On-line learning and Boosting

Overview of “A Decision-Theoretic Generalization of On-Line Learning and an Application to Boosting,” by

Freund and Schapire (1997).

Tim Miller

University of Minnesota

Department of Computer Science and Engineering

Hedge - Motivation

Generalization of Weighted Majority Algorithm

Given a set of expert predictions, minimize mistakes over time

Slight emphasis in motivation on possibility of treating wt as a prior.

Hedge Algorithm

Parameters , w, T For 1..T

1. Choose allocation p (probability distribution formed from weights)

2. Receive loss vector l3. Suffer loss p l

4. Set new weight vector to w l

Hedge Analysis

Does not perform “too much worse” than best strategy: LHedge() ( - ln (w1) – Li ln ) · Z Z = 1 / (1 - )

Is it possible to do better?

Boosting

If we have n classifiers, possibly looking at the problem from different perspectives, how can we optimally combine them

Example: We have a collection of “rules of thumb” for predicting horse races, how to weight them

Definitions

Given labeled data < x, c(x) >, where c is the target concept, c: X {0, 1}.

c CC,, the concept class Strong PAC-learning algorithm: For

parameters ,, hypothesis has error less than with probability (1-)

Weak algorithm: (0.5 - ), > 0

AdaBoost Algorithm

Input: Sequence of N labeled examples Distribution D over the N examples Weak learning algorithm (called WeakLearn) Number of iterations T

AdaBoost contd.

Initialize: w1 = D For t =1..T

1. Form probability distribution p from w

2. Call WeakLearn with distribution p

3. Calculate error t = i=1..N pi | ht(xi) – yi |

4. Set t = t / (1 - t)

5. Multiplicatively adjust weights (w)by

t 1-|ht(xi)–yi|

AdaBoost Output

Output (+1) if: t=1..T (log 1/t) ht(x) ½ t=1..T log 1/t 0 otherwise Computes a weighted average

AdaBoost Analysis

Note of “dual” relationship with Hedge Strategies Examples Trials Weak hypotheses Hedge increases weight for successful strategies,

AdaBoost increases weight for difficult examples

AdaBoost has dynamic

AdaBoost Bounds

2T t=1..T sqrt(t(1 - t))

Previous bounds depended on maximum error of weakest hypothesis (weak link syndrome)

AdaBoost takes advantage of gains from best hypotheses

Multi-class Setting

k > 2 output labels, i.e. Y = {1, 2, …, k} Error: Probability of incorrect prediction Two algorithms:

AdaBoost.M1 – More direct AdaBoost.M2 – Somewhat complex constraints on weak

learners

Could also just divide into “one vs. one” or “one vs. all” categories

AdaBoost.M1

Requires each classifier to have error less than 50% (stronger requirement than binary case)

Similar to regular AdaBoost algorithm except: Error is 1 if ht(xi) yi

Can’t use algorithms with error > 0.5 Algorithm outputs vector of length k with values

between 0 & 1

AdaBoost.M1 Analysis

2T t=1..T sqrt(t(1 - t))

Same as bounds for regular AdaBoost Proof converts multi-class problem to a

binary setup Can we improve this algorithm?

AdaBoost.M2

More expressive, more complex constraints on weak hypotheses

Defines idea of “Pseudo-Loss” Pseudo-loss of each weak hypothesis must be

better than chance Benefit: Allows contributions from

hypotheses with accuracy < 0.5

Pseudo-loss

Replaces straightforward loss of AdaBoost.M1 plossq(h,i) =

0.5 ( 1 – h(xi,yi) + yyi q(i,y) h(xi,y)

Intuition: For each incorrect label, pit it against known label in binary classification (second term), then take a weighted average.

Makes use of information in entire hypothesis vector, not just prediction

AdaBoost.M2 Details

Extra init: wti,y = D(i) / (k-1)

For each iteration t = 1 to T Wt

i = yyi wti,y

qt(i,y) = wti,y / Wt

i

Dt(i) = Wti / i=1..N Wt

i

WeakLearn gets D as well as q Calculate t as shown above t = t / (1 - t) wt

i,y· t (0.5)(1 + ht(xi,yi) – ht(xi,y))

Error Bounds

(k – 1) 2Tt=1..T sqrt(t(1 - t)) Where is traditional error and the t are pseudo-

losses

Regression Setting

Instead of picking from a discrete set of output labels, choose a continuous value

More formally Y = [0, 1] Minimize the mean squared error:

E[(h(x) – y)2]

Reduce to binary classification and use AdaBoost!

How it works (roughly)

For each example in training set, create continuum of associated instances xtilde(xi, y) where y [0, 1].

Label is 1 if y yi

Mapping to an infinite training set – need to convert discrete distributions to density functions

AdaBoost.R Bounds

2T t=1..T sqrt(t (1 - t))

Conclusions

Starting from a on-line learning perspective, it is possible to generalize to boosting

Boosting can take weak learners and convert them to strong learners

This paper presented several algorithms to do boosting, with proofs of error bounds