. We experimentally show that the proposed model achieves competitive performance compared to various advanced RNNs in both the single-cell and seq2seq architectures.

We revisit the finite time analysis of policy gradient methods in the one of the simplest settings: finite state and action MDPs with a policy class consisting of all stochastic policies and with exact gradient evaluations. There has been some recent work viewing this setting as an instance of smooth non-linear optimization problems, to show sub-linear convergence rates with small step-sizes. Here, we take a completely different perspective based on illuminating connections with policy iteration, to show how many variants of policy gradient algorithms succeed with large step-sizes and attain a linear rate of convergence.

Model-based Reinforcement Learning (MBRL) is a promising framework for learning control in a data-efficient manner. MBRL algorithms can be fairly complex due to the separate dynamics modeling and the subsequent planning algorithm, and as a result, they often possess tens of hyperparameters and architectural choices. For this reason, MBRL typically requires significant human expertise before it can be applied to new problems and domains. To alleviate this problem, we propose to use automatic hyperparameter optimization (HPO). We demonstrate that this problem can be tackled effectively with automated HPO, which we demonstrate to yield significantly improved performance compared to human experts. In addition, we show that tuning of several MBRL hyperparameters dynamically, i.e. during the training itself, further improves the performance compared to using static hyperparameters which are kept fix for the whole training. Finally, our experiments provide valuable insights into the effects of several hyperparameters, such as plan horizon or learning rate and their influence on the stability of training and resulting rewards.

Recent years have witnessed the success of adaptive (or unified) approaches in estimating symmetric properties of discrete distributions, where the learner first obtains a distribution estimator independent of the target property, and then plugs the estimator into the target property as the final estimator. Several such approaches have been proposed and proved to be adaptively optimal, i.e. they achieve the optimal sample complexity for a large class of properties within a low accuracy, especially for a large estimation error $\varepsilon\gg n^{-1/3}$ where $n$ is the sample size. In this paper, we characterize the high accuracy limitation, or the penalty for adaptation, for general adaptive approaches. Specifically, we obtain the first known adaptation lower bound that under a mild condition, any adaptive approach cannot achieve the optimal sample complexity for every $1$-Lipschitz property within accuracy $\varepsilon \ll n^{-1/3}$. In particular, this result disproves a conjecture in [Acharya et al. 2017] that the profile maximum likelihood (PML) plug-in approach is optimal in property estimation for all ranges of $\varepsilon$, and confirms a conjecture in [Han and Shiragur 2020] that their competitive analysis of the PML is tight.

Modern machine learning and deep learning models are shown to be vulnerable when testing data are slightly perturbed. Theoretical studies of adversarial training algorithms mostly focus on their adversarial training losses or local convergence properties. In contrast, this paper studies the generalization performance of a generic adversarial training algorithm. Specifically, we consider linear regression models and two-layer neural networks (with lazy training) using squared loss under low-dimensional regime and high-dimensional regime. In the former regime, after overcoming the non-smoothness of adversarial training, the adversarial risk of the trained models will converge to the minimal adversarial risk. In the latter regime, we discover that data interpolation prevents the adversarial robust estimator from being consistent (i.e. converge in probability). Therefore, inspired by successes of the least absolute shrinkage and selection operator (LASSO), we incorporate the $\mathcal{L}_1$ penalty in the high dimensional adversarial learning, and show that it leads to consistent adversarial robust estimation. A series of numerical studies are conducted to demonstrate that how the smoothness and $\mathcal{L}_1$ penalization help to improve the adversarial robustness of DNN models.

We study alternating least-squares (ALS) for canonical correlation analysis (CCA). Recent research shows that the alternating least-squares solver for k-CCA can be directly accelerated with momentum and prominent performance gain has been observed in practice for the resulting simple algorithm. However, despite the simplicity, it is difficult for the accelerated rate to be analyzed in theory in order to explain and match the empirical performance gain. By looking into two neighboring iterations, in this work, we propose an even simpler variant of the faster alternating least-squares solver. Instead of applying momentum to each update for acceleration, the proposed variant only leverages momentum at every other iteration and can converge at a provably faster linear rate of nearly square-root dependence on the singular value gap of the whitened cross-covariance matrix. In addition to the high consistency between theory and practice, experimental studies also show that our variant of the alternating least-squares algorithm as a block CCA solver is even more pass efficient than other variants.

