We introduce several novel change of measure inequalities for two families of divergences: $f$-divergences and $α$-divergences. We show how the variational representation for $f$-divergences leads to novel change of measure inequalities. We also present a multiplicative change of measure inequality for $α$-divergences and a generalized version of Hammersley-Chapman-Robbins inequality. Finally, we present several applications of our change of measure inequalities, including PAC-Bayesian bounds for various classes of losses and non-asymptotic intervals for Monte Carlo estimates.
The R package walker extends standard Bayesian general linear models to the case where the effects of the explanatory variables can vary in time. This allows, for example, to model the effects of interventions such as changes in tax policy which gradually increases their effect over time. The Markov chain Monte Carlo algorithms powering the Bayesian inference are based on Hamiltonian Monte Carlo provided by Stan software, using a state space representation of the model to marginalise over the regression coefficients for efficient low-dimensional sampling.
In this article, we describe a {\tt R} package for sampling from an empirical likelihood-based posterior using a Hamiltonian Monte Carlo method. Empirical likelihood-based methodologies have been used in Bayesian modeling of many problems of interest in recent times. This semiparametric procedure can easily combine the flexibility of a non-parametric distribution estimator together with the interpretability of a parametric model. The model is specified by estimating equations-based constraints. Drawing an inference from a Bayesian empirical likelihood (BayesEL) posterior is challenging. The li…
Approximate Bayesian computation (ABC) is a widely used inference method in Bayesian statistics to bypass the point-wise computation of the likelihood. In this paper we develop theoretical bounds for the distance between the statistics used in ABC. We show that some versions of ABC are inherently robust to misspecification. The bounds are given in the form of oracle inequalities for a finite sample size. The dependence on the dimension of the parameter space and the number of statistics is made explicit. The results are shown to be amenable to oracle inequalities in parameter space. We apply o…
We present an R package bssm for Bayesian non-linear/non-Gaussian state space modelling. Unlike the existing packages, bssm allows for easy-to-use approximate inference based on Gaussian approximations such as the Laplace approximation and the extended Kalman filter. The package accommodates also discretely observed latent diffusion processes. The inference is based on fully automatic, adaptive Markov chain Monte Carlo (MCMC) on the hyperparameters, with optional importance sampling post-correction to eliminate any approximation bias. The package implements also a direct pseudo-marginal MCMC a…
We prove semi-empirical concentration inequalities for random variables which are given as possibly nonlinear functions of independent random variables. These inequalities describe concentration of random variable in terms of the data/distribution-dependent Efron-Stein (ES) estimate of its variance and they do not require any additional assumptions on the moments. In particular, this allows us to state semi-empirical Bernstein type inequalities for general functions of unbounded random variables, which gives user-friendly concentration bounds for cases where related methods (e.g. bounded diffe…
We study PAC-Bayesian generalization bounds for Multilayer Perceptrons (MLPs) with the cross entropy loss. Above all, we introduce probabilistic explanations for MLPs in two aspects: (i) MLPs formulate a family of Gibbs distributions, and (ii) minimizing the cross-entropy loss for MLPs is equivalent to Bayesian variational inference, which establish a solid probabilistic foundation for studying PAC-Bayesian bounds on MLPs. Furthermore, based on the Evidence Lower Bound (ELBO), we prove that MLPs with the cross entropy loss inherently guarantee PAC- Bayesian generalization bounds, and minimizin…
We present a new concentration of measure inequality for sums of independent bounded random variables, which we name a split-kl inequality. The inequality is particularly well-suited for ternary random variables, which naturally show up in a variety of problems, including analysis of excess losses in classification, analysis of weighted majority votes, and learning with abstention. We demonstrate that for ternary random variables the inequality is simultaneously competitive with the kl inequality, the Empirical Bernstein inequality, and the Unexpected Bernstein inequality, and in certain regim…
INRIA Paris - École Normale Supérieure（法国国家信息与自动化研究所巴黎分部 - 巴黎高等师范学院）；Google（谷歌）
We exhibit a strong link between frequentist PAC-Bayesian risk bounds and the Bayesian marginal likelihood. That is, for the negative log-likelihood loss function, we show that the minimization of PAC-Bayesian generalization risk bounds maximizes the Bayesian marginal likelihood. This provides an alternative explanation to the Bayesian Occam's razor criteria, under the assumption that the data is generated by an i.i.d distribution. Moreover, as the negative log-likelihood is an unbounded loss function, we motivate and propose a PAC-Bayesian theorem tailored for the sub-gamma loss family, and w…
Inria Lille - Nord Europe（法国国家信息与自动化研究所里尔-北欧中心）；University College London（伦敦大学学院）
Generalised Bayesian learning algorithms are increasingly popular in machine learning, due to their PAC generalisation properties and flexibility. The present paper aims at providing a self-contained survey on the resulting PAC-Bayes framework and some of its main theoretical and algorithmic developments.
