Paul Cohen Ian P. Gent Toby Walsh cohen@cs.umass.edu ipg@dcs.st-and.ac.uk tw@cs.york.ac.uk

 Introduction
 What are empirical methods?
 Why use them?
 Case Study
 Eight Basic Lessons
 Experiment design
 Data analysis
 How not to do it
 Supplementary material
2

 Web www.cs.york.ac.uk/~tw/empirical.html
www.cs.amherst.edu/~dsj/methday.html
 Books
“Empirical Methods for AI”, Paul Cohen, MIT Press, 1995
 Journals
Journal of Experimental Algorithmics, www.jea.acm.org
 Conferences
Workshop on Empirical Methods in AI (last Saturday, ECAI-02?)
Workshop on Algorithm Engineering and Experiments, ALENEX 01
(alongside SODA)
3

 Relying on observations, data, experiments
 Empirical work should complement theoretical work
 Theories often have holes (e.g., How big is the constant term? Is the current problem a “bad” one?)
 Theories are suggested by observations
 Theories are tested by observations
 Conversely, theories direct our empirical attention
 In addition (in this tutorial at least) empirical means “wanting to understand behavior of complex systems”
5

 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
6

 Theory based science need not be all theorems
 otherwise science would be mathematics
 Consider theory of QED
 based on a model of behaviour of particles
 predictions accurate to many decimal places (9?)
 most accurate theory in the whole of science?
 success derived from accuracy of predictions
 not the depth or difficulty or beauty of theorems
 QED is an empirical theory!
7

 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 Horn clauses 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?
8

 Treat computer programs as natural objects
 like fundamental particles, chemicals, living organisms
 Build (approximate) theories about them
 construct hypotheses e.g. greedy hill-climbing is important to GSAT
 test with empirical experiments e.g. compare GSAT with other types of hill-climbing
 refine hypotheses and modelling assumptions e.g. greediness not important, but hill-climbing is!
9

 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
10

 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 underconstrained systems even better as we can extrapolate to new situations?
11

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?
...
12

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
13

...
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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 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
15

 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
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 “An Empirical Study of Dynamic
Scheduling on Rings of
Processors”
Gregory, Gao, Rosenberg &
Cohen
Proc. of 8th IEEE Symp. on
Parallel & Distributed
Processing, 1996
Linked to from www.cs.york.ac.uk/~tw/empirical.html
18

 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
19

 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!
Thm: Using KOSO on a ring of p processors, a binary tree of height n is executed within (2^n-1)/p + low order terms
20

 More realistic trees
 probabilistic generator that makes shallow trees, which are
“bushy” near root but quickly get “scrawny”
 similar to trees generated when performing Trapezoid or
Simpson’s Rule calculations
 binary trees correspond to interval bisection
 Startup costs
 network must be loaded
21

 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
22

 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
23

 T-test on time to complete jobs: t = (2825-2935)/587 = -.19
 KOSO* apparently no faster than KOSO (as theory predicted)
 Why? Look more closely at the data:
80
70
60
50
40
30
20
10
KOSO
70
60
50
40
30
20
10
KOSO*
10000 20000 10000 20000
 Outliers create excessive variance, so test isn’t significant
24

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.
Algorithm (KOSO/KOSO*)
Number of processors run-time
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.
25

 Why does the KOSO/KOSO* choice account for so little of the variance in run time?
30
20
10
Queue length at processor i
50
40
KOSO
10
Queue length at processor i
30
KOSO*
20
100 200 300 100 200 300
 Unless processors starve, there will be no effect of load balancing. In most conditions in this experiment, processors never starved. (This is why we run pilot experiments!)
26

magnitude of effect t = sample size x s
N m background variance
This intimate relationship holds for all statistics
27

Let N = num-jobs, P = num-processors, T = run time
Then T = k (N / P), or k multiples of the theoretical best time
And k = 1 / (N / P T) t =
1.61
1.4
.08
= 2.42, p  .02
70
60
50
40
30
20
10
90
80
70
60
50
40
30
20
10
2 3 k(KOSO)
4 5 2 3 4 k(KOSO*)
5
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 KOSO* is significantly better than KOSO when the dependent variable is recoded as percentage of optimal run time
 The difference between KOSO* and KOSO explains very little of the variance in either dependent variable
 Exploratory data analysis tells us that processors aren’t starving so we shouldn’t be surprised
 Prediction: The effect of algorithm on run time (or k) increases as the number of jobs increases or the number of processors increases
 This prediction is about interactions between factors
29

3 multiples of optimal run-time
2
1
3 6 10 number of processors
20
KOSO
KOSO*
 Data confirm prediction
 KOSO* is superior on larger rings where starvation is an issue
 Interaction of independent variables
 choice of algorithm
 number of processors
 Interaction effects are essential to explaining how things work
30

