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NIPS 2003 Workshop on Information Theory and Learning: The Bottleneck and Distortion Approach Organizers: Thomas Gedeon Naftali Tishby http://www.cs.huji.ac.il/~tishby/NIPS-Workshop. Saturday, December 13 Morning session

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NIPS 2003 Workshop onInformation Theory and Learning:The Bottleneck and Distortion ApproachOrganizers:Thomas Gedeon Naftali Tishbyhttp://www.cs.huji.ac.il/~tishby/NIPS-Workshop

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Saturday, December 13

Morning session

  • 7:30-8:20 N. TishbyIntroduction – A new look at Shannon’s Information Theory
  • 8:20-9:00 T. GedeonMathematical structure of Information Distortion methods
  • 9:00-9:30 D. Miller Deterministic annealing
  • 9:30-10:00 N. SlonimMultivariate Information Bottleneck
  • 10:00-10:30 B. MumeyOptimal Mutual Information Quantization is NP-complete

Afternoon session

  • 16:00-16:20 G. Elidan The Information Bottleneck EM algorithm
  • 16:20-16:40 A. GlobersonSufficient Dimensionality Reduction with Side Information
  • 16:40-17:00 A. ParkerPhase Transitions in the Information Distortion
  • 17:00-17:20 A. Dimitrov Information Distortion as a model of sensory processing

break

  • 17:40-18:00 J. SinkkonenIB-type clustering for continuous finite data
  • 18:00-18:20 G. ChechikGIB: Gaussian Information Bottleneck
  • 18:20-18:40 Y. CrammerIB and data-representation – Bregman to the rescue
  • 18:40-19:00 B. WestoverAchievable rates for Pattern Recognition
outline
Outline:

The fundamental dilemma: Simplicity/Complexity versus Accuracy

    • Lessons from Statistical Physics for Machine Learning
    • Is there a “right level” of description?
  • A Variational Principle
    • Shannon’s source and channel coding – dual problems
    • [Mutual] Information as theFundamental Quantity
    • Information Bottleneck (IB) andEfficient Representations
    • Finite Data Issues
  • Algorithms, Applications and Extensions
    • Words, documents and meaning…
    • Understanding neural codes …
    • Quantization of side-information
    • Relevant linear dimension reduction: Gaussian IB
    • Using “irrelevant” side information
    • Multivariate-IB and Graphical Models
    • Sufficient Dimensionality Reduction (SDR)
  • The importance of being principled…
learning theory the fundamental dilemma
Learning Theory:The fundamental dilemma…

y=f(x)

Tradeoff between accuracy and simplicity

Y

Good models should enable Prediction of new data…

X

lessons from statistical physics
Lessons from Statistical Physics

Physical systems with many degrees of freedom:

What is the right level of description?

  • “Microscopic Variables” – dynamical variables (positions, momenta,…)
  • “Variational Functions” – Thermodynamic potentials

(Energy, Entropy, Free-energy,…)

  • Competition between order (energy) and disorder (entropy)…

Emergent “relevant” description

Order Parameters

(magnetization,…)

statistical models in learning and ai
Statistical Models in Learning and AI

Statistical models of complex systems:

What is the right level of description?

  • “Microscopic Variables” – probability distributions (over all observed and hidden variables)
  • “Variational Functions” – Log-likelihood, Entropy,… ?(Multi-Information, “Free-energy”,… )
  • Competition between accuracy and complexity

Emerged “relevant” description

Features

(clusters, sufficient statistics,…)

the fundamental dilemma of science model complexity vs prediction accuracy

Limited data

Bounded

Computation

The Fundamental Dilemma (of science): Model Complexity vs Prediction Accuracy

Accuracy

Survivability

Possible Models/representations

Complexity

Efficiency

can we quantify it
Can we quantify it…?

When there is a (relevant) prediction or distortion measure

Accuracy small average distortion (good predictions)

Complexity short description (high compression)

A general tradeoff between distortion and compression:

Shannon’s Information Theory

shannon s information theory

decoder

encoder

receiver

source

channel

Shannon’s Information Theory

Sent

messages

Received

messages

symbols

Source

coding

Channel

coding

Channel

decoding

Source

decoding

0110101001110…

Compression

Error Correction

Decompression

Source Entropy

Channel Capacity

Rate vs Distortion

Capacity vs Efficiency

compression and mutual information
Compression and Mutual Information

We need to index the max number of non-overlapping green blobs inside the blue blob:

(mutual information!)

axioms for multi information w i nemenman
Axioms for “(multi) information”(w I Nemenman)
  • M1: For two nodes, X and Y, the shared information,I[X;Y], is a continuous function of p(x,y).
  • M2: when one of the variables defines the other uniquely (1-1 mapping from X to Y) and both p(x)=p(y)=1/K , then I[X;Y] is a monotonic function of K.
  • M3: If X is partitioned into two subsets, X1 and X2, then the amount of shared information is additive in the following sense: I[X;Y]=I[(X1,X2);Y]+p(x1)I[X;Y|X1]+p(x2)I[X;Y|X2]
  • M4: Shared information is symmetric: I[X;Y]=I[Y;X].
  • M5:

Theorem:

the dual variational problems of it rate vs distortion and capacity vs efficiency
The Dual Variational problems of IT:Rate vs Distortion and Capacity vs Efficiency

Q:What is the simplest representation – fewer bits(/sec) (Rate) for a given expected distortion (accuracy)?