LTS2；EPFL（洛桑联邦理工学院）；IDCOM；University of Edinburgh（爱丁堡大学）
Recently the generalization error of deep neural networks has been analyzed through the PAC-Bayesian framework, for the case of fully connected layers. We adapt this approach to the convolutional setting.
Empirically, the PAC-Bayesian analysis is known to produce tight risk bounds for practical machine learning algorithms. However, in its naive form, it can only deal with stochastic predictors while such predictors are rarely used and deterministic predictors often performs well in practice. To fill this gap, we develop a new generalization error bound, the PAC-Bayesian transportation bound, unifying the PAC-Bayesian analysis and the chaining method in view of the optimal transportation. It is the first PAC-Bayesian bound that relates the risks of any two predictors according to their distance,…
Aalborg University（奥尔堡大学）；Laboratoire Signal et Automatique de Lille (CRIStAL)（里尔信号与自动化实验室（CRIStAL））
In this paper we derive a PAC-Bayesian error bound for autonomous stochastic LTI state-space models. The motivation for deriving such error bounds is that they will allow deriving similar error bounds for more general dynamical systems, including recurrent neural networks. In turn, PACBayesian error bounds are known to be useful for analyzing machine learning algorithms and for deriving new ones.
The Bayesian posterior minimizes the "inferential risk" which itself bounds the "predictive risk". This bound is tight when the likelihood and prior are well-specified. However since misspecification induces a gap, the Bayesian posterior predictive distribution may have poor generalization performance. This work develops a multi-sample loss (PAC$^m$) which can close the gap by spanning a trade-off between the two risks. The loss is computationally favorable and offers PAC generalization guarantees. Empirical study demonstrates improvement to the predictive distribution.
The limit of infinite width allows for substantial simplifications in the analytical study of over-parameterised neural networks. With a suitable random initialisation, an extremely large network exhibits an approximately Gaussian behaviour. In the present work, we establish a similar result for a simple stochastic architecture whose parameters are random variables, holding both before and during training. The explicit evaluation of the output distribution allows for a PAC-Bayesian training procedure that directly optimises the generalisation bound. For a large but finite-width network, we sho…
Univ Lyon, UJM-Saint-Etienne, CNRS, Institut d Optique Graduate School, Laboratoire Hubert Curien UMR 5516（里昂大学，圣艾蒂安大学，法国国家科学研究中心，光学研究所研究生院，于贝尔·居里安实验室）；Université Laval（拉瓦尔大学）
In the PAC-Bayesian literature, the C-Bound refers to an insightful relation between the risk of a majority vote classifier (under the zero-one loss) and the first two moments of its margin (i.e., the expected margin and the voters' diversity). Until now, learning algorithms developed in this framework minimize the empirical version of the C-Bound, instead of explicit PAC-Bayesian generalization bounds. In this paper, by directly optimizing PAC-Bayesian guarantees on the C-Bound, we derive self-bounding majority vote learning algorithms. Moreover, our algorithms based on gradient descent are s…
Université Laval（拉瓦尔大学）；Inria（法国国家信息与自动化研究所）；University College London（伦敦大学学院）
We present a comprehensive study of multilayer neural networks with binary activation, relying on the PAC-Bayesian theory. Our contributions are twofold: (i) we develop an end-to-end framework to train a binary activated deep neural network, (ii) we provide nonvacuous PAC-Bayesian generalization bounds for binary activated deep neural networks. Our results are obtained by minimizing the expected loss of an architecture-dependent aggregation of binary activated deep neural networks. Our analysis inherently overcomes the fact that binary activation function is non-differentiable. The performance…
Norwegian University of Science and Technology（挪威科技大学）
We study the problem of matrix completion in this paper. A spectral scaled Student prior is exploited to favour the underlying low-rank structure of the data matrix. We provide a thorough theoretical investigation for our approach through PAC-Bayesian bounds. More precisely, our PAC-Bayesian approach enjoys a minimax-optimal oracle inequality which guarantees that our method works well under model misspecification and under general sampling distribution. Interestingly, we also provide efficient gradient-based sampling implementations for our approach by using Langevin Monte Carlo. More specifi…