While auxiliary tasks play a key role in shaping the representations learnt by reinforcement learning agents, much is still unknown about the mechanisms through which this is achieved. This work develops our understanding of the relationship between auxiliary tasks, environment structure, and representations by analysing the dynamics of temporal difference algorithms. Through this approach, we establish a connection between the spectral decomposition of the transition operator and the representations induced by a variety of auxiliary tasks. We then leverage insights from these theoretical results to inform the selection of auxiliary tasks for deep reinforcement learning agents in sparse-reward environments.

We consider the maximum mean discrepancy MMD GAN problem and propose a parametric kernelized gradient flow that mimics the min-max game in gradient regularized MMD GAN. We show that this flow provides a descent direction minimizing the MMD on a statistical manifold of probability distributions. We then derive an explicit condition which ensures that gradient descent on the parameter space of the generator in gradient regularized MMD GAN is globally convergent to the target distribution. Under this condition , we give non asymptotic convergence results for MMD GAN. Another contribution of this paper is the introduction of a dynamic formulation of a regularization of MMD and demonstrating that the parametric kernelized descent for MMD is the gradient flow of this functional with respect to the new Riemannian structure. Our obtained theoretical result allows ones to treat gradient flows for quite general functionals and thus has potential applications to other types of variational inferences on a statistical manifold beyond GANs. Finally, numerical experiments suggest that our parametric kernelized gradient flow stabilizes GAN training and guarantees convergence.

We establish the consistency of K-medoids in the context of metric spaces. We start by proving that K-medoids is asymptotically equivalent to K-means restricted to the support of the underlying distribution under general conditions, including a wide selection of loss functions. This asymptotic equivalence, in turn, enables us to apply the work of Parna (1986) on the consistency of K-means. This general approach applies also to non-metric settings where only an ordering of the dissimilarities is available. We consider two types of ordinal information: one where all quadruple comparisons are available; and one where only triple comparisons are available. We provide some numerical experiments to illustrate our theory.

The restricted isometry property (RIP) is a well-known condition that guarantees the absence of spurious local minima in low-rank matrix recovery problems with linear measurements. In this paper, we introduce a novel property named bound difference property (BDP) to study low-rank matrix recovery problems with nonlinear measurements. Using RIP and BDP jointly, we propose a new criterion to certify the nonexistence of spurious local minima in the rank-1 case, and prove that it leads to a much stronger theoretical guarantee than the existing bounds on RIP.

This paper studies fixed step-size stochastic approximation (SA) schemes, including stochastic gradient schemes, in a Riemannian framework. It is motivated by several applications, where geodesics can be computed explicitly, and their use accelerates crude Euclidean methods. A fixed step-size scheme defines a family of time-homogeneous Markov chains, parametrized by the step-size. Here, using this formulation, non-asymptotic performance bounds are derived, under Lyapunov conditions. Then, for any step-size, the corresponding Markov chain is proved to admit a unique stationary distribution, and to be geometrically ergodic. This result gives rise to a family of stationary distributions indexed by the step-size, which is further shown to converge to a Dirac measure, concentrated at the solution of the problem at hand, as the step-size goes to $0$. Finally, the asymptotic rate of this convergence is established, through an asymptotic expansion of the bias, and a central limit theorem.

