Nicolò Cesa-Bianchi and Gábor Lugosi
Cambridge University Press, 2006
ISBN 0521841089
Published March 2006, 406 pages

...beware of mathematicians, and all those who make empty prophecies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell.    St. Augustine, De Genesi ad Litteram libri duodecim. Liber Secundus, 17, 37.

Errata (as of September 8, 2006)

A review (in French) appeared on the Gazette of Mathématiciens, 110, October 2006.

This book offers the first comprehensive treatment of the problem of predicting "individual sequences". Unlike standard statistical approaches to forecasting, prediction of individual sequences does not impose any probabilistic assumption on the data-generating mechanism. Yet, prediction algorithms can be constructed that work well for all possible sequences, in the sense that their performance is always nearly as good as the best forecasting strategy in a given reference class.
The central theme is the model of "prediction using expert advice", a general framework within which many related problems can be cast and discussed. Repeated game playing, adaptive data compression, sequential investment in the stock market, sequential pattern analysis, and several other problems are viewed as instances of the experts' framework and analyzed from a common nonstochastic standpoint that often reveals new and intriguing connections. Old and new forecasting methods are described in a mathematically precise way with the purpose of characterizing their theoretical limitations and possibilities.

Table of Contents

1  Introduction
2  Prediction with expert advice
  2.1  Weighted average prediction
  2.2  An optimal bound
  2.3  Bounds that hold uniformly over time
  2.4  An improvement for small losses
  2.5  Forecasters using the gradient of the loss
  2.6  Scaled losses and signed games
  2.7  The multilinear forecaster
  2.8  The exponential forecaster for signed games
  2.9  Simulatable experts
  2.10 Minimax regret
  2.11 Discounted regret
  2.12 Bibliographic remarks
  2.13 Exercises
3  Tight bounds for specific losses
  3.1  Introduction
  3.2  Follow the best expert
  3.3  Exp-concave loss functions
  3.4  The greedy forecaster
  3.5  The aggregating forecaster
  3.6  Mixability for certain losses
  3.7  General lower bounds
  3.8  Bibliographic remarks
  3.9  Exercises
4  Randomized prediction
  4.1  Introduction
  4.2  Weighted average forecasters
  4.3  Follow the perturbed leader
  4.4  Internal regret
  4.5  Calibration
  4.6  Generalized regret
  4.7  Calibration with checking rules
  4.8  Bibliographic remarks
  4.9  Exercises
5  Efficient forecasters for large classes of experts
  5.1  Introduction
  5.2  Tracking the best expert
  5.3  Tree experts
  5.4  The shortest path problem
  5.5  Tracking the best of many actions
  5.6  Bibliographic remarks
  5.7  Exercises
6  Prediction with limited feedback
  6.1  Introduction
  6.2  Label efficient prediction
  6.3  Lower bounds
  6.4  Partial monitoring
  6.5  A general forecaster for partial monitoring
  6.6  Hannan consistency and partial monitoring
  6.7  Multi-armed bandit problems
  6.8  An improved bandit strategy
  6.9  Lower bounds for the bandit problem
  6.10 How to select the best action
  6.11 Bibliographic remarks
  6.12 Exercises
7  Prediction and playing games
  7.1  Games and equilibria
  7.2  Minimax theorems
  7.3  Repeated two-player zero-sum games
  7.4  Correlated equilibrium and internal regret
  7.5  Unknown games: game-theoretic bandits
  7.6  Calibration and correlated equilibrium
  7.7  Blackwell's approachability theorem
  7.8  Potential-based approachability
  7.9  Convergence to Nash equilibria
  7.10 Convergence in unknown games
  7.11 Playing against opponents that react
  7.12 Bibliographic remarks
  7.13 Exercises
8  Absolute loss
  8.1  Simulatable experts
  8.2  Optimal algorithm for simulatable experts
  8.3  Static experts
  8.4  A simple example
  8.5  Bounds for classes of static experts
  8.6  Bounds for general classes
  8.7  Bibliographic remarks
  8.8  Exercises
9  Logarithmic loss
  9.1  Sequential probability assignment
  9.2  Mixture forecasters
  9.3  Gambling and data compression
  9.4  The minimax optimal forecaster
  9.5  Examples
  9.6  The Laplace mixture
  9.7  A refined mixture forecaster
  9.8  Lower bounds for most sequences
  9.9  Prediction with side information
  9.10 A general upper bound
  9.11 Further examples
  9.12 Bibliographic remarks
  9.13 Exercises
10  Sequential investment
  10.1  Portfolio selection
  10.2  The minimax wealth ratio
  10.3  Prediction and investment
  10.4  Universal portfolios
  10.5  The EG investment strategy
  10.6  Investment with side information
  10.7  Bibliographic remarks
  10.8  Exercises
11  Linear pattern recognition
  11.1  Prediction with side information
  11.2  Bregman divergences
  11.3  Potential-based gradient descent
  11.4  The transfer function
  11.5  Forecasters using Bregman projections
  11.6  Time-varying potentials
  11.7  The elliptic potential
  11.8  A nonlinear forecaster
  11.9  Lower bounds
  11.10 Mixture forecasters
  11.11 Bibliographic remarks
  11.12 Exercises
12  Linear classification
  12.1  The zero-one loss
  12.2  The hinge loss
  12.3  Maximum margin classifiers
  12.4  Label efficient classifiers
  12.5  Kernel-based classifiers
  12.6  Bibliographic remarks
  12.7  Exercises
13  Appendix
  13.1  Inequalities from probability theory
    13.1.1  Hoeffding's inequality
    13.1.2  Bernstein's inequality
    13.1.3  Hoeffding-Azuma inequality and related results
    13.1.4  Khinchine's inequality
    13.1.5  Slud's inequality
    13.1.6  A simple limit theorem
    13.1.7  Proof of Theorem 8.3
    13.1.8  Rademacher averages
    13.1.9  The Beta distribution
  13.2  Basic information theory
  13.3  Basics of classification
Author index