Equivariant networks capture the inductive bias about the symmetry of the learning task by building those symmetries into the model. In this paper, we study how equivariance relates to generalization error utilizing PAC Bayesian analysis for equivariant networks, where the transformation laws of feature spaces are determined by group representations. By using perturbation analysis of equivariant networks in Fourier domain for each layer, we derive norm-based PAC-Bayesian generalization bounds. The bound characterizes the impact of group size, and multiplicity and degree of irreducible represen…
Université Laval（拉瓦尔大学）；Univ. of St-Etienne（圣艾蒂安大学）
This paper provides a theoretical analysis of domain adaptation based on the PAC-Bayesian theory. We propose an improvement of the previous domain adaptation bound obtained by Germain et al. in two ways. We first give another generalization bound tighter and easier to interpret. Moreover, we provide a new analysis of the constant term appearing in the bound that can be of high interest for developing new algorithmic solutions.
We propose an extensive analysis of the behavior of majority votes in binary classification. In particular, we introduce a risk bound for majority votes, called the C-bound, that takes into account the average quality of the voters and their average disagreement. We also propose an extensive PAC-Bayesian analysis that shows how the C-bound can be estimated from various observations contained in the training data. The analysis intends to be self-contained and can be used as introductory material to PAC-Bayesian statistical learning theory. It starts from a general PAC-Bayesian perspective and e…
INRIA, SIERRA Project-Team（法国国家信息与自动化研究所，SIERRA项目组）；D.I., Ecole Normale Superieure（巴黎高等师范学校计算机科学系）；Univ Lyon, UJM-Saint-Etienne, CNRS, IOGS, Laboratoire Hubert Curien UMR 5516（里昂大学，圣艾蒂安让·莫内大学，法国国家科学研究中心，IOGS，于贝尔·居里安实验室）；Université Laval（拉瓦尔大学）
We study the issue of PAC-Bayesian domain adaptation: We want to learn, from a source domain, a majority vote model dedicated to a target one. Our theoretical contribution brings a new perspective by deriving an upper-bound on the target risk where the distributions' divergence---expressed as a ratio---controls the trade-off between a source error measure and the target voters' disagreement. Our bound suggests that one has to focus on regions where the source data is informative.From this result, we derive a PAC-Bayesian generalization bound, and specialize it to linear classifiers. Then, we i…
PAC-Bayesian Analysis of the Exploration-Exploitation Trade-off
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Università degli Studi di Milano（米兰大学）；Université Laval（拉瓦尔大学）；University of Leoben（莱奥本大学）；University College London（伦敦大学学院）
Aalborg University（奥尔堡大学）；Laboratoire Signal et Automatique de Lille (CRIStAL)（里尔信号与自动化实验室（CRIStAL））
In this paper we derive a Probably Approxilmately Correct(PAC)-Bayesian error bound for linear time-invariant (LTI) stochastic dynamical systems with inputs. Such bounds are widespread in machine learning, and they are useful for characterizing the predictive power of models learned from finitely many data points. In particular, with the bound derived in this paper relates future average prediction errors with the prediction error generated by the model on the data used for learning. In turn, this allows us to provide finite-sample error bounds for a wide class of learning/system identificatio…
University of Hawaii Manoa（夏威夷大学马诺阿分校）；Coveo Solutions Inc.（Coveo解决方案公司）
We present a proof of a combinatorial conjecture from the second author's Ph.D. thesis. The proof relies on binomial and multinomial sums identities. We also discuss the relevance of the conjecture in the context of PAC-Bayesian machine learning.
INRIA Lille Nord-Europe（法国国家信息与自动化研究所里尔-北欧分部）；Univ. Lille（里尔大学）；CNRS（法国国家科学研究中心）；Centrale Lille（里尔中央理工学院）；CRIStAL - Centre de Recherche en Informatique Signal et Automatique de Lille（里尔信息、信号与自动化研究中心）
In this paper, we improve the PAC-Bayesian error bound for linear regression derived in Germain et al. [10]. The improvements are twofold. First, the proposed error bound is tighter, and converges to the generalization loss with a well-chosen temperature parameter. Second, the error bound also holds for training data that are not independently sampled. In particular, the error bound applies to certain time series generated by well-known classes of dynamical models, such as ARX models.