A: (RDT, Shannon 1960) solve:

Q: What is the maximum number of bits(/sec) that can be sent reliably (prediction rate) for a given efficiency of the channel (power, cost)?

A: (Capacity-power tradeoff, Shannon 1956) solve:

rate distortion theory
Rate-Distortion theory
  • The tradeoff between expected representation size (Rate) and the expected distortion is expressed by the rate-distortion function:
  • By introducing a Lagrange multiplier, , we have a variational principle:
  • with:
slide15
The Rate-Distortion function, R(D), is the optimal ratefor agiven distortionand is a convex function.

R(D)

Achievable region

-bits/accuracy

impossible

D

slide16

0

The Capacity – Efficiency Tradeoff

bits/erg

Achievable Region

C(E)

Capacity at E

E (Efficiency (power))

double matching of source to channel eliminates the need for coding
Double matching of source to channel- eliminates the need for coding (?!)

R(D)

C(E)

E

D

In the case of a Gaussian channel (w. power constraint) and square distortion, double matching means:

And No coding!

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“There isa curious and provocative duality between the properties of a source with a distortion measure and those of a channel … if we consider channels in which there is a “cost” associated with the different input letters…

The solution to this problem amounts, mathematically, to maximizing a mutual information under linear inequality constraint… which leads to a capacity-cost function C(E) for the channel…

In a somewhat dual way, evaluating the rate-distortion function R(D) for source amounts, mathematically, to minimizing a mutual Information … again with a linear inequality constraint.

This duality can be pursued further and is related to the duality between past and future and the notions of control and knowledge. Thus we may have knowledge of the past but cannot control it; we may control the future but have no knowledge of it.”

C. E. Shannon (1959)

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Bottlenecks and Neural Nets

  • Auto association: forcing compact representations
  • provide a “relevant code” of w.r.t.

Input

Output

Sample 1

Sample 2

Past

Future

slide20

Bottlenecks in Nature…

(w E Schniedman, Rob de Ruyter, W Bialek, N Brenner)

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The idea: find a compressed variable

that enables short encoding of ( small )

while preserving as much as possible the information on the relevant signal ( )

a variational principle
A Variational Principle

We want a short representation of X that keeps the information about another variable, Y, if possible.

self consistent equations
Self Consistent Equations
  • Marginal:
  • Markov condition:
  • Bayes’ rule:
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The emerged effective distortion measure:

  • Regular if is absolutely continuous w.r.t.
  • Small if predicts y as well as x:
i projections on a set of distributions csiszar 75 84
I-Projections on a set of distributions (Csiszar 75,84)
  • The I-projection of a distribution q(x) on a convex set of distributions L:
key theorems
Key Theorems

IB Coding Theorem(Wyner 1975, Bachrach, Navot, Tishby 2003):

The IB variational problem provides a tight convex lower bound on the expected size of the achievable representations of X, that maintain at least of mutual information on the variable Y. The bound can be achieved by: .

Formally equivalent to “Channel coding with Side-Information” (introduced in a very different context by Wyner, 75).

The same bound is true for the dual channel coding problem and the optimal representation constitutes a “perfectly matched source-channel” and requires “no further coding”.

IB Algorithm Convergence(Tishby 99, Friedman, Slonim,Tishby 01):

The GeneralizedArimoto-Blauht IB algorithm converges to a local minima of the IB functional, (including its generalized multivariate case).

summary
Summary
  • Shannon’s Information Theory suggests a compelling framework

for quantifying the “Complexity-Accuracy” dilemma, by unifying

source and channel coding as a Min Max of mutual information.

  • The IB method turns this unified coding principle into algorithms

for extracting “informative” relevant structures from empirical

joint probability distributions P(X1,X2)…

  • The Multivariate IB extends this framework to extract

“informative” structures from multivariate distributions,

P(X1,…,Xn), via Min Max I(X1,…,Xn),and Graphical models

  • IB can be further extended for

Sufficient Dimensionality Reduction

a method for finding informative continuous

low dimensional representations via Min Max I(X1,X2)

slide32
Many thanks to:

Fernando Pereira

William Bialek

Nir Friedman

Noam Slonim

Gal Chechik

Amir Globerson

Ran Bachrach

Amir Navot

Ilya Nemenman

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