Optimal transport (OT) distances are increasingly used as loss functions for statistical inference, notably in the learning of generative models or supervised learning. Yet, the behavior of minimum Wasserstein estimators is poorly understood, notably in high-dimensional regimes or under model misspecification. In this work we adopt the viewpoint of projection robust (PR) OT, which seeks to maximize the OT cost between two measures by choosing a $k$-dimensional subspace onto which they can be projected. Our first contribution is to establish several fundamental statistical properties of PR Wasserstein distances, complementing and improving previous literature that has been restricted to one-dimensional and well-specified cases. Next, we propose the integral PR Wasserstein (IPRW) distance as an alternative to the PRW distance, by averaging rather than optimizing on subspaces. Our complexity bounds can help explain why both PRW and IPRW distances outperform Wasserstein distances empirically in high-dimensional inference tasks. Finally, we consider parametric inference using the PRW distance. We provide an asymptotic guarantee of two types of minimum PRW estimators and formulate a central limit theorem for max-sliced Wasserstein estimator under model misspecification. To enable our analysis on PRW with projection dimension larger than one, we devise a novel combination of variational analysis and statistical theory.

Standard variational schemes for training deep latent variable models rely on biased gradient estimates of the target objective. Techniques based on the Evidence Lower Bound (ELBO), and tighter variants obtained via importance sampling, produce biased gradient estimates of the true log-likelihood. The family of Reweighted Wake-Sleep (RWS) methods further relies on a biased estimator of the inference objective, which biases training of the encoder also. In this work, we show how Multilevel Monte Carlo (MLMC) can provide a natural framework for debiasing these methods with two different estimators. We prove rigorously that this approach yields unbiased gradient estimators with finite variance under reasonable conditions. Furthermore, we investigate methods that can reduce variance and ensure finite variance in practice. Finally, we show empirically that the proposed unbiased estimators outperform IWAE and other debiasing method on a variety of applications at the same expected cost.

We consider learning a sparse pairwise Markov Random Field (MRF) with continuous-valued variables from i.i.d samples. We adapt the algorithm of Vuffray et al. (2019) to this setting and provide finite-sample analysis revealing sample complexity scaling logarithmically with the number of variables, as in the discrete and Gaussian settings. Our approach is applicable to a large class of pairwise MRFs with continuous variables and also has desirable asymptotic properties, including consistency and normality under mild conditions. Further, we establish that the population version of the optimization criterion employed in Vuffray et al. (2019) can be interpreted as local maximum likelihood estimation (MLE). As part of our analysis, we introduce a robust variation of sparse linear regression a‘ la Lasso, which may be of interest in its own right.

Consider the sequential optimization of an expensive to evaluate and possibly non-convex objective function $f$ from noisy feedback, that can be considered as a continuum-armed bandit problem. Upper bounds on the regret performance of several learning algorithms (GP-UCB, GP-TS, and their variants) are known under both a Bayesian (when $f$ is a sample from a Gaussian process (GP)) and a frequentist (when $f$ lives in a reproducing kernel Hilbert space) setting. The regret bounds often rely on the maximal information gain $\gamma_T$ between $T$ observations and the underlying GP (surrogate) model. We provide general bounds on $\gamma_T$ based on the decay rate of the eigenvalues of the GP kernel, whose specialisation for commonly used kernels improves the existing bounds on $\gamma_T$, and subsequently the regret bounds relying on $\gamma_T$ under numerous settings. For the Matérn family of kernels, where the lower bounds on $\gamma_T$, and regret under the frequentist setting, are known, our results close a huge polynomial in $T$ gap between the upper and lower bounds (up to logarithmic in $T$ factors).

Meta-learning has enabled learning statistical models that can be quickly adapted to new prediction tasks. Motivated by use-cases in personalized federated learning, we study the often overlooked aspect of the modern meta-learning algorithms—their data efficiency. To shed more light on which methods are more efficient, we use techniques from algorithmic stability to derive bounds on the transfer risk that have important practical implications, indicating how much supervision is needed and how it must be allocated for each method to attain the desired level of generalization. Further, we introduce a new simple framework for evaluating meta-learning methods under a limit on the available supervision, conduct an empirical study of MAML, Reptile, andProtoNets, and demonstrate the differences in the behavior of these methods on few-shot and federated learning benchmarks. Finally, we propose active meta-learning, which incorporates active data selection into learning-to-learn, leading to better performance of all methods in the limited supervision regime.