a statistical analysis of the precision-recall graph

Microsoft Research 1

A Statistical Analysis of the Precision-Recall Graph

Ralf Herbrich, Hugo Zaragoza, Simon Hill.

Microsoft Research,

Cambridge University,

UK.

Hugo Zaragoza. AMS-IMS-SIAM Joint Summer Research Conference on Machine Learning, Statistics, and Discovery. June 2003. 2

Overview

2-class ranking Average-Precision From points to curves Generalisation bound Discussion

Hugo Zaragoza. AMS-IMS-SIAM Joint Summer Research Conference on Machine Learning, Statistics, and Discovery. June 2003. 3

“Search” cost-functions

Maximise the number of relevant documents found in the top 10.

Maximise the number of relevant documents at the top (e.g. weight inversely proportional to rank)

Minimise the number of documents seen by the user until he is satisfied.

Hugo Zaragoza. AMS-IMS-SIAM Joint Summer Research Conference on Machine Learning, Statistics, and Discovery. June 2003. 4

Motivation

Why should 45 August, 2003 work for document categorisation?

Why should any algorithm obtain good generalisation average-precision?

How to devise algorithms to optimise rank dependant loss-functions?

Hugo Zaragoza. AMS-IMS-SIAM Joint Summer Research Conference on Machine Learning, Statistics, and Discovery. June 2003. 5

2-class ranking problem

{y=1}

X,YMapping: X R

Relevancy: P(y=1|x) P(y=1|f(x))

Hugo Zaragoza. AMS-IMS-SIAM Joint Summer Research Conference on Machine Learning, Statistics, and Discovery. June 2003. 6

Collection samples

A collection is a sample:

z= ((x1,y1),...,(xm,ym)) (X x {0,1})m

where: y = 1 if the document x is relevant to a particular topic, z is drawn from the (unknown) distribution πXY

let k denote the number of positive examples

Hugo Zaragoza. AMS-IMS-SIAM Joint Summer Research Conference on Machine Learning, Statistics, and Discovery. June 2003. 7

Ranking the collection

We are given a scoring function f :XR This function imposes an order in the

collection: (x(1) ,…, x(m)) such that : f(x(1)) > … > f(x(m))

Hits (i1,…, ik) are the indices of the positive y(j).

f(x(i))

y(i) = 1 1 0 1 0 0 1 0 0 0

ij = 1 2 4 7

Hugo Zaragoza. AMS-IMS-SIAM Joint Summer Research Conference on Machine Learning, Statistics, and Discovery. June 2003. 8

Classification setting

If we threshold the function f, we obtain a classification:

Recall:

Precision:

f(x(i)) t

))(|1(1

1txfyPy

ip

i

j(j)i

)1|)((1

1

ytxfPy

kr

i

j(j)i

Hugo Zaragoza. AMS-IMS-SIAM Joint Summer Research Conference on Machine Learning, Statistics, and Discovery. June 2003. 9

Precision .vs. PGC

kmk kmk PGC PGC

PRECISION PRECISION

1

)1|)(()0|)((1

1

ytxfPytxfP

kmk

p

Hugo Zaragoza. AMS-IMS-SIAM Joint Summer Research Conference on Machine Learning, Statistics, and Discovery. June 2003. 10

The Precision-Recall Graph

After reordering:

f(x(i))

Hugo Zaragoza. AMS-IMS-SIAM Joint Summer Research Conference on Machine Learning, Statistics, and Discovery. June 2003. 11

Graph Summarisations

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Recall

Pre

cisi

on

Break-Even point

Hugo Zaragoza. AMS-IMS-SIAM Joint Summer Research Conference on Machine Learning, Statistics, and Discovery. June 2003. 12

Precision-Recall Example

Hugo Zaragoza. AMS-IMS-SIAM Joint Summer Research Conference on Machine Learning, Statistics, and Discovery. June 2003. 13

0

0.2

0.4

0.6

0.8

1

0.4 0.6 0.8 1

overfitting?

0.6

0.7

0.8

0.96 0.97 0.98 0.99 1Ave

rage

Pre

cisi

on (

TE

ST

SE

T)

Average Precision (TAIN SET)

Hugo Zaragoza. AMS-IMS-SIAM Joint Summer Research Conference on Machine Learning, Statistics, and Discovery. June 2003. 14

Overview

2-class ranking Average-Precision From points to curves Generalisation bound Discussion

Hugo Zaragoza. AMS-IMS-SIAM Joint Summer Research Conference on Machine Learning, Statistics, and Discovery. June 2003. 15

From point to curve bounds

There exist SVM margin-bounds [Joachims 2000] for precision and recall.

