# FEO3350 Information Theory for Statistics and Learning, Fall 2020

Information theory, machine learning and artificial intelligence have been overlapping fields during their whole existence as academic disciplines. These areas, in turn, overlap significantly with applied and theoretical statistics.

Arguably the most central concepts in information theory are: entropy, mutual information and relative entropy (Kullback-Leibler divergence). These entities are important also in inference and learning, for example via their manifestation in the evidence lower bound (variational free energy). Entropy and mutual information also play important parts in a class of general bounds to error probability in estimation and decision-making, where the most basic special case is known as Fano's inequality. Relative entropy was introduced in parallel in the statistics and information theory literature, and is a special case of the more general concept of f-divergence. Divergence is in general an important measure of "statistical dissimilarity," and plays an fundamental part in several bounding techniques. A more recent framework that has caught considerable attention is the information bottleneck principle, which in turn has several interesting connections to traditional rate-distortion theory.

This course will explore these, and several other, relations and tools at some depth. The goal is to give PhD students in decision and control, learning, AI, network science, and information theory a solid introduction to how information-theoretic concepts and tools can be applied to problems in statistics, decision and learning well beyond their more traditional use in, for example, communication theory.

The course is registered as FEO3350 and is worth 12 cu's.

## Teachers

Mikael Skoglund and Tobias Oechtering

## Prerequisites

Required: Solid working knowledge (at the "advanced undergrad level") in analysis, linear algebra and probability

Recommended: Information theory, corresponding to FEO3210; (measure theoretic) probability, corresponding to FEO3230; optimization, corresponding to SF3847

## Material

Teaching the course will draw from several different sources. The following is a partial list of recommended textbooks, tutorials, lecture-notes and papers.
• [CT] Cover & Thomas, "Elements of Information Theory"
• [MK] MacKay, "Information Theory, Inference and Learning Algorithms"
• [CK] Csiszar & Körner, "Information Theory"
• [CS] Csiszar & Shields, "Information Theory and Statistics: A Tutorial"
• [W] Y. Wu, "Information-theoretic methods for high-dimensional statistics" (lecture notes, Yale)
• [PW] Polyanskiy & Wu, "Lecture notes on information theory" (lecture notes, MIT)
• [D] Duchi, "Information theory and statistics" (lecture notes, Stanford)
• [WV] Wu & Verdú, "Rényi information dimension: Fundamental limits of almost lossless analog compression,'' IEEE Trans. on IT Aug. 2010
• [V] Vapnik, "The nature of statistical learning theory," Springer 1995. (Can be accessed from KTH via Springer Link)
• [XR] Xu & Raginsky, "Information-theoretic analysis of generalization capability of learning algorithms," in Proc. NIPS 2017 (se also this lecture by Raginsky)
• [RWY] G. Raskutti, M. J. Wainwright and B. Yu, "Minimax rates of estimation for high-dimensional linear regression over $\ell_q$ balls," IEEE Trans. on IT, Oct. 2011
• [CRT] Candés, Romberg & Tao, "Stable signal recovery from incomplete and inaccurate measurements," Commun. Pure and Applied Mathematics, August 2006
• [GP] Z. Goldfeld and Y. Polyanskiy, "The information bottleneck problem and its applications in machine learning," IEEE J. Select. Areas in Information Theory, May 2020

## Preliminary Schedule 2020

• Lecture 1, November 6: Information theory fundamentals: Entropy, mutual information, relative entropy, and f-divergence. Total variation and other distance metrics. Inequalities. [CT,PW,W]
• Lecture 2, November 13: Rate-Distortion theory: Cost versus information. Bounds. The Blahut algorithm. [CT,PW]
• Lecture 3: Limits on information flow and processing: Conditional mutual information and relative entropy. Data processing inequalities. Sufficient statistic and the information bottleneck. Rate-distortion interpretation [CT,PW,W,GP]
• Lecture 4: Foundations of statistical decision theory: Parameter estimation. Bayes and minimax risk. Binary hypothesis testing [PW,W]
• Lecture 5: Information bounds on error probability and risk: Sample complexity. The mutual information method and rate-distortion. Fano inequalities. [W,PW,D]
• Lecture 6: Learning and generalization: Information bounds on generalization error. VC dimension and complexity. [XR,D,V]
• Lecture 7: Variational methods: Variational characterization of divergence, Donsker-Varadhan [PW,W]. Variational inference and the ELBO [MK]
• Lecture 8: Classical estimation theory: Maximum likelihood, Fischer information, information bounds, Cramér-Rao, Hammersley-Chapman-Robbins. [CT,W,PW,D,MK]
• Lecture 9: Packing, covering, Fano & minimax risk, metric entropy [W,D]
• Lecture 10: Le Cam's method, Assouad's method, mutual information method continued. Density estimation. Functional estimation. [W,D]
• Lecture 11: Dimension compression and denoising: Sparse denoising, compressed sensing, almost lossless analog compression [W,D,RWY,CRT,WV]
• Lecture 12: The method of types [CT,CK,CS]
• Lecture 13: Information theory and large deviations, Stein, Chernoff and Sanov. Total variation and hypothesis testing. [CT,CK,CS,PW]
• Lecture 14: The geometry of information: Information geometry, information projection, iterative methods, Expectation-Maximization [CK,CS,W,PW,D]

The first meeting (on November 6) is held in Room Q2 (Malvinas Väg 10), starting at 9:30

Nov 16: WE ARE PAUSING THE COURSE AFTER LEC 2 BECAUSE OF THE DEVELOPMENT OF THE PANDEMIC, AND THE NEW CAP ON NUMBER OF PEOPLE PER GATHERING

The course is paused until further notice. The assumption is that we can hopefully hold another 1-2 meetings before Christmas, but we may need to pause until after the Holidays. Check back here for updates. (The new deadline for HW 2 corresponds to the new date for Lec 3, when known.)