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Many types of regularization schemes have been employed in statistical learning, each one motivated by some assumption about the problem domain. In this paper, we present a unified asymptotic analysis of smooth regularizers, which allows us to see how the validity of these assumptions impacts the success of a particular regularizer. In addition, our analysis motivates an algorithm for optimizing regularization parameters, which in turn can be analyzed within our framework. We apply our analysis to several examples, including hybrid generative-discriminative learning and multi-task learning.
Author Information
Percy Liang (Stanford University)

Percy Liang is an Assistant Professor of Computer Science at Stanford University (B.S. from MIT, 2004; Ph.D. from UC Berkeley, 2011). His research spans machine learning and natural language processing, with the goal of developing trustworthy agents that can communicate effectively with people and improve over time through interaction. Specific topics include question answering, dialogue, program induction, interactive learning, and reliable machine learning. His awards include the IJCAI Computers and Thought Award (2016), an NSF CAREER Award (2016), a Sloan Research Fellowship (2015), and a Microsoft Research Faculty Fellowship (2014).
Francis Bach (INRIA - Ecole Normale Superieure)
Guillaume Bouchard (Xerox Research Center Europe)
Michael Jordan (UC Berkeley)
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2016 Poster: Stochastic Optimization for Large-scale Optimal Transport »
Aude Genevay · Marco Cuturi · Gabriel Peyré · Francis Bach -
2016 Tutorial: Large-Scale Optimization: Beyond Stochastic Gradient Descent and Convexity »
Suvrit Sra · Francis Bach -
2015 : Sharing the "How" (and not the "What") »
Percy Liang -
2015 : Structured Sparsity and convex optimization »
Francis Bach -
2015 Workshop: Non-convex Optimization for Machine Learning: Theory and Practice »
Anima Anandkumar · Niranjan Uma Naresh · Kamalika Chaudhuri · Percy Liang · Sewoong Oh -
2015 : Sharp Analysis of Random Feature Expansions »
Francis Bach -
2015 : Convergence Rates of Kernel Quadrature Rules »
Francis Bach -
2015 Poster: Variational Consensus Monte Carlo »
Maxim Rabinovich · Elaine Angelino · Michael Jordan -
2015 Demonstration: CodaLab Worksheets for Reproducible, Executable Papers »
Percy Liang · Evelyne Viegas -
2015 Poster: On-the-Job Learning with Bayesian Decision Theory »
Keenon Werling · Arun Tejasvi Chaganty · Percy Liang · Christopher Manning -
2015 Spotlight: On-the-Job Learning with Bayesian Decision Theory »
Keenon Werling · Arun Tejasvi Chaganty · Percy Liang · Christopher Manning -
2015 Poster: On the Accuracy of Self-Normalized Log-Linear Models »
Jacob Andreas · Maxim Rabinovich · Michael Jordan · Dan Klein -
2015 Poster: Linear Response Methods for Accurate Covariance Estimates from Mean Field Variational Bayes »
Ryan Giordano · Tamara Broderick · Michael Jordan -
2015 Spotlight: Linear Response Methods for Accurate Covariance Estimates from Mean Field Variational Bayes »
Ryan Giordano · Tamara Broderick · Michael Jordan -
2015 Poster: Estimating Mixture Models via Mixtures of Polynomials »
Sida Wang · Arun Tejasvi Chaganty · Percy Liang -
2015 Poster: Rethinking LDA: Moment Matching for Discrete ICA »
Anastasia Podosinnikova · Francis Bach · Simon Lacoste-Julien -
2015 Poster: Spectral Norm Regularization of Orthonormal Representations for Graph Transduction »
Rakesh Shivanna · Bibaswan K Chatterjee · Raman Sankaran · Chiranjib Bhattacharyya · Francis Bach -