They only apply to a single (unknown a priori) point of the curve!

Pre

cisi

on

Recall

Hugo Zaragoza. AMS-IMS-SIAM Joint Summer Research Conference on Machine Learning, Statistics, and Discovery. June 2003. 16

Max-Min precision-recall

Hugo Zaragoza. AMS-IMS-SIAM Joint Summer Research Conference on Machine Learning, Statistics, and Discovery. June 2003. 17

Max-Min precision-recall (2)

Hugo Zaragoza. AMS-IMS-SIAM Joint Summer Research Conference on Machine Learning, Statistics, and Discovery. June 2003. 18

Features of Ranking Learning

We cannot take differences of ranks. We cannot ignore the order of ranks. Point-wise loss functions do not capture the

ranking performance! ROC or precision-recall curves do capture

the ranking performance. We need generalisation error bounds for

ROC and precision-recall curves

Hugo Zaragoza. AMS-IMS-SIAM Joint Summer Research Conference on Machine Learning, Statistics, and Discovery. June 2003. 19

Generalisation and Avg.Prec.

How far can the observed Avg.Prec. A(f,z)be from the expected average A(f) ?

How far can train and test Avg.Prec.?

?)(),(~

fAzfAP mXYz

),()(~

zfAfAXYz E

?),(),(~,

zfAzfAP mXYzz

Hugo Zaragoza. AMS-IMS-SIAM Joint Summer Research Conference on Machine Learning, Statistics, and Discovery. June 2003. 20

Approach

1. McDiarmid’s inequality: For any function g:ZnR with stability c, for all probability measures P with probability at least 1-δ over the IID draw of Z

Hugo Zaragoza. AMS-IMS-SIAM Joint Summer Research Conference on Machine Learning, Statistics, and Discovery. June 2003. 21

Approach (cont.)

2. Set n= 2m and call the two m-halves Z1 and Z2.

Define gi (Z):=A(f,Zi). Then, by IID :

Hugo Zaragoza. AMS-IMS-SIAM Joint Summer Research Conference on Machine Learning, Statistics, and Discovery. June 2003. 22

Bounding A(f,z) - A(f,zi)

1. How much does A(f,z) change if we can alter one sample (xi,yi)?

We need to fix the number of positive examples in order to answer this question!

e.g. if k=1, the change can be from 0 to 1.

Hugo Zaragoza. AMS-IMS-SIAM Joint Summer Research Conference on Machine Learning, Statistics, and Discovery. June 2003. 23

Stability Analysis

Case 1: yi=0

Case 2: yi=1

Hugo Zaragoza. AMS-IMS-SIAM Joint Summer Research Conference on Machine Learning, Statistics, and Discovery. June 2003. 24

Main Result

Theorem: For all probability measures, for all f:XR, with probability at least 1- δ over the IID draw of a training and test sample both of size m, if both training sample z and test sample z contain at least αm positive examples for all α(0,1), then:

Hugo Zaragoza. AMS-IMS-SIAM Joint Summer Research Conference on Machine Learning, Statistics, and Discovery. June 2003. 25

Hugo Zaragoza. AMS-IMS-SIAM Joint Summer Research Conference on Machine Learning, Statistics, and Discovery. June 2003. 26

Positive results

First bound which shows that asymptotically training and test set performance (in terms of average precision) converge!

The effective sample size is only the number of positive examples.

The proof can be generalised to arbitrary test sample sizes.

The constants can be improved.

Hugo Zaragoza. AMS-IMS-SIAM Joint Summer Research Conference on Machine Learning, Statistics, and Discovery. June 2003. 27

Open questions

How can we let k change, so as to investigate:

What algorithms could be used to directly maximise A(f,z) ?

)|)',(),((| zfAzfAPz

Hugo Zaragoza. AMS-IMS-SIAM Joint Summer Research Conference on Machine Learning, Statistics, and Discovery. June 2003. 28

Conclusions

Many problems require ranking objects in some degree.

Ranking learning requires to consider non-point-wise loss functions.

In order to study the complexity of algorithms we need to have large deviation inequalities for ranking performance measures.