 KOSO* is significantly better than KOSO, but can you use the result?
 Suppose you wanted to use the knowledge that the ring is controlled by
KOSO or KOSO* for some prediction.
 Grand median k = 1.11; Pr(trial i has k > 1.11) = .5
 Pr(trial i under KOSO has k > 1.11) = 0.57
 Pr(trial i under KOSO* has k > 1.11) = 0.43
 Predict for trial i whether it’s k is above or below the median:
 If it’s a KOSO* trial you’ll say no with (.43 * 150) = 64.5 errors
 If it’s a KOSO trial you’ll say yes with ((1 - .57) * 160) = 68.8 errors
 If you don’t know you’ll make (.5 * 310) = 155 errors
 155 - (64.5 + 68.8) = 22
 Knowing the algorithm reduces error rate from .5 to .43. Is this enough???
31

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!
32

 Evaluation begins with claims
 Demonstrations of simple main effects are good, understanding the effects is better
 Exploratory data analysis means using your eyes to find explanatory patterns in data
 The task of empirical work is to explain variablitity
 Control variability before increasing sample size
 Interaction effects are essential to explanations
 Significant ≠ meaningful
 Keep the big picture in mind
33

 Exploration
 Hypothesis construction
 Experiment
 Data analysis
 Drawing of conclusions
35

*
 Consider the experimental procedure
 making it explicit helps to identify spurious effects and sampling biases
 Consider a sample data table
 identifies what results need to be collected
 clarifies dependent and independent variables
 shows whether data pertain to hypothesis
 Consider an example of the data analysis
 helps you to avoid collecting too little or too much data
 especially important when looking for interactions
*From Chapter 3, “Empirical Methods for Artificial Intelligence”, Paul Cohen, MIT Press
36

 Consider possible results and their interpretation
 may show that experiment cannot support/refute hypotheses under test
 unforeseen outcomes may suggest new hypotheses
 What was the question again?
 easy to get carried away designing an experiment and lose the BIG picture
 Run a pilot experiment to calibrate parameters (e.g., number of processors in Rosenberg experiment)
37

 Manipulation experiment
 Observation experiment
 Factorial experiment
38

 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
39

 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
40

 Several independent variables, x i
 there may be no simple causal links
 data may come that way e.g. individuals will have different sexes, ages, ...
 Factorial experiment
 every possible combination of x i considered
 expensive as its name suggests!
41

 In general, stick to 2 to 3 independent variables
 Solve same set of problems in each case
 reduces variance due to differences between problem sets
 If this not possible, use same sample sizes
 simplifies statistical analysis
 As usual, default hypothesis is that no influence exists
 much easier to fail to demonstrate influence than to demonstrate an influence
42

 Control
 Ceiling and Floor effects
 Sampling Biases
43

 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!
44

 Macaque monkeys given vaccine based on human T-cells infected with SIV (relative of HIV)
 macaques gained immunity from SIV
 Later, macaques given uninfected human T-cells
 and macaques still gained immunity!
 Control experiment not originally done
 and not always obvious (you can’t control for all variables)
45

 MYCIN was a medial expert system
 recommended therapy for blood/meningitis infections
 How to evaluate its recommendations?
 Shortliffe used
 10 sample problems, 8 therapy recommenders
 5 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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 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
47

 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
48

 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?
49

 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
Rob Holte, Machine Learning, vol. 3, pp. 63-91, 1993 www.site.uottawa.edu/~holte/Publications/simple_rules.ps
50

DataSet : BC CH GL G2 HD HE … Mean
C4
1R*
72
72.5
99.2
69.2
63.2
56.4
74.3
77
73.6
78
81.2
85.1
...
...
85.9
83.8
Is 1R* above the floor of performance?
How would we tell?
51

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
Baseline 70.3
52.2
35.5
53.4
54.5
79.4
… 59.9
“Baseline rule” puts all items in more popular category.
1R* is above baseline on most datasets
A bit like the prime number joke?
1 is prime. 3 is prime. 5 is prime. So, baseline rule is that all odd numbers are prime.
52

DataSet : BC GL HY LY MU … Mean
C4
1R*
72
72.5
63.2
56.4
99.1
97.2
77.5
70.7
100.0 ...
98.4
...
85.9
83.8
 How do we know that C4 and 1R* are not near the ceiling of performance?
 Do the datasets have enough attributes to make perfect classification?
 Obviously for MU, but what about the rest?
53