2015 Poster: Learning with Relaxed Supervision »
Jacob Steinhardt · Percy Liang -
2015 Poster: Calibrated Structured Prediction »
Volodymyr Kuleshov · Percy Liang -
2014 Workshop: Advances in Variational Inference »
David Blei · Shakir Mohamed · Michael Jordan · Charles Blundell · Tamara Broderick · Matthew D. Hoffman -
2014 Workshop: Challenges in Machine Learning workshop (CiML 2014) »
Isabelle Guyon · Evelyne Viegas · Percy Liang · Olga Russakovsky · Rinat Sergeev · Gábor Melis · Michele Sebag · Gustavo Stolovitzky · Jaume Bacardit · Michael S Kim · Ben Hamner -
2014 Poster: Communication-Efficient Distributed Dual Coordinate Ascent »
Martin Jaggi · Virginia Smith · Martin Takac · Jonathan Terhorst · Sanjay Krishnan · Thomas Hofmann · Michael Jordan -
2014 Poster: Spectral Methods meet EM: A Provably Optimal Algorithm for Crowdsourcing »
Yuchen Zhang · Xi Chen · Denny Zhou · Michael Jordan -
2014 Poster: Altitude Training: Strong Bounds for Single-Layer Dropout »
Stefan Wager · William S Fithian · Sida Wang · Percy Liang -
2014 Poster: Parallel Double Greedy Submodular Maximization »
Xinghao Pan · Stefanie Jegelka · Joseph Gonzalez · Joseph K Bradley · Michael Jordan -
2014 Spotlight: Spectral Methods meet EM: A Provably Optimal Algorithm for Crowdsourcing »
Yuchen Zhang · Xi Chen · Denny Zhou · Michael Jordan -
2014 Poster: Metric Learning for Temporal Sequence Alignment »
Rémi Lajugie · Damien Garreau · Francis Bach · Sylvain Arlot -
2014 Poster: SAGA: A Fast Incremental Gradient Method With Support for Non-Strongly Convex Composite Objectives »
Aaron Defazio · Francis Bach · Simon Lacoste-Julien -
2014 Poster: Simple MAP Inference via Low-Rank Relaxations »
Roy Frostig · Sida Wang · Percy Liang · Christopher D Manning -
2014 Poster: On the Convergence Rate of Decomposable Submodular Function Minimization »
Robert Nishihara · Stefanie Jegelka · Michael Jordan -
2013 Workshop: Big Learning : Advances in Algorithms and Data Management »
Xinghao Pan · Haijie Gu · Joseph Gonzalez · Sameer Singh · Yucheng Low · Joseph Hellerstein · Derek G Murray · Raghu Ramakrishnan · Michael Jordan · Christopher Ré -
2013 Workshop: Discrete Optimization in Machine Learning: Connecting Theory and Practice »
Stefanie Jegelka · Andreas Krause · Pradeep Ravikumar · Kazuo Murota · Jeffrey A Bilmes · Yisong Yue · Michael Jordan -
2013 Session: Oral Session 10 »
Michael Jordan -
2013 Poster: Non-strongly-convex smooth stochastic approximation with convergence rate O(1/n) »
Francis Bach · Eric Moulines -
2013 Poster: A Comparative Framework for Preconditioned Lasso Algorithms »
Fabian L Wauthier · Nebojsa Jojic · Michael Jordan -
2013 Poster: Information-theoretic lower bounds for distributed statistical estimation with communication constraints »
Yuchen Zhang · John Duchi · Michael Jordan · Martin J Wainwright -
2013 Oral: Information-theoretic lower bounds for distributed statistical estimation with communication constraints »
Yuchen Zhang · John Duchi · Michael Jordan · Martin J Wainwright -
2013 Spotlight: Non-strongly-convex smooth stochastic approximation with convergence rate O(1/n) »
Francis Bach · Eric Moulines -
2013 Poster: Dropout Training as Adaptive Regularization »
Stefan Wager · Sida Wang · Percy Liang -
2013 Poster: Optimistic Concurrency Control for Distributed Unsupervised Learning »
Xinghao Pan · Joseph Gonzalez · Stefanie Jegelka · Tamara Broderick · Michael Jordan -
2013 Spotlight: Dropout Training as Adaptive Regularization »
Stefan Wager · Sida Wang · Percy Liang -
2013 Session: Oral Session 2 »
Francis Bach -
2013 Poster: Convex Relaxations for Permutation Problems »
