Empirical Methods for AI (and CS)

CS1573: AI Application Development

(after a tutorial by Paul Cohen, Ian P. Gent, and Toby Walsh)

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Overview

Introduction What are empirical methods? Why use them?

Case Study Experiment design Data analysis

Empirical Methods for CS

Part I : Introduction

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What does “empirical” mean?

Relying on observations, data, experiments Empirical work should complement theoretical work

Theories often have holes Theories are suggested by observations Theories are tested by observations Conversely, theories direct our empirical attention

In addition (in this course at least) empirical means “wanting to understand behavior of complex systems”

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Why We Need Empirical Methods Cohen, 1990 Survey of 150 AAAI Papers

Roughly 60% of the papers gave no evidence that the work they described had been tried on more than a single example problem.

Roughly 80% of the papers made no attempt to explain performance, to tell us why it was good or bad and under which conditions it might be better or worse.

Only 16% of the papers offered anything that might be interpreted as a question or a hypothesis.

Theory papers generally had no applications or empirical work to support them, empirical papers were demonstrations, not experiments, and had no underlying theoretical support.

The essential synergy between theory and empirical work was missing

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Empirical CS/AI (Theory, not Theorems)

Computer programs are formal objects so let’s reason about them entirely formally?

Two reasons why we can’t or won’t: theorems are hard some questions are empirical in nature

e.g. are finite state automata adequate to represent the sort of knowledge met in practice?

e.g. even though our problem is intractable in general, are the instances met in practice easy to solve?

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Empirical CS/AI

Treat computer programs as natural objects like fundamental particles, chemicals, living organisms

Build (approximate) theories about them construct hypotheses

e.g. greedy search is important to chess test with empirical experiments

e.g. compare search with planning refine hypotheses and modelling assumptions

e.g. greediness not important, but search is!

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Empirical CS/AI

Many advantage over other sciences Cost

no need for expensive super-colliders Control

unlike the real world, we often have complete command of the experiment

Reproducibility in theory, computers are entirely deterministic

Ethics no ethics panels needed before you run experiments

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Types of hypothesis

My search program is better than yours

not very helpful beauty competition? Search cost grows exponentially with number of variables for

this kind of problem

better as we can extrapolate to data not yet seen? Constraint systems are better at handling over-constrained

systems, but OR systems are better at handling under-constrained systems

even better as we can extrapolate to new situations?

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A typical conference conversation

What are you up to these days?

I’m running an experiment to compare the Davis-Putnam algorithm with GSAT?

Why?

I want to know which is faster

Why?

Lots of people use each of these algorithms

How will these people use your result?...

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Keep in mind the BIG picture

What are you up to these days?

I’m running an experiment to compare the Davis-Putnam algorithm with GSAT?

Why?

I have this hypothesis that neither will dominate

What use is this?

A portfolio containing both algorithms will be more robust than either algorithm on its own

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Keep in mind the BIG picture

...

Why are you doing this?

Because many real problems are intractable in theory but need to be solved in practice.

How does your experiment help?

It helps us understand the difference between average and worst case results

So why is this interesting?

Intractability is one of the BIG open questions in CS!

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Why is empirical CS/AI in vogue?

Inadequacies of theoretical analysis problems often aren’t as hard in practice as theory predicts

in the worst-case average-case analysis is very hard (and often based on

questionable assumptions) Some “spectacular” successes

phase transition behaviour local search methods theory lagging behind algorithm design

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Why is empirical CS/AI in vogue?

Compute power ever increasing even “intractable” problems coming into range easy to perform large (and sometimes meaningful)

experiments Empirical CS/AI perceived to be “easier” than theoretical

CS/AI often a false perception as experiments easier to mess up

than proofs

Empirical Methods for CS

Part II: A Case Study

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Case Study

Scheduling processors on ring network jobs spawned as binary trees

KOSO keep one, send one to my left

or right arbitrarily KOSO*

keep one, send one to my least heavily loaded neighbour

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Theory

On complete binary trees, KOSO is asymptotically optimal

So KOSO* can’t be any better?

But assumptions unrealistic tree not complete asymptotically not necessarily

the same as in practice!

