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Industrywide successes of machine learning at the dawn of the (socalled) big data era has led to an increasing gap between practitioners and theoreticians. The former are using offtheshelf statistical and machine learning methods, while the latter are designing and studying the mathematical properties of such algorithms. The tradeoff between those two movements is somewhat addressed by Bayesian researchers, where sound mathematical guarantees often meet efficient implementation and provide model selection criteria. In the late 90s, a new paradigm has emerged in the statistical learning community, used to derive probably approximately correct (PAC) bounds on Bayesianflavored estimators. This PACBayesian theory has been pioneered by ShaweTaylor and Willamson (1997), and McAllester (1998, 1999). It has been extensively formalized by Catoni (2004, 2007) and has triggered, slowly but surely, increasing research efforts during last decades.
We believe it is time to pinpoint the current PACBayesian trends relatively to other modern approaches in the (statistical) machine learning community. Indeed, we observe that, while the field grows by its own, it took some undesirable distance from some related areas. Firstly, it seems to us that the relation to Bayesian methods has been forsaken in numerous works, despite the potential of PACBayesian theory to bring new insights to the Bayesian community and to go beyond the classical Bayesian/frequentist divide. Secondly, the PACBayesian methods share similarities with other quasiBayesian (or pseudoBayesian) methods studying Bayesian practices from a frequentist standpoint, such as the Minimum Description Length (MDL) principle (Grünwald, 2007). Last but not least, even if some practical and theory grounded learning algorithm has emerged from PACBayesian works, these are almost unused for realworld problems.
In short, this workshop aims at gathering statisticians and machine learning researchers to discuss current trends and the future of {PAC,quasi}Bayesian learning. From a broader perspective, we aim to bridge the gap between several communities that can all benefit from sharper statistical guarantees and sound theorydriven learning algorithms.
References
[1] J. ShaweTaylor and R. Williamson. A PAC analysis of a Bayes estimator. In Proceedings of COLT, 1997.
[2] D. A. McAllester. Some PACBayesian theorems. In Proceedings of COLT, 1998.
[3] D. A. McAllester. PACBayesian model averaging. In Proceedings of COLT, 1999.
[4] O. Catoni. Statistical Learning Theory and Stochastic Optimization. SaintFlour Summer School on Probability Theory 2001 (Jean Picard ed.), Lecture Notes in Mathematics. Springer, 2004.
[5] O. Catoni. PACBayesian supervised classification: the thermodynamics of statistical learning. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 56. Institute of Mathematical Statistics, 2007.
[6] P. D. Grünwald. The Minimum Description Length Principle. The MIT Press, 2007.
Sat 8:30 a.m.  8:30 a.m.
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Overture
(Opening remarks)

Benjamin Guedj, Francis Bach, Pascal Germain 
Sat 8:30 a.m.  9:30 a.m.
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François Laviolette  A Tutorial on PACBayesian Theory (Talk)  Francois Laviolette 
Sat 9:30 a.m.  10:15 a.m.
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Peter Grünwald  A Tight Excess Risk Bound via a Unified PACBayesianRademacherShtarkovMDL Complexity
(Talk)
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Over the last 15 years, machine learning theorists have bounded the performance of empirical risk minimization by (localized) Rademacher complexity; Bayesians with frequentist sympathies have studied Bayesian consistency and rate of convergence theorems, sometimes under misspecification, and PACBayesians have studied convergence properties of generalized Bayesian and Gibbs posteriors. We show that, amazingly, most such bounds readily follow from essentially a single result that bounds excess risk in terms of a novel complexity COMP$(\eta,w)$. which depends on a learning rate $\eta$ and a luckiness function $w$, the latter generalizing the concept of a 'prior'. Depending on the choice of $w$, COMP$(\eta,w)$ specializes to PACBayesian (KL(posteriorprior) complexity, MDL (normalized maximum likelihood) complexity and Rademacher complexity, and the bounds obtained are optimized for generalized Bayes, ERM, penalized ERM (such as Lasso) or other methods. Tuning $\eta$ leads to optimal excess risk convergence rates, even for very large (polynomial entropy) classes which have always been problematic for the PACBayesian approach; the optimal $\eta$ depends on 'fast rate' properties of the domain, such as central, Bernstein and Tsybakov conditions.
Joint work with Nishant Mehta, University of Victoria. See https://arxiv.org/abs/1710.07732

Peter Grünwald 
Sat 11:00 a.m.  11:45 a.m.
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JeanMichel Marin  Some recent advances on Approximate Bayesian Computation techniques
(Talk)
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In an increasing number of application domains, the statistical model is so complex that the pointwise computation of the likelihood is intractable. That is typically the case when the underlying probability distribution involves numerous latent variables. Approximate Bayesian Computation (ABC) is a widely used technique to bypass that difficulty. I will review some recent developments on ABC techniques, emphazing the fact that modern machine learning approaches are useful in this field. Although intrinsically very different of PACBayesian strategies  the choice of a generative model is essential within the ABC paradigm  I will highlight some links between these two methodologies. 
JeanMichel Marin 
Sat 11:45 a.m.  12:05 p.m.
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Contributed talk 1  A PACBayesian Approach to SpectrallyNormalized Margin Bounds for Neural Networks
(Talk)