Fajwel Fogel · Rodolphe Jenatton · Francis Bach · Alexandre d'Aspremont -
2013 Poster: Reflection methods for user-friendly submodular optimization »
Stefanie Jegelka · Francis Bach · Suvrit Sra -
2013 Poster: Local Privacy and Minimax Bounds: Sharp Rates for Probability Estimation »
John Duchi · Martin J Wainwright · Michael Jordan -
2013 Poster: Streaming Variational Bayes »
Tamara Broderick · Nicholas Boyd · Andre Wibisono · Ashia C Wilson · Michael Jordan -
2013 Poster: Estimation, Optimization, and Parallelism when Data is Sparse »
John Duchi · Michael Jordan · Brendan McMahan -
2013 Session: Tutorial Session B »
Francis Bach -
2012 Workshop: Bayesian Nonparametric Models For Reliable Planning And Decision-Making Under Uncertainty »
Jonathan How · Lawrence Carin · John Fisher III · Michael Jordan · Alborz Geramifard -
2012 Workshop: Analysis Operator Learning vs. Dictionary Learning: Fraternal Twins in Sparse Modeling »
Martin Kleinsteuber · Francis Bach · Remi Gribonval · John Wright · Simon Hawe -
2012 Poster: Multiple Operator-valued Kernel Learning »
Hachem Kadri · Alain Rakotomamonjy · Francis Bach · philippe preux -
2012 Poster: Privacy Aware Learning »
John Duchi · Michael Jordan · Martin J Wainwright -
2012 Poster: Ancestor Sampling for Particle Gibbs »
Fredrik Lindsten · Michael Jordan · Thomas Schön -
2012 Poster: Identifiability and Unmixing of Latent Parse Trees »
Percy Liang · Sham M Kakade · Daniel Hsu -
2012 Oral: Privacy Aware Learning »
John Duchi · Michael Jordan · Martin J Wainwright -
2012 Poster: Finite Sample Convergence Rates of Zero-Order Stochastic Optimization Methods »
John Duchi · Michael Jordan · Martin J Wainwright · Andre Wibisono -
2012 Poster: A Stochastic Gradient Method with an Exponential Convergence
Rate for Finite Training Sets »
Nicolas Le Roux · Mark Schmidt · Francis Bach -
2012 Poster: Small-Variance Asymptotics for Exponential Family Dirichlet Process Mixture Models »
Ke Jiang · Brian Kulis · Michael Jordan -
2012 Oral: A Stochastic Gradient Method with an Exponential Convergence
Rate for Finite Training Sets »
Nicolas Le Roux · Mark Schmidt · Francis Bach -
2011 Workshop: Choice Models and Preference Learning »
Jean-Marc Andreoli · Cedric Archambeau · Guillaume Bouchard · Shengbo Guo · Kristian Kersting · Scott Sanner · Martin Szummer · Paolo Viappiani · Onno Zoeter -
2011 Workshop: Big Learning: Algorithms, Systems, and Tools for Learning at Scale »
Joseph E Gonzalez · Sameer Singh · Graham Taylor · James Bergstra · Alice Zheng · Misha Bilenko · Yucheng Low · Yoshua Bengio · Michael Franklin · Carlos Guestrin · Andrew McCallum · Alexander Smola · Michael Jordan · Sugato Basu -
2011 Workshop: Sparse Representation and Low-rank Approximation »
Ameet S Talwalkar · Lester W Mackey · Mehryar Mohri · Michael W Mahoney · Francis Bach · Mike Davies · Remi Gribonval · Guillaume R Obozinski -
2011 Poster: Bayesian Bias Mitigation for Crowdsourcing »
Fabian L Wauthier · Michael Jordan -
2011 Poster: Convergence Rates of Inexact Proximal-Gradient Methods for Convex Optimization »
Mark Schmidt · Nicolas Le Roux · Francis Bach -
2011 Poster: Divide-and-Conquer Matrix Factorization »
Lester W Mackey · Ameet S Talwalkar · Michael Jordan -
2011 Oral: Convergence Rates of Inexact Proximal-Gradient Methods for Convex Optimization »
Mark Schmidt · Nicolas Le Roux · Francis Bach -
2011 Poster: Non-Asymptotic Analysis of Stochastic Approximation Algorithms for Machine Learning »
Francis Bach · Eric Moulines -
2011 Poster: Trace Lasso: a trace norm regularization for correlated designs »
Edouard Grave · Guillaume R Obozinski · Francis Bach -