Bar-Ilan University（巴伊兰大学）；Technion（以色列理工学院）；University of Illinois（伊利诺伊大学）
Generalization bounds which assess the difference between the true risk and the empirical risk have been studied extensively. However, to obtain bounds, current techniques use strict assumptions such as a uniformly bounded or a Lipschitz loss function. To avoid these assumptions, in this paper, we propose a new generalization bound for Bayesian deep nets by exploiting the contractivity of the Log-Sobolev inequalities. Using these inequalities adds an additional loss-gradient norm term to the generalization bound, which is intuitively a surrogate of the model complexity. Empirically, we analyze…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；University College London（伦敦大学学院）；Université Laval（拉瓦尔大学）；Università degli Studi di Milano（米兰大学）；Montanuniversität Leoben（莱奥本矿业大学）
Recent studies have empirically investigated different methods to train stochastic neural networks on a classification task by optimising a PAC-Bayesian bound via stochastic gradient descent. Most of these procedures need to replace the misclassification error with a surrogate loss, leading to a mismatch between the optimisation objective and the actual generalisation bound. The present paper proposes a novel training algorithm that optimises the PAC-Bayesian bound, without relying on any surrogate loss. Empirical results show that this approach outperforms currently available PAC-Bayesian tra…
Abstract We present PAC-Bayesian bounds for the generalisation error of the K-nearest-neighbour classifier (K-NN). This is achieved by casting the K-NN classifier into a kernel space framework in the limit of vanishing kernel bandwidth. We establish a relation between prior measures over the coefficients in the kernel expansion and the induced measure on the weight vectors in kernel space. Defining a sparse prior over the coefficients allows the application of a PAC-Bayesian folk theorem that leads to a generalisation bound that is a function of the number of redundant training examples: those…
A General Framework for the Practical Disintegration of PAC-Bayesian Bounds
Inria（法国国家信息与自动化研究所）；CNRS（法国国家科学研究中心）；Ecole Normale Supérieure（巴黎高等师范学院）；PSL Research University（巴黎文理研究大学）；Université Laval（拉瓦尔大学）；Université Jean Monnet Saint-Étienne（圣艾蒂安让·莫内大学）；Institut d Optique Graduate School（光学研究所研究生院）；Institut Universitaire de France（法兰西大学研究院）
We apply the PAC-Bayes theory to the setting of learning-to-optimize. To the best of our knowledge, we present the first framework to learn optimization algorithms with provable generalization guarantees (PAC-bounds) and explicit trade-off between a high probability of convergence and a high convergence speed. Even in the limit case, where convergence is guaranteed, our learned optimization algorithms provably outperform related algorithms based on a (deterministic) worst-case analysis. Our results rely on PAC-Bayes bounds for general, unbounded loss-functions based on exponential families. By…
The Australian National University（澳大利亚国立大学）；Data61；Inria（法国国家信息与自动化研究所）；University College London（伦敦大学学院）
Conditional Value at Risk (CVaR) is a family of "coherent risk measures" which generalize the traditional mathematical expectation. Widely used in mathematical finance, it is garnering increasing interest in machine learning, e.g., as an alternate approach to regularization, and as a means for ensuring fairness. This paper presents a generalization bound for learning algorithms that minimize the CVaR of the empirical loss. The bound is of PAC-Bayesian type and is guaranteed to be small when the empirical CVaR is small. We achieve this by reducing the problem of estimating CVaR to that of merel…
In this paper, we provide two main contributions in PAC-Bayesian theory for domain adaptation where the objective is to learn, from a source distribution, a well-performing majority vote on a different target distribution. On the one hand, we propose an improvement of the previous approach proposed by Germain et al. (2013), that relies on a novel distribution pseudodistance based on a disagreement averaging, allowing us to derive a new tighter PAC-Bayesian domain adaptation bound for the stochastic Gibbs classifier. We specialize it to linear classifiers, and design a learning algorithm which …