Behnam Neyshabur 
Sat 2:00 p.m.  2:40 p.m.
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Olivier Catoni  Dimensionfree PACBayesian Bounds
(Talk)
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PACBayesian inequalities have already proved to be a great tool to obtain dimension free generalization bounds, such as margin bounds for Support Vector Machines. In this talk, we will play with PACBayesian inequalities and influence functions to present new robust estimators for the mean of random vectors and random matrices, as well as for linear least squares regression. A common theme of the presentation will be to establish dimension free bounds and to work under mild polynomial moment assumptions regarding the tail of the sample distribution. Joint work with Ilaria Giulini. 
Olivier Catoni 
Sat 2:40 p.m.  3:00 p.m.
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Contributed talk 2  Dimension free PACBayesian bounds for the estimation of the mean of a random vector
(Talk)

Olivier Catoni 
Sat 3:30 p.m.  4:15 p.m.
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Yevgeny Seldin  A Strongly Quasiconvex PACBayesian Bound
(Talk)
»
We propose a new PACBayesian bound and a way of constructing a hypothesis space, so that the bound is convex in the posterior distribution and also convex in a tradeoff parameter between empirical performance of the posterior distribution and its complexity. The complexity is measured by the KullbackLeibler divergence to a prior. We derive an alternating procedure for minimizing the bound. We show that the bound can be rewritten as a onedimensional function of the tradeoff parameter and provide sufficient conditions under which the function has a single global minimum. When the conditions are satisfied the alternating minimization is guaranteed to converge to the global minimum of the bound. We provide experimental results demonstrating that rigorous minimization of the bound is competitive with crossvalidation in tuning the tradeoff between complexity and empirical performance. In all our experiments the tradeoff turned to be quasiconvex even when the sufficient conditions were violated. Joint work with Niklas Thiemann, Christian Igel, and Olivier Wintenberger. 
Yevgeny Seldin 
Sat 4:15 p.m.  5:00 p.m.
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John ShaweTaylor  Distribution Dependent Priors for Stable Learning (Talk)  John ShaweTaylor 
Sat 5:00 p.m.  5:30 p.m.
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Daniel Roy  Deep Neural Networks: From Flat Minima to Numerically Nonvacuous Generalization Bounds via PACBayes
(Talk)
»
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. I will discuss recent work using PACBayesian bounds and optimization to arrive at nonvacuous generalization bounds for neural networks 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 MDLbased explanations of generalization, as well as to variational inference for deep learning. Time permitting, I'll discuss new work interpreting EntropySGD as a PACBayesian method. Joint work with Gintare Karolina Dziugaite, based on https://arxiv.org/abs/1703.11008 
Dan Roy 
Sat 5:30 p.m.  6:25 p.m.
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Neil Lawrence, Francis Bach and François Laviolette (Discussion)  Neil Lawrence, Francis Bach, Francois Laviolette 
Sat 6:25 p.m.  6:25 p.m.
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Concluding remarks

Francis Bach, Benjamin Guedj, Pascal Germain 
Author Information
Benjamin Guedj (Inria & University College London)
Benjamin Guedj is a tenured research scientist at Inria since 2014, member of the MODAL projectteam (MOdels for Data Analysis and Learning) of the Lille  Nord Europe research centre in France. He is also affiliated with the mathematics department of the University of Lille. He obtained his Ph.D. in mathematics in 2013 from UPMC (Université Pierre & Marie Curie, France) under the supervision of Gérard Biau and Éric Moulines. Prior to that, he was a research assistant at DTU Compute (Denmark). His main line of research is in statistical machine learning, both from theoretical and algorithmic perspectives. He is primarily interested in the design, analysis and implementation of statistical machine learning methods for high dimensional problems, mainly using the PACBayesian theory.
Pascal Germain (INRIA Paris)
Francis Bach (Inria)
Francis Bach is a researcher at INRIA, leading since 2011 the SIERRA projectteam, which is part of the Computer Science Department at Ecole Normale Supérieure in Paris, France. After completing his Ph.D. in Computer Science at U.C. Berkeley, he spent two years at Ecole des Mines, and joined INRIA and Ecole Normale Supérieure in 2007. He is interested in statistical machine learning, and especially in convex optimization, combinatorial optimization, sparse methods, kernelbased learning, vision and signal processing. He gave numerous courses on optimization in the last few years in summer schools. He has been program cochair for the International Conference on Machine Learning in 2015.
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