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(Almost) 50 shades of Bayesian Learning: PAC-Bayesian trends and insights
Benjamin Guedj · Pascal Germain · Francis Bach

Sat Dec 09 08:00 AM -- 06:30 PM (PST) @ 101 A
Event URL: https://bguedj.github.io/nips2017/50shadesbayesian.html »

Industry-wide successes of machine learning at the dawn of the (so-called) big data era has led to an increasing gap between practitioners and theoreticians. The former are using off-the-shelf 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 Bayesian-flavored estimators. This PAC-Bayesian theory has been pioneered by Shawe-Taylor 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 PAC-Bayesian 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 PAC-Bayesian theory to bring new insights to the Bayesian community and to go beyond the classical Bayesian/frequentist divide. Secondly, the PAC-Bayesian methods share similarities with other quasi-Bayesian (or pseudo-Bayesian) 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 PAC-Bayesian works, these are almost unused for real-world 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 theory-driven learning algorithms.

[1] J. Shawe-Taylor and R. Williamson. A PAC analysis of a Bayes estimator. In Proceedings of COLT, 1997.
[2] D. A. McAllester. Some PAC-Bayesian theorems. In Proceedings of COLT, 1998.
[3] D. A. McAllester. PAC-Bayesian model averaging. In Proceedings of COLT, 1999.
[4] O. Catoni. Statistical Learning Theory and Stochastic Optimization. Saint-Flour Summer School on Probability Theory 2001 (Jean Picard ed.), Lecture Notes in Mathematics. Springer, 2004.
[5] O. Catoni. PAC-Bayesian 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. [iCal]
Overture (Opening remarks)
Benjamin Guedj, Francis Bach, Pascal Germain
Sat 8:30 a.m. - 9:30 a.m. [iCal]
François Laviolette - A Tutorial on PAC-Bayesian Theory (Talk) Francois Laviolette
Sat 9:30 a.m. - 10:15 a.m. [iCal]
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 PAC-Bayesians 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 PAC-Bayesian (KL(posterior||prior) 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 PAC-Bayesian 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. [iCal]

In an increasing number of application domains, the statistical model is so complex that the point-wise 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 PAC-Bayesian strategies - the choice of a generative model is essential within the ABC paradigm - I will highlight some links between these two methodologies.

Jean-Michel Marin
Sat 11:45 a.m. - 12:05 p.m. [iCal]
Contributed talk 1 - A PAC-Bayesian Approach to Spectrally-Normalized Margin Bounds for Neural Networks (Talk)
Behnam Neyshabur
Sat 2:00 p.m. - 2:40 p.m. [iCal]

PAC-Bayesian 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 PAC-Bayesian 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. [iCal]
Contributed talk 2 - Dimension free PAC-Bayesian bounds for the estimation of the mean of a random vector (Talk)
Olivier Catoni
Sat 3:30 p.m. - 4:15 p.m. [iCal]

We propose a new PAC-Bayesian bound and a way of constructing a hypothesis space, so that the bound is convex in the posterior distribution and also convex in a trade-off parameter between empirical performance of the posterior distribution and its complexity. The complexity is measured by the Kullback-Leibler divergence to a prior. We derive an alternating procedure for minimizing the bound. We show that the bound can be rewritten as a one-dimensional function of the trade-off 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 cross-validation in tuning the trade-off between complexity and empirical performance. In all our experiments the trade-off 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. [iCal]
John Shawe-Taylor - Distribution Dependent Priors for Stable Learning (Talk) John Shawe-Taylor
Sat 5:00 p.m. - 5:30 p.m. [iCal]

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 PAC-Bayesian 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 MDL-based explanations of generalization, as well as to variational inference for deep learning. Time permitting, I'll discuss new work interpreting Entropy-SGD as a PAC-Bayesian 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. [iCal]
Neil Lawrence, Francis Bach and François Laviolette (Discussion) Neil Lawrence, Francis Bach, Francois Laviolette
Sat 6:25 p.m. - 6:25 p.m. [iCal]
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 project-team (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 PAC-Bayesian theory.

Pascal Germain (INRIA Paris)
Francis Bach (Inria)

Francis Bach is a researcher at INRIA, leading since 2011 the SIERRA project-team, 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, kernel-based learning, vision and signal processing. He gave numerous courses on optimization in the last few years in summer schools. He has been program co-chair for the International Conference on Machine Learning in 2015.

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