The lottery ticket hypothesis (LTH) has attracted attention because it can explain why over-parameterized models often show high generalization ability. It is known that when we use iterative magnitude pruning (IMP), which is an algorithm to find sparse networks with high generalization ability that can be trained from the initial weights independently, called winning tickets, the initial large learning rate does not work well in deep neural networks such as ResNet. However, since the initial large learning rate generally helps the optimizer to converge to flatter minima, we hypothesize that t…
Université Laval（拉瓦尔大学）；Laboratoire Hubert Curien UMR CNRS 5516（于贝尔·居里安实验室 CNRS UMR 5516）；Univ. Jean Monnet（让·莫内大学）；Aix-Marseille Univ.（艾克斯-马赛大学）；CNRS（法国国家科学研究中心）
The notion of flat minima has played a key role in the generalization studies of deep learning models. However, existing definitions of the flatness are known to be sensitive to the rescaling of parameters. The issue suggests that the previous definitions of the flatness might not be a good measure of generalization, because generalization is invariant to such rescalings. In this paper, from the PAC-Bayesian perspective, we scrutinize the discussion concerning the flat minima and introduce the notion of normalized flat minima, which is free from the known scale dependence issues. Additionally,…
Bosch Center for Artificial Intelligence（博世人工智能中心）
Gaussian Processes (GPs) are a generic modelling tool for supervised learning. While they have been successfully applied on large datasets, their use in safety-critical applications is hindered by the lack of good performance guarantees. To this end, we propose a method to learn GPs and their sparse approximations by directly optimizing a PAC-Bayesian bound on their generalization performance, instead of maximizing the marginal likelihood. Besides its theoretical appeal, we find in our evaluation that our learning method is robust and yields significantly better generalization guarantees than …
United States CCDC Army Research Laboratory（美国陆军作战能力发展司令部研究实验室）；Indian Institute of Technology Palakkad（印度理工学院帕拉卡德校区）
We examine the role of information geometry in the context of classical Cramér-Rao (CR) type inequalities. In particular, we focus on Eguchi's theory of obtaining dualistic geometric structures from a divergence function and then applying Amari-Nagoaka's theory to obtain a CR type inequality. The classical deterministic CR inequality is derived from Kullback-Leibler (KL)-divergence. We show that this framework could be generalized to other CR type inequalities through four examples: $α$-version of CR inequality, generalized CR inequality, Bayesian CR inequality, and Bayesian $α$-CR inequality.…
Toyota Technological Institute at Chicago（芝加哥丰田理工学院）
We present a generalization bound for feedforward neural networks in terms of the product of the spectral norm of the layers and the Frobenius norm of the weights. The generalization bound is derived using a PAC-Bayes analysis.
University of Technology Sydney（悉尼科技大学）；University of California San Diego（加州大学圣地亚哥分校）
PAC-Bayesian is an analysis framework where the training error can be expressed as the weighted average of the hypotheses in the posterior distribution whilst incorporating the prior knowledge. In addition to being a pure generalization bound analysis tool, PAC-Bayesian bound can also be incorporated into an objective function to train a probabilistic neural network, making them a powerful and relevant framework that can numerically provide a tight generalization bound for supervised learning. For simplicity, we call probabilistic neural network learned using training objectives derived from P…
A PAC-Bayesian Analysis of Graph Clustering and Pairwise Clustering
Max Planck Institute for Biological Cybernetics（马克斯·普朗克生物控制论研究所）
我们制定加权图聚类作为一个预测问题：给定一个子集的边缘权重，我们分析的能力图聚类预测剩余的边缘权重。该公式使实际和理论上的比较不同的方法图聚类，以及比较图聚类与其他可能的方式来建模的图。我们采用了PAC-Bayesian协聚类分析（Seldin and Tishby，2008; Seldin，2009）来推导图聚类的PAC-Bayesian泛化界。该界限表明，图聚类应该优化经验数据拟合和聚类在图节点上保留的互信息之间的权衡。从信息理论考虑导出的类似权衡已经显示在实践中产生最先进的结果（Slonim等人，2005年; Yom-Tov和Slonim，2009年）。本文通过提供更好的理论基础，提出形式化的推广保证，并提供更准确的方法来处理有限样本问题，从而支持了经验证据。我们推导出一个界最小化算法，并表明它在现实生活中的问题提供了很好的结果，所得出的PAC贝叶斯界是合理的紧。