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Lesson: Evaluation begins with claimsLesson: Demonstration is good, understanding better

Hypothesis (or claim): KOSO takes longer than KOSO* because KOSO* balances loads better The “because phrase” indicates a hypothesis about why it

works. This is a better hypothesis than the beauty contest demonstration that KOSO* beats KOSO

Experiment design Independent variables: KOSO v KOSO*, no. of

processors, no. of jobs, probability(job will spawn), Dependent variable: time to complete jobs

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Criticism: This experiment design includes no direct measure of the hypothesized effect

Hypothesis: KOSO takes longer than KOSO* because KOSO* balances loads better

But experiment design includes no direct measure of load balancing: Independent variables: KOSO v KOSO*, no. of

processors, no. of jobs, probability(job will spawn), Dependent variable: time to complete jobs

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Lesson: The task of empirical work is to explain variability

run-time

Algorithm (KOSO/KOSO*)

Number of processors

Number of jobs

“random noise” (e.g., outliers)

Number of processors and number of jobs explain 74% of the variance in run time. Algorithm explains almost none.

Empirical work assumes the variability in a dependent variable (e.g., run time) is the sum of causal factors and random noise. Statistical methods assign parts of this variability to the factors and the noise.

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Lesson: Keep the big picture in mind

Why are you studying this?

Load balancing is important to get good performance out of parallel computers

Why is this important?

Parallel computing promises to tackle many of our computational bottlenecks

How do we know this? It’s in the first paragraph of the paper!

Empirical Methods for CS

Part III : Experiment design

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Experimental Life Cycle

Exploration Hypothesis construction Experiment Data analysis Drawing of conclusions

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Types of experiment designs

Manipulation experiment Observation experiment Factorial experiment

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Manipulation experiment

Independent variable, x x=identity of parser, size of dictionary, …

Dependent variable, y y=accuracy, speed, …

Hypothesis x influences y

Manipulation experiment change x, record y

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Observation experiment

Predictor, x x=volatility of stock prices, …

Response variable, y y=fund performance, …

Hypothesis x influences y

Observation experiment classify according to x, compute y

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Factorial experiment

Several independent variables, xi

there may be no simple causal links data may come that way

e.g. programs have different languages, algorithms, ... Factorial experiment

every possible combination of xi considered expensive as its name suggests!

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Some problem issues

Control Ceiling and Floor effects Sampling Biases

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Control

A control is an experiment in which the hypothesised variation does not occur so the hypothesized effect should not occur either

BUT remember placebos cure a large percentage of patients!

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Control: MYCIN case study

MYCIN was a medial expert system recommended therapy for blood/meningitis infections

How to evaluate its recommendations? Shortliffe used

10 sample problems, 8 therapy recommenders5 faculty, 1 resident, 1 postdoc, 1 student

8 impartial judges gave 1 point per problem max score was 80 Mycin 65, faculty 40-60, postdoc 60, resident 45, student 30

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Control: MYCIN case study

What were controls? Control for judge’s bias for/against computers

judges did not know who recommended each therapy Control for easy problems

medical student did badly, so problems not easy Control for our standard being low

e.g. random choice should do worse Control for factor of interest

e.g. hypothesis in MYCIN that “knowledge is power” have groups with different levels of knowledge

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Ceiling and Floor Effects

Well designed experiments (with good controls) can still go wrong

What if all our algorithms do particularly well Or they all do badly?

We’ve got little evidence to choose between them

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Ceiling and Floor Effects

Ceiling effects arise when test problems are insufficiently challenging floor effects the opposite, when problems too challenging

A problem in AI because we often repeatedly use the same benchmark sets most benchmarks will lose their challenge eventually? but how do we detect this effect?

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Ceiling Effects: machine learning

14 datasets from UCI corpus of benchmarks used as mainstay of ML community

Problem is learning classification rules each item is vector of features and a classification measure classification accuracy of method (max 100%)

Compare C4 with 1R*, two competing algorithms

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Ceiling Effects: machine learning

DataSet: BC CH GL G2 HD HE … Mean

C4 72 99.2 63.2 74.3 73.6 81.2 ... 85.9

1R* 72.5 69.2 56.4 77 78 85.1 ... 83.8

Max 72.5 99.2 63.2 77 78 85.1 … 87.4

C4 achieves only about 2% better than 1R* Best of the C4/1R* achieves 87.4% accuracy We have only weak evidence that C4 better Both methods performing near ceiling of possible so

comparison hard!

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Ceiling Effects: machine learning

In fact 1R* only uses one feature (the best one) C4 uses on average 6.6 features 5.6 features buy only about 2% improvement Conclusion?

