Information Geometry in Learning and Optimization  PhD COURSE  September 2226 2014 University of Copenhagen

Lectures overview
Contents of Lectures by Shunichi Amari
I. Introduction to Information Geometry  without knowledge on differential geometry Divergence function on a manifold
 Flat divergence and dual affine structures with Riemannian metric derived from it
 Two types of geodesics and orthogonality
 Pythagorean theorem and projection theorem
 Examples of dually flat manifold: Manifold of probability distributions (exponential families), positive measures and positivedefinite matrices
II. Geometrical Structure Derived from Invariance
 Invariance and information monotonicity in manifold of probability distributions
 fdivergence : unique invariant divergence
 Dual affine connections with Riemannian metric derived from divergence: Tangent space, parallel transports and duality
 Alphageometry induced from invariant geometry
 Geodesics, curvatures and dually flat manifold:
 Canonical divergence: KL and alphadivergence
III. Applications of Information Geometry to Statistical Inference
 Higherorder asymptotic theory of statistical inference – estimation and hypothesis testing
 NeymanScott problem and semiparametric model
 em (EM) algorithm and hidden variables
IV. Applications of Information Geometry to Machine Learning
 Belief propagation and CCCP algorithm in graphical model
 Support vector machine and Riemannian modification of kernels
 Bayesian information geometry and geometry of restricted Boltzmann machine: Towards deep learning
 Natural gradient learning and its dynamics: singular statistical model and manifold
 Clustering with divergence
 Sparse signal analysis
 Convex optimization
Suggested reading:
Amari, ShunIchi. Natural gradient works efficiently in learning. Neural Computation 10, 2 (1998): 251276.
Amari, Shunichi, and Hiroshi Nagaoka. Methods of information geometry. Vol. 191. American Mathematical Soc., 2007.
Contents of Lectures by Nihat Ay
I. Differential Equations: Vector and Covector Fields
 FisherShahshahani Metric, Gradient Fields
 m and eLinearity of Differential Equations
II. Applications to Evolution:
 LotkaVolterra and Replicator Differential Equations
 "Fisher's Fundamental Theorem of Natural Selection"
 The Hypercycle Model of Eigen and Schuster
III. Applications to Learning:
 Information Geometry of Conditional Models
 Amari's Natural Gradient Method
 InformationGeometric Design of Learning Systems
Contents of Lectures by Nikolaus Hansen
I. A short introduction to continuous optimization
II. Continuous optimization using natural gradients
III. The Covariance Matrix Adaptation Evolution Strategy (CMAES)
IV. A short introduction into Python (practice session, see also here)
V. A practical approach to continuous optimization using cma.py (practice session)
Suggested reading:
Hansen, Nikolaus. The CMA Evolution Strategy: A Tutorial, 2011
Ollivier, Yann, Ludovic Arnold, Anne Auger, and Nikolaus Hansen. InformationGeometric Optimization Algorithms: A Unifying Picture via Invariance Principles. arXiv:1106.3708Contents of Lectures by Jan Peters
Suggested reading:
Peters, Jan, and Stefan Schaal. Natural actor critic. Neurocomputing 71, 79 (2008):11801190Contents of Luigi Malagò
Stochastic Optimization in Discrete Domains
I. Stochastic Relaxation of Discrete Optimization Problems
II. Information Geometry of Hierarchical Models
III. Stochastic Natural Gradient Descent
IV. Graphical Models and Model Selection
V. Examples of Natural Gradientbased Algorithms in Stochastic Optimization
For the gradient flow movie click here.
Suggested reading:
Amari, ShunIchi. Information geometry on hierarchy of probability distributions IEEE Transactions on Information Theory 47, 5 (2001):17011711Contents of Lectures by Aasa Feragen and François Lauze
I. Aasa's lectures Recap of Differential Calculus
 Differential manifolds
 Tangent space
 Vector fields
 Submanifolds of R^n
 Riemannian metrics
 Invariance of Fisher information metric
 If time: Metric geometry view of Riemannian manifolds, their curvature and consequences thereof
 Riemannian metrics
 Gradient, gradient descent, duality
 Distances
 Connections and Christoffel symbols
 Parallelism
 LeviCivita Connections
 Geodesics, exponential and log maps
 Fréchet Means and Gradient Descent
Suggested reading:
Sueli I. R. Costa, Sandra A. Santos, and Joao E. Strapasson. Fisher information distance: a geometrical reading. arXiv:1106.3708Contents of Tutorial by Stefan Sommer
In the tutorial on numerics for Riemannian geometry on Tuesday morning, we will discuss computational representations and numerical solutions of some differential geometry problems. The goal is to be able to implement geodesic equations numerically for simple probability distributions, to visualize the computed geodesics, to compute Riemannian logarithms, and to find mean distributions. We will follow the presentation in the paper Fisher information distance: a geometrical reading from a computational viewpoint.