2011 Spotlight: Non-Asymptotic Analysis of Stochastic Approximation Algorithms for Machine Learning »
Francis Bach · Eric Moulines -
2011 Poster: Shaping Level Sets with Submodular Functions »
Francis Bach -
2010 Workshop: New Directions in Multiple Kernel Learning »
Marius Kloft · Ulrich Rueckert · Cheng Soon Ong · Alain Rakotomamonjy · Soeren Sonnenburg · Francis Bach -
2010 Oral: Tree-Structured Stick Breaking for Hierarchical Data »
Ryan Adams · Zoubin Ghahramani · Michael Jordan -
2010 Invited Talk: Statistical Inference of Protein Structure and Function »
Michael Jordan -
2010 Spotlight: Online Learning for Latent Dirichlet Allocation »
Matthew D. Hoffman · David Blei · Francis Bach -
2010 Poster: Tree-Structured Stick Breaking for Hierarchical Data »
Ryan Adams · Zoubin Ghahramani · Michael Jordan -
2010 Poster: Variational bounds for mixed-data factor analysis »
Mohammad Emtiyaz Khan · Benjamin Marlin · Guillaume Bouchard · Kevin Murphy -
2010 Poster: Efficient Optimization for Discriminative Latent Class Models »
Armand Joulin · Francis Bach · Jean A Ponce -
2010 Poster: Online Learning for Latent Dirichlet Allocation »
Matthew D. Hoffman · David Blei · Francis Bach -
2010 Spotlight: Variational Inference over Combinatorial Spaces »
Alexandre Bouchard-Côté · Michael Jordan -
2010 Oral: Structured sparsity-inducing norms through submodular functions »
Francis Bach -
2010 Poster: Variational Inference over Combinatorial Spaces »
Alexandre Bouchard-Côté · Michael Jordan -
2010 Poster: Unsupervised Kernel Dimension Reduction »
Meihong Wang · Fei Sha · Michael Jordan -
2010 Poster: Structured sparsity-inducing norms through submodular functions »
Francis Bach -
2010 Poster: Network Flow Algorithms for Structured Sparsity »
Julien Mairal · Rodolphe Jenatton · Guillaume R Obozinski · Francis Bach -
2010 Poster: Heavy-Tailed Process Priors for Selective Shrinkage »
Fabian L Wauthier · Michael Jordan -
2010 Poster: Random Conic Pursuit for Semidefinite Programming »
Ariel Kleiner · ali rahimi · Michael Jordan -
2009 Workshop: Nonparametric Bayes »
Dilan Gorur · Francois Caron · Yee Whye Teh · David B Dunson · Zoubin Ghahramani · Michael Jordan -
2009 Workshop: The Generative and Discriminative Learning Interface »
Simon Lacoste-Julien · Percy Liang · Guillaume Bouchard -
2009 Workshop: Understanding Multiple Kernel Learning Methods »
Brian McFee · Gert Lanckriet · Francis Bach · Nati Srebro -
2009 Poster: Sharing Features among Dynamical Systems with Beta Processes »
Emily Fox · Erik Sudderth · Michael Jordan · Alan S Willsky -
2009 Oral: Sharing Features among Dynamical Systems with Beta Processes »
Emily Fox · Erik Sudderth · Michael Jordan · Alan S Willsky -
2009 Poster: Data-driven calibration of linear estimators with minimal penalties »
Sylvain Arlot · Francis Bach -
2009 Poster: Nonparametric Latent Feature Models for Link Prediction »
Kurt T Miller · Tom Griffiths · Michael Jordan -
2009 Spotlight: Nonparametric Latent Feature Models for Link Prediction »
Kurt T Miller · Tom Griffiths · Michael Jordan -
2009 Tutorial: Sparse Methods for Machine Learning: Theory and Algorithms »
Francis Bach -
2008 Workshop: Speech and Language: Unsupervised Latent-Variable Models »
Slav Petrov · Aria Haghighi · Percy Liang · Dan Klein -
2008 Oral: Shared Segmentation of Natural Scenes Using Dependent Pitman-Yor Processes »
Erik Sudderth · Michael Jordan -
2008 Poster: Nonparametric Bayesian Learning of Switching Linear Dynamical Systems »
Emily Fox · Erik Sudderth · Michael Jordan · Alan S Willsky -
2008 Poster: Clustered Multi-Task Learning: A Convex Formulation »
