Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Notation
- Contributors
- 1 Introduction to Information Theory and Data Science.
- 2 An Information-Theoretic Approach to Analog-to-Digital Compression
- 3 Compressed Sensing via Compression Codes
- 4 Information-Theoretic Bounds on Sketching
- 5 Sample Complexity Bounds for Dictionary Learning from Vector- and Tensor-Valued Data
- 6 Uncertainty Relations and Sparse Signal Recovery
- 7 Understanding Phase Transitions via Mutual Information and MMSE
- 8 Computing Choice: Learning Distributions over Permutations
- 9 Universal Clustering
- 10 Information-Theoretic Stability and Generalization
- 11 Information Bottleneck and Representation Learning
- 12 Fundamental Limits in Model Selection for Modern Data Analysis
- 13 Statistical Problems with Planted Structures: Information-Theoretical and Computational Limits
- 14 Distributed Statistical Inference with Compressed Data
- 15 Network Functional Compression
- 16 An Introductory Guide to Fano’s Inequality with Applications in Statistical Estimation
- Index
- References
10 - Information-Theoretic Stability and Generalization
Published online by Cambridge University Press: 22 March 2021
Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Notation
- Contributors
- 1 Introduction to Information Theory and Data Science.
- 2 An Information-Theoretic Approach to Analog-to-Digital Compression
- 3 Compressed Sensing via Compression Codes
- 4 Information-Theoretic Bounds on Sketching
- 5 Sample Complexity Bounds for Dictionary Learning from Vector- and Tensor-Valued Data
- 6 Uncertainty Relations and Sparse Signal Recovery
- 7 Understanding Phase Transitions via Mutual Information and MMSE
- 8 Computing Choice: Learning Distributions over Permutations
- 9 Universal Clustering
- 10 Information-Theoretic Stability and Generalization
- 11 Information Bottleneck and Representation Learning
- 12 Fundamental Limits in Model Selection for Modern Data Analysis
- 13 Statistical Problems with Planted Structures: Information-Theoretical and Computational Limits
- 14 Distributed Statistical Inference with Compressed Data
- 15 Network Functional Compression
- 16 An Introductory Guide to Fano’s Inequality with Applications in Statistical Estimation
- Index
- References
Summary
Machine-learning algorithms can be viewed as stochastic transformations that map training data to hypotheses. Following Bousquet and Elisseeff, we say such an algorithm is stable if its output does not depend too much on any individual training example. Since stability is closely connected to generalization capabilities of learning algorithms, it is of interest to obtain sharp quantitative estimates on the generalization bias of machine-learning algorithms in terms of their stability properties. We describe several information-theoretic measures of algorithmic stability and illustrate their use for upper-bounding the generalization bias of learning algorithms. Specifically, we relate the expected generalization error of a learning algorithm to several information-theoretic quantities that capture the statistical dependence between the training data and the hypothesis. These include mutual information and erasure mutual information, and their counterparts induced by the total variation distance. We illustrate the general theory through examples, including the Gibbs algorithm and differentially private algorithms, and discuss strategies for controlling the generalization error.
Keywords
- Type
- Chapter
- Information
- Information-Theoretic Methods in Data Science , pp. 302 - 329Publisher: Cambridge University PressPrint publication year: 2021
References
Shalev-Shwartz, S., Shamir, O., Srebro, N., and Sridharan, K., “Learnability, stability, and uniform convergence,” J. Machine Learning Res., vol. 11, pp. 2635–2670, 2010.Google Scholar
Taylor, J. and Tibshirani, R. J., “Statistical learning and selective inference,” Proc. Natl. Acad. Sci. USA, vol. 112, no. 25, pp. 7629–7634, 2015.Google Scholar
Devroye, L. and Wagner, T., “Distribution-free performance bounds for potential function rules,” IEEE Trans. Information Theory, vol. 25, no. 5, pp. 601–604, 1979.CrossRefGoogle Scholar
Rogers, W. H. and Wagner, T. J., “A finite sample distribution-free performance bound for local discrimination rules,” Annals Statist., vol. 6, no. 3, pp. 506–514, 1978.Google Scholar
Ron, D. and Kearns, M., “Algorithmic stability and sanity-check bounds for leave-one-out crossvalidation,” Neural Computation, vol. 11, no. 6, pp. 1427–1453, 1999.Google Scholar
Bousquet, O. and Elisseeff, A., “Stability and generalization,” J. Machine Learning Res., vol. 2, pp. 499–526, 2002.Google Scholar
Poggio, T., Rifkin, R., Mukherjee, S., and Niyogi, P., “General conditions for predictivity in learning theory,” Nature, vol. 428, no. 6981, pp. 419–422, 2004.CrossRefGoogle ScholarPubMed
Kutin, S. and Niyogi, P., “Almost -everywhere algorithmic stability and generalization error,” in Proc. 18th Conference on Uncertainty in Artificial Intelligence (UAI 2002), 2002, pp. 275–282.Google Scholar
Rakhlin, A., Mukherjee, S., and Poggio, T., “Stability results in learning theory,” Analysis Applications, vol. 3, no. 4, pp. 397–417, 2005.CrossRefGoogle Scholar
Dwork, C., McSherry, F., Nissim, K., and Smith, A., “Calibrating noise to sensitivity in private data analysis,” in Proc. Theory of Cryptography Conference, 2006, pp. 265–284.Google Scholar
