The developments of Rademacher complexity and PAC-Bayesian theory have been largely independent. One exception is the PAC-Bayes theorem of Kakade, Sridharan, and Tewari (2008), which is established via Rademacher complexity theory by viewing Gibbs classifiers as linear operators. The goal of this paper is to extend this bridge between Rademacher complexity and state-of-the-art PAC-Bayesian theory. We first demonstrate that one can match the fast rate of Catoni's PAC-Bayes bounds (Catoni, 2007) using shifted Rademacher processes (Wegkamp, 2003; Lecu\'{e} and Mitchell, 2012; Zhivotovskiy and Hanneke, 2018). We then derive a new fast-rate PAC-Bayes bound in terms of the "flatness" of the empirical risk surface on which the posterior concentrates. Our analysis establishes a new framework for deriving fast-rate PAC-Bayes bounds and yields new insights on PAC-Bayesian theory.
We develop constructions for exchangeable sequences of point processes that are rendered conditionally-i.i.d. negative binomial processes by a (possibly unknown) random measure called the base measure. Negative binomial processes are useful in Bayesian nonparametrics as models for random multisets, and in applications we are often interested in cases when the base measure itself is difficult to construct (for example when it has countably infinite support). While a finitary construction for an important case (corresponding to a beta process base measure) has appeared in the literature, our constructions generalize to any random base measure, requiring only an exchangeable sequence of Bernoulli processes rendered conditionally-i.i.d. by the same underlying random base measure. Because finitary constructions for such Bernoulli processes are known for several different classes of random base measures--including generalizations of the beta process and hierarchies thereof--our results immediately provide constructions for negative binomial processes with a random base measure from any member of these classes.
As the size and complexity of models and datasets grow, so does the need for communication-efficient variants of stochastic gradient descent that can be deployed on clusters to perform model fitting in parallel. Alistarh et al. (2017) describe two variants of data-parallel SGD that quantize and encode gradients to lessen communication costs. For the first variant, QSGD, they provide strong theoretical guarantees. For the second variant, which we call QSGDinf, they demonstrate impressive empirical gains for distributed training of large neural networks. Building on their work, we propose an alternative scheme for quantizing gradients and show that it yields stronger theoretical guarantees than exist for QSGD while matching the empirical performance of QSGDinf.
Recent work on the "lottery ticket hypothesis" proposes that randomly-initialized, dense neural networks contain much smaller, fortuitously initialized subnetworks ("winning tickets") capable of training to similar accuracy as the original network at a similar speed. While strong evidence exists for the hypothesis across many settings, it has not yet been evaluated on large, state-of-the-art networks and there is even evidence against the hypothesis on deeper networks. We modify the lottery ticket pruning procedure to make it possible to identify winning tickets on deeper networks. Rather than set the weights of a winning ticket to their original initializations, we set them to the weights obtained after a small number of training iterations ("late resetting"). Using late resetting, we identify the first winning tickets for Resnet-50 on Imagenet To understand the efficacy of late resetting, we study the "stability" of neural network training to pruning, which we define as the consistency of the optimization trajectories followed by a winning ticket when it is trained in isolation and as part of the larger network. We find that later resetting produces stabler winning tickets and that improved stability correlates with higher winning ticket accuracy. This analysis offers new insights into the lottery ticket hypothesis and the dynamics of neural network learning.
We show that Entropy-SGD (Chaudhari et al., 2017), when viewed as a learning algorithm, optimizes a PAC-Bayes bound on the risk of a Gibbs (posterior) classifier, i.e., a randomized classifier obtained by a risk-sensitive perturbation of the weights of a learned classifier. Entropy-SGD works by optimizing the bound's prior, violating the hypothesis of the PAC-Bayes theorem that the prior is chosen independently of the data. Indeed, available implementations of Entropy-SGD rapidly obtain zero training error on random labels and the same holds of the Gibbs posterior. In order to obtain a valid generalization bound, we rely on a result showing that data-dependent priors obtained by stochastic gradient Langevin dynamics (SGLD) yield valid PAC-Bayes bounds provided the target distribution of SGLD is $\epsilon$-differentially private. We observe that test error on MNIST and CIFAR10 falls within the (empirically nonvacuous) risk bounds computed under the assumption that SGLD reaches stationarity. In particular, Entropy-SGLD can be configured to yield relatively tight generalization bounds and still fit real labels, although these same settings do not obtain state-of-the-art performance.