Laurent Jacob · Francis Bach · Jean-Philippe Vert -
2008 Poster: High-dimensional union support recovery in multivariate regression »
Guillaume R Obozinski · Martin J Wainwright · Michael Jordan -
2008 Poster: Sparse probabilistic projections »
Cedric Archambeau · Francis Bach -
2008 Poster: Shared Segmentation of Natural Scenes Using Dependent Pitman-Yor Processes »
Erik Sudderth · Michael Jordan -
2008 Spotlight: Sparse probabilistic projections »
Cedric Archambeau · Francis Bach -
2008 Spotlight: High-dimensional union support recovery in multivariate regression »
Guillaume R Obozinski · Martin J Wainwright · Michael Jordan -
2008 Spotlight: Clustered Multi-Task Learning: A Convex Formulation »
Laurent Jacob · Francis Bach · Jean-Philippe Vert -
2008 Spotlight: Nonparametric Bayesian Learning of Switching Linear Dynamical Systems »
Emily Fox · Erik Sudderth · Michael Jordan · Alan S Willsky -
2008 Poster: Posterior Consistency of the Silverman g-prior in Bayesian Model Choice »
Zhihua Zhang · Michael Jordan · Dit-Yan Yeung -
2008 Poster: Exploring Large Feature Spaces with Hierarchical Multiple Kernel Learning »
Francis Bach -
2008 Poster: DiscLDA: Discriminative Learning for Dimensionality Reduction and Classification »
Simon Lacoste-Julien · Fei Sha · Michael Jordan -
2008 Spotlight: Posterior Consistency of the Silverman g-prior in Bayesian Model Choice »
Zhihua Zhang · Michael Jordan · Dit-Yan Yeung -
2008 Poster: Efficient Inference in Phylogenetic InDel Trees »
Alexandre Bouchard-Côté · Michael Jordan · Dan Klein -
2008 Poster: Spectral Clustering with Perturbed Data »
Ling Huang · Donghui Yan · Michael Jordan · Nina Taft -
2008 Poster: Kernel Change-point Analysis »
Zaid Harchaoui · Francis Bach · Eric Moulines -
2008 Poster: SDL: Supervised Dictionary Learning »
Julien Mairal · Francis Bach · Jean A Ponce · Guillermo Sapiro · Andrew Zisserman -
2008 Spotlight: Efficient Inference in Phylogenetic InDel Trees »
Alexandre Bouchard-Côté · Michael Jordan · Dan Klein -
2008 Spotlight: Spectral Clustering with Perturbed Data »
Ling Huang · Donghui Yan · Michael Jordan · Nina Taft -
2007 Poster: Agreement-Based Learning »
Percy Liang · Dan Klein · Michael Jordan -
2007 Spotlight: Agreement-Based Learning »
Percy Liang · Dan Klein · Michael Jordan -
2007 Poster: Testing for Homogeneity with Kernel Fisher Discriminant Analysis »
Zaid Harchaoui · Francis Bach · Moulines Eric -
2007 Poster: DIFFRAC: a discriminative and flexible framework for clustering »
Francis Bach · Zaid Harchaoui -
2007 Session: Session 2: Probabilistic Optimization »
Francis Bach -
2007 Spotlight: Resampling Methods for Protein Structure Prediction with Rosetta »
Ben Blum · David Baker · Michael Jordan · Philip Bradley · Rhiju Das · David Kim -
2007 Spotlight: Estimating divergence functionals and the likelihood ratio by penalized convex risk minimization »
XuanLong Nguyen · Martin J Wainwright · Michael Jordan -
2007 Poster: Resampling Methods for Protein Structure Prediction with Rosetta »
Ben Blum · David Baker · Michael Jordan · Philip Bradley · Rhiju Das · David Kim -
2007 Poster: Estimating divergence functionals and the likelihood ratio by penalized convex risk minimization »
XuanLong Nguyen · Martin J Wainwright · Michael Jordan -
2007 Poster: A Probabilistic Approach to Language Change »
Alexandre Bouchard-Côté · Percy Liang · Tom Griffiths · Dan Klein -
2006 Poster: Active learning for misspecified generalized linear models »
Francis Bach -
2006 Poster: Distributed PCA and Network Anomaly Detection »
Ling Huang · XuanLong Nguyen · Minos Garofalakis · Michael Jordan · Anthony D Joseph · Nina Taft