Dwork, C., Feldman, V., Hardt, M., Pitassi, T., Reingold, O., and Roth, A., “Generalization in adaptive data analysis and holdout reuse,” arXiv:1506.02629, 2015.Google Scholar
Dwork, C. and Roth, A., “The algorithmic foundatons of differential privacy,” Foundations and Trends in Theoretical Computer Sci., vol. 9, nos. 3–4, pp. 211–407, 2014.Google Scholar
Kairouz, P., Oh, S., and Viswanath, P., “The composition theorem for differential privacy,” in Proc. 32nd International Conference on Machine Learning (ICML), 2015, pp. 1376–1385.Google Scholar
Raginsky, M., Rakhlin, A., Tsao, M., Wu, Y., and Xu, A., “Information -theoretic analysis of stability and bias of learning algorithms,” in Proc. IEEE Information Theory Workshop, 2016.Google Scholar
Xu, A. and Raginsky, M., “Information –theoretic analysis of generalization capability of learning algorithms,” in Proc. Conference on Neural Information Processing Systems, 2017.Google Scholar
Strasser, H., Mathematical theory of statistics: Statistical experiments and asymptotic decision Theory. Walter de Gruyter, 1985.Google Scholar
Polyanskiy, Y. and Wu, Y., “Dissipation of information in channels with input constraints,” IEEE Trans. Information Theory, vol. 62, no. 1, pp. 35–55, 2016.Google Scholar
Verdú, S. and Weissman, T., “The information lost in erasures,” IEEE Trans. Information Theory, vol. 54, no. 11, pp. 5030–5058, 2008.Google Scholar
Hoeffding, W., “Probability inequalities for sums of bounded random variables,” J. Amer. Statist. Soc., vol. 58, no. 301, pp. 13–30, 1963.Google Scholar
Boucheron, S., Lugosi, G., and Massart, P., Concentration inequalities: A nonasymptotic theory of independence. Oxford University Press, 2013.Google Scholar
Bassily, R., Nissim, K., Smith, A., Steinke, T., Stemmer, U., and Ullman, J., “Algorithmic stability for adaptive data analysis,” in Proc. 48th ACM Symposium on Theory of Computing (STOC), 2016, pp. 1046–1059.Google Scholar
McAllester, D., “PAC -Bayesian model averaging,” in Proc. 1999 Conference on Learning Theory, 1999.Google Scholar
McAllester, D., “A PAC-Bayesian tutorial with a dropout bound,” arXiv:1307.2118, 2013, http://arxiv.org/abs/1307.2118.Google Scholar
Russo, D. and Zou, J., “Controlling bias in adaptive data analysis using information theory,” in Proc. 19th International Conference on Artificial Intelligence and Statistics (AISTATS), 2016, pp. 1232–1240.Google Scholar
Jiao, J., Han, Y., and Weissman, T., “Dependence measures bounding the exploration bias for general measurements,” in Proc. IEEE International Symposium on Information Theory, 2017, pp. 1475–1479.CrossRefGoogle Scholar
Dwork, C., Feldman, V., Hardt, M., Pitassi, T., Reingold, O., and Roth, A., “Preserving statistical validity in adaptive data analysis,” in Proc. 47th ACM Symposium on Theory of Computing (STOC), 2015.CrossRefGoogle Scholar
Dwork, C., Feldman, V., Hardt, M., Pitassi, T., Reingold, O., and Roth, A., “Generalization in adaptive data analysis and holdout reuse,” in 28th Annual Conference on Neural Information Processing Systems (NIPS), 2015.Google Scholar
Bassily, R., Moran, S., Nachum, I., Shafer, J., and Yehudayoff, A., “Learners that use little information,” in Proc. Conference on Algorithmic Learning Theory (ALT), 2018.Google Scholar
Vershynin, R., High-dimensional probability: An introduction with applications in data science. Cambridge University Press, 2018.Google Scholar
Devroye, L., Györfi, L., and Lugosi, G., A probabilistic theory of pattern recognition. Springer, 1996.Google Scholar
Buescher, K. L. and Kumar, P. R., “Learning by canonical smooth estimation. I. Simultaneous estimation,” IEEE Tran. Automatic Control, vol. 41, no. 4, pp. 545–556, 1996.Google Scholar
Cohen, J. E., Kemperman, J. H. B., and Zbǎganu, G., Comparisons of stochastic matrices, with applications in information theory, statistics, economics, and population sciences. Birkhäuser, 1998.Google Scholar
Polyanskiy, Y., “Channel coding: Non-asymptotic fundamental limits,” Ph.D. dissertation, Princeton University, 2010.Google Scholar
Sason, I. and Verdú, S., “f-divergence inequalities,” IEEE Trans. Information Theory, vol. 62, no. 11, pp. 5973–6006, 2016.CrossRefGoogle Scholar
McSherry, F. and Talwar, K., “Mechanism design via differential privacy,” in Proc. 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2007.Google Scholar
Polyanskiy, Y. and Wu, Y., “Lecture notes on information theory,” Lecture Notes for ECE563 (UIUC) and 6.441 (MIT), 2012–2016, http://people.lids.mit.edu/yp/homepage/ data/itlectures_v4. pdf.Google Scholar
Verdú, S., “The exponential distribution in information theory,” Problems of Information Transmission, vol. 32, no. 1, pp. 86–95, 1996.Google Scholar
Raginsky, M., “Strong data processing inequalities and Φ-Sobolev inequalities for discrete channels,” IEEE Trans. Information Theory, vol. 62, no. 6, pp. 3355–3389, 2016.Google Scholar
Feldman, V. and Steinke, T., “Calibrating noise to variance in adaptive data analysis,” in Proc. 2018 Conference on Learning Theory, 2018.Google Scholar
Cuff, P. and Yu, L., “Differential privacy as a mutual information constraint,” in Proc. 2016 ACM SIGSAC Conference on Computer and Communication Security (CCS), 2016, pp. 43–54.Google Scholar
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