The Probably Approximately Correct (PAC) Bayes framework (McAllester, 1999) can incorporate knowledge about the learning algorithm and data distribution through the use of distribution-dependent priors, yielding tighter generalization bounds on data-dependent posteriors. Using this flexibility, however, is difficult, especially when the data distribution is presumed to be unknown. We show how an {\epsilon}-differentially private data-dependent prior yields a valid PAC-Bayes bound, and then show how non-private mechanisms for choosing priors obtain the same generalization bound provided they converge weakly to the private mechanism. As an application of this result, we show that a Gaussian prior mean chosen via stochastic gradient Langevin dynamics (SGLD; Welling and Teh, 2011) leads to a valid PAC-Bayes bound, despite SGLD only converging weakly to an {\epsilon}-differentially private mechanism. As the bounds are data-dependent, we study the bounds empirically on synthetic data and standard neural network benchmarks in order to illustrate the gains of data-dependent priors over existing distribution-dependent PAC-Bayes bound.
A variety of machine learning tasks---e.g., matrix factorization, topic modelling, and feature allocation---can be viewed as learning the parameters of a probability distribution over bipartite graphs. Recently, a new class of models for networks, the sparse exchangeable graphs, have been introduced to resolve some important pathologies of traditional approaches to statistical network modelling; most notably, the inability to model sparsity (in the asymptotic sense). The present paper explains some practical insights arising from this work. We first show how to check if sparsity is relevant for modelling a given (fixed size) dataset by using network subsampling to identify a simple signature of sparsity. We discuss the implications of the (sparse) exchangeable subsampling theory for test-train dataset splitting; we argue common approaches can lead to biased results, and we propose a principled alternative. Finally, we study sparse exchangeable Poisson matrix factorization as a worked example. In particular, we show how to adapt mean field variational inference to the sparse exchangeable setting, allowing us to scale inference to huge datasets.
One of the defining properties of deep learning is that models are chosen to have many more parameters than available training data. In light of this capacity for overfitting, it is remarkable that simple algorithms like SGD reliably return solutions with low test error. One roadblock to explaining these phenomena in terms of implicit regularization, structural properties of the solution, and/or easiness of the data is that many learning bounds are quantitatively vacuous when applied to networks learned by SGD in this "deep learning" regime. Logically, in order to explain generalization, we need nonvacuous bounds. We return to an idea by Langford and Caruana (2001), who used PAC-Bayes bounds to compute nonvacuous numerical bounds on generalization error for stochastic two-layer two-hidden-unit neural networks via a sensitivity analysis. By optimizing the PAC-Bayes bound directly, we are able to extend their approach and obtain nonvacuous generalization bounds for deep stochastic neural network classifiers with millions of parameters trained on only tens of thousands of examples. We connect our findings to recent and old work on flat minima and MDL-based explanations of generalization.
Neural network image classifiers are known to be vulnerable to adversarial images, i.e., natural images which have been modified by an adversarial perturbation specifically designed to be imperceptible to humans yet fool the classifier. Not only can adversarial images be generated easily, but these images will often be adversarial for networks trained on disjoint subsets of data or with different architectures. Adversarial images represent a potential security risk as well as a serious machine learning challenge---it is clear that vulnerable neural networks perceive images very differently from humans. Noting that virtually every image classification data set is composed of JPG images, we evaluate the effect of JPG compression on the classification of adversarial images. For Fast-Gradient-Sign perturbations of small magnitude, we found that JPG compression often reverses the drop in classification accuracy to a large extent, but not always. As the magnitude of the perturbations increases, JPG recompression alone is insufficient to reverse the effect.
We characterize the class of exchangeable feature allocations assigning probability $V_{n,k}\prod_{l=1}^{k}W_{m_{l}}U_{n-m_{l}}$ to a feature allocation of $n$ individuals, displaying $k$ features with counts $(m_{1},\ldots,m_{k})$ for these features. Each element of this class is parametrized by a countable matrix $V$ and two sequences $U$ and $W$ of non-negative weights. Moreover, a consistency condition is imposed to guarantee that the distribution for feature allocations of $n-1$ individuals is recovered from that of $n$ individuals, when the last individual is integrated out. In Theorem 1.1, we prove that the only members of this class satisfying the consistency condition are mixtures of the Indian Buffet Process over its mass parameter $\gamma$ and mixtures of the Beta--Bernoulli model over its dimensionality parameter $N$. Hence, we provide a characterization of these two models as the only, up to randomization of the parameters, consistent exchangeable feature allocations having the required product form.