Chapter 6 The Structural Risk Minimization Principle Junping Zhang jpzhang@fudan.edu.cn Intelligent Information Processing Laboratory, Fudan University March 23, 2004
Objectives
Structural risk minimization
Two other induction principles
The Scheme of the SRM induction principle
Real-Valued functions
Principle of SRM
SRM
Minimum Description Length and SRM inductive principles The idea about the Nature of Random Phenomena Minimum Description Length Principle for the Pattern Recognition Problem Bounds for the MDL SRM for the simplest Model and MDL The Shortcoming of the MDL
The idea about the Nature of Random Phenomena Probability theory (1930s, Kolmogrov) Formal inference Axiomatization hasn t considered nature of randomness Axioms: given probability measures
The idea about the Nature of Random Phenomena The model of randomness Solomonoff (1965), Kolmogrov (1965), Chaitin (1966). Algorithm (descriptive) complexity The length of the shortest binary computer program Up to an additive constant does not depend on the type of computer. Universal characteristic of the object.
A relatively large string describing an object is random If algorithm complexity of an object is high If the given description of an object cannot be compressed significantly. MML (Wallace and Boulton, 1968)& MDL (Rissanen, 1978) Algorithm Complexity as a main tool of induction inference of learning machines
Minimum Description Length Principle for the Pattern Recognition Problem Given l pairs containing the vector x and the binary value ω Consider two strings: the binary string
Question Q: Given (147), is the string (146) a random object? A: to analyze the complexity of the string (146) in the spirit of Solomonoff- Kolmogorov-Chaitin ideas
Compress its description Since ω i i=1, l are binary values, the string (146) is described by l bits. Since training pairs were drawn randomly and independently. The value ω i depend on the vector x i but not on the vector x j.
Model
General Case: not contain the perfect table.
Randomness
Bounds for the MDL Q: A: Does the compression coefficient K(T) determine the probability of the test error in classification (decoding) vectors x by the table T? Yes
Comparison between the MDL and ERM in the simplest model
SRM for the simplest Model and MDL
SRM for the simplest Model and MDL
The power of compression coefficient To obtain bound for the probability of error Only information about the coefficient need to be known.
The power of compression coefficient How many examples we used How the structure of code books was organized Which code book was used and how many tables were in this code book. How many errors were made by the table from the code book we used.
MDL principle To minimize the probability of error One has to minimize the coefficient of compression
The shortcoming of the MDL MDL uses code books with a finite number of tables. Continuously depends on parameters, one has to first quantize that set to make the tables.
Quantization How do we make the smart quantization for a given number of observations. For a given set of functions, how can we construct a code book with a small number of tables but with good approximation ability?
The shortcoming of the MDL Finding a good quantization is extremely difficult and determines the main shortcoming of MDL principle. The MDL principle works well when the problem of constructing reasonable code books has a good solution.
Consistency of the SRM principle and asymptotic bounds on the rate of convergence Q: Is the SRM consistent? What is the bound on the (asymptotic) rate of convergence?
Consistency of the SRM principle.
Simplification version
Remark To avoid choosing the minimum of functional (156) over the infinite number of elements of the structure. Additional constraint Choose the minimum from the first l elements of the structure where l is equal to the number of observations.
Discussions and Example
The rate of convergence is determined by two contradictory requirements on the rule n=n(l). The first summand: The larger n=n(l), the smaller is the deviation The second summand: The larger n=n(l), the larger deviation For structures with a known bound on the rate of approximation, select the rule that assures the largest rate of convergence.
Bounds for the regression estimation problem
The model of regression estimation by series expansion
Example
The problem of approximating functions
To get high asymptotic rate of approximation the only constraint is that the kernel should be a bounded function which can be described as a family of functions possessing finite VC dimension.
Problem of local risk minimization
Local Risk Minimization Model
Note Using local risk minimization methods, one probably does not need rich sets of approximating functions. Whereas the classical semi-local methods are based on using a set of constant functions.
Note For local estimation functions in the one-dimensional case, it is probably enough to consider elements S k, k=0,1,2,3 containing the polynomials of degree 0,1,2,3
Summary MDL SRM Local Risk Functional