Either real world learning problems are easy (use 1R*) Or we need more challenging datasets We need to be aware of ceiling effects in results

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Sampling bias

Data collection is biased against certain data e.g. teacher who says “Girls don’t

answer maths question” observation might suggest:

girls don’t answer many questions

but that the teacher doesn’t ask them many questions

Experienced AI researchers don’t do that, right?

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Sampling bias: Phoenix case study

AI system to fight (simulated) forest fires

Experiments suggest that wind speed uncorrelated with time to put out fire obviously incorrect as high

winds spread forest fires

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Sampling bias: Phoenix case study

Wind Speed vs containment time (max 150 hours):

3: 120 55 79 10 140 26 15 110 12 54 10 103

6: 78 61 58 81 71 57 21 32 70

9: 62 48 21 55 101 What’s the problem?

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Sampling bias: Phoenix case study

The cut-off of 150 hours introduces sampling bias many high-wind fires get cut off, not many low wind

On remaining data, there is no correlation between wind speed and time (r = -0.53)

In fact, data shows that: a lot of high wind fires take > 150 hours to contain those that don’t are similar to low wind fires

You wouldn’t do this, right? you might if you had automated data analysis.

Empirical Methods for CS

Part IV: Data analysis

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Kinds of data analysis

Exploratory (EDA) – looking for patterns in data Statistical inferences from sample data

Testing hypotheses Estimating parameters

Building mathematical models of datasets Machine learning, data mining…

We will introduce hypothesis testing and computer-intensive methods

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The logic of hypothesis testing

Example: toss a coin ten times, observe eight heads. Is the coin fair (i.e., what is it’s long run behavior?) and what is your residual uncertainty?

You say, “If the coin were fair, then eight or more heads is pretty unlikely, so I think the coin isn’t fair.”

Like proof by contradiction: Assert the opposite (the coin is fair) show that the sample result (≥ 8 heads) has low probability p, reject the assertion, with residual uncertainty related to p.

Estimate p with a sampling distribution.

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The logic of hypothesis testing

Establish a null hypothesis: H0: p = .5, the coin is fair Establish a statistic: r, the number of heads in N tosses Figure out the sampling distribution of r given H0

The sampling distribution will tell you the probability p of a result at least as extreme as your sample result, r = 8

If this probability is very low, reject H0 the null hypothesis Residual uncertainty is p

0 1 2 3 4 5 6 7 8 9 10

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A common statistical test: The Z test for different means

A sample N = 25 computer science students has mean IQ m=135. Are they “smarter than average”?

Population mean is 100 with standard deviation 15 The null hypothesis, H0, is that the CS students are “average”,

i.e., the mean IQ of the population of CS students is 100. What is the probability p of drawing the sample if H0 were true?

If p small, then H0 probably false. Find the sampling distribution of the mean of a sample of size

25, from population with mean 100

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Reject the null hypothesis?

Commonly we reject the H0 when the probability of obtaining a sample statistic (e.g., mean = 135) given the null hypothesis is low, say < .05.

A test statistic value, e.g. Z = 11.67, recodes the sample statistic (mean = 135) to make it easy to find the probability of sample statistic given H0.

We find the probabilities by looking them up in tables, or statistics packages provide them.

For example, Pr(Z ≥ 1.67) = .05; Pr(Z ≥ 1.96) = .01.

Pr(Z ≥ 11) is approximately zero, reject H0.

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Summary of hypothesis testing

H0 negates what you want to demonstrate; find probability p of sample statistic under H0 by comparing test statistic to sampling distribution; if probability is low, reject H0 with residual uncertainty proportional to p.

Example: Want to demonstrate that CS graduate students are smarter than average. H0 is that they are average. t = 2.89, p ≤ .022

Have we proved CS students are smarter? NO! We have only shown that mean = 135 is unlikely if they aren’t. We

never prove what we want to demonstrate, we only reject H0, with residual uncertainty.

And failing to reject H0 does not prove H0, either!

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Computer-intensive Methods

Basic idea: Construct sampling distributions by simulating on a computer the process of drawing samples.

Three main methods: Monte carlo simulation when one knows population parameters; Bootstrap when one doesn’t; Randomization, also assumes nothing about the population.

Enormous advantage: Works for any statistic and makes no strong assumptions (e.g., normality)

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Summary

Empirical CS and AI are exacting sciences There are many ways to do experiments wrong

We are experts in doing experiments badly As you perform experiments, you’ll make many mistakes Learn from those mistakes, and ours!