The tutorial is based on an ipython notebook that is available here. Please click here for details.Background
Principles of Information Geometry have been successfully applied in all major areas of machine learning, including supervised, unsupervised, and reinforcement learning, as well as in stochastic optimization. Information Geometry comes into play when we consider parametrized probabilistic models (e.g., in the context of stochastic behavioral policies, search distributions, stochastic neural networks, ...) and their adaptation. Technically speaking, in Information Geometry the space of probability distributions that can be represented by a parametrized probabilistic model is described as a manifold, on which the Fisher information metric defines a Riemannian structure. Through the geometry of the Riemannian manifold of distributions, optimization and statistics can be done directly on the space of distributions.
Information geometry was founded and pioneered by Shun'ichi Amari in the 1980s, with statistical learning as one of the first applications. Due to the nonlinear nature of the space of distributions, the steepest ascent direction for adapting a probability distribution parametrized by a set of realvalued parameters (e.g., the mean and the covariance of a Gaussian distribution) is not the ordinary gradient in Euclidean space, but the so called natural gradient, defined with respect to the Riemannian structure of the space of distributions. The natural gradient is natural in the sense that it renders the adaptation invariant under reparametrization and changing representations, and it is closely linked to the KullbackLeibler divergence often used for quantifying the similarity of distributions.
The natural gradient for adapting probabilistic models has been successfully used in all major areas of machine learning, from supervised learning of neural networks over independent component analysis to reinforcement learning. In this PhD course there will, in particular, be lectures on supervised learning, reinforcement learning and stochastic optimization. Reinforcement learning refers to machine learning algorithms that improve their behavior based on interaction with the environment, whereas stochastic optimization refers to stochastic solutions to complex optimization problems for which we do not have an analytical description. Both in stochastic optimization and reinforcement learning, (intermediate) solutions are best described by probability distributions. In the one case, we consider distributions over potential actions to be taken in a certain situation. In the other case, we consider the search distribution describing which candidate solution to probe next. Thus, both the learning as well as the optimization process are best described by an iterative update of probability distributions.Confirmed Speakers
 Shun'ichi Amari, RIKEN Brain Science Institute
 Nihat Ay, Max Planck Institute for Mathematics in the Sciences and Universität Leipzig
 Nikolaus Hansen, Université ParisSud and Inria Saclay – ÎledeFrance
 Jan Peters, Technische Universität Darmstadt and MaxPlanck Institute for Intelligent Systems
 Luigi Malagò, Shinshu University, Nagano
 Aasa Feragen, University of Copenhagen
 Francois Lauze, University of Copenhagen
 Stefan Sommer, University of Copenhagen
Scientific content
The course will consist of 5 days of lectures and exercises. In addition, students will be expected to read a predefined set of scientific articles on information geometry prior to the course, and write a report on information geometry and its potential use in their own research field after the course. The course will consist of three modules: A crash course on Riemannian geometry and numerical tools for applications of Riemannian geometry
 Introduction to Information Geometry and its role in Machine Learning and Stochastic Optimization
 Applications of Information Geometry
Learning goals
After participating in this course, the participant should Understand basic differential geometric concepts (manifolds, Riemannian metric, geodesics, manifold statistics) to the point where they can apply differential geometric concepts in their own research;
 Be able to implement basic numerical tools for differential geometric computations;
 Have a strong knowledge of information geometry and its role in machine learning and stochastic optimization;
 Be able to apply information theoretic approaches to machine learning and stochastic optimization in their own research;
Have a basic knowledge of existing applications of information geometry.
Organizers
 Christian Igel, University of Copenhagen
 Aasa Feragen, University of Copenhagen
Place
Københavns Universitet, Njalsgade 128, Bygning (building) 27, Lokal (room): 27.0.17The lectures are at the south campus of the University of Copenhagen, very close to the Metro station Islands Brygge. Room 27.0.17 in building 27 is on the ground floor. Click here for a map. See also Google maps.