DataSet :
C4
1R* max(C4,1R*) max([Buntine])
BC GL HY LY MU … Mean
72 63.2
99.1
77.5
100.0 ...
85.9
72.5
56.4
97.2
70.7
98.4
...
83.8
72.5
63.2
99.1
77.5
100.0
… 87.4
72.8
60.4
99.1
66.0
98.6
… 82.0
 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 appear to be near ceiling of possible so comparison hard!
54

 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
55

 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?
56

 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
57

 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?
58

 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.
59

 Assume gender (G) is an independent variable and number of siblings (S) is a noise variable.
 If S is truly a noise variable then under random sampling, no dependency should exist between G and S in samples.
 Parents have children until they get at least one boy. They don't feel the same way about girls. In a sample of 1000 girls the number with S = 0 is smaller than in a sample of 1000 boys.
 The frequency distribution of S is different for different genders. S and G are not independent.
 Girls do better at math than boys in random samples at all levels of education.
 Is this because of their genes or because they have more siblings?
 What else might be systematically associated with G that we don't know about?
60

 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
62

 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 .
63

 If the coin were fair (p= .5, the null hypothesis ) what is the probability distribution of r, the number of heads, obtained in
N tosses of a fair coin? Get it analytically or estimate it by simulation (on a computer):
 Loop K times
 r := 0 ;; r is num.heads in N tosses
 Loop N times ;; simulate the tosses
• Generate a random 0 ≤ x ≤ 1.0
• If x < p increment r ;; p is the probability of a head
 Push r onto sampling_distribution
 Print sampling_distribution
64

Frequency (K = 1000)
70
60
50
40
30
20
10
Probability of r = 8 or more heads in N = 10 tosses of a fair coin is 54 / 1000 = .054
0 1 2 3 4 5 6 7 8 9 10
Number of heads in 10 tosses
This is the estimated sampling distribution of r under the null hypothesis that p = .5. The estimation is constructed by
Monte Carlo sampling .
65

 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
0 1 2 3 4 5 6 7 8 9 10
 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
66

 Sampling distributions can be derived...
 Exactly, e.g., binomial probabilities for coins are given by the formula N !
r !( N r )!
 p N
 Analytically, e.g., the central limit theorem tells us that the sampling distribution of the mean approaches a Normal distribution as samples grow to infinity
 Estimated by Monte Carlo simulation of the null hypothesis process
67

 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
68

The sampling distribution of the mean is given by the Central Limit Theorem
The sampling distribution of the mean of samples of size N approaches a normal (Gaussian) distribution as N approaches infinity.
If the samples are drawn from a population with mean and m

  N m standard deviation , then the mean of the sampling distribution
These statements hold irrespective of the shape of the original distribution.
69

 If sample of N = 25 students were drawn from a population with mean 100 and standard deviation 15 (the null hypothesis) then the sampling distribution of the mean would asymptotically be normal with mean 100 and standard deviation 15 25 = 3
100 135
The mean of the CS students falls nearly 12 standard deviations away from the mean of the sampling distribution
Only ~1% of a normal distribution falls more than two standard deviations away from the mean
The probability that the students are
“average” is roughly zero
70

Mean of sampling distribution std=3
Sample statistic
Mean of sampling distribution std=1.0
Test statistic
100 135 0
Z = x -
 m
N
=
135 100
15
25
=
35
3
= 11.67
11.67
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 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.
72

 Same logic as the Z test, but appropriate when population standard deviation is unknown, samples are small, etc.
 Sampling distribution is t, not normal, but approaches normal as samples size increases
 Test statistic has very similar form but probabilities of the test statistic are obtained by consulting tables of the t distribution, not the normal
73

Suppose N = 5 students have mean IQ = 135, std = 27
Estimate the standard deviation of sampling distribution using the sample standard deviation t = x s m
N
=
135 100
27
5
=
35
12.1
= 2.89
Mean of sampling distribution
Sample statistic std=12.1
Mean of sampling distribution
Test statistic std=1.0
100 135 0 2.89
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 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!
75

 Tests that means are equal
 Tests that samples are uncorrelated or independent
 Tests that slopes of lines are equal
 Tests that predictors in rules have predictive power
 Tests that frequency distributions (how often events happen) are equal
 Tests that classification variables such as smoking history and heart disease history are unrelated
...
 All follow the same basic logic
76

 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 parametric assumptions (e.g., normality)
77

 Suppose you want to buy stocks in a mutual fund; for simplicity assume there are just N = 50 funds to choose from and you’ll base your decision on the proportion of J=30 stocks in each fund that increased in value
 Suppose Pr(a stock increasing in price) = .75
 You are tempted by the best of the funds, F, which reports price increases in 28 of its 30 stocks.
 What is the probability of this performance?
78

Loop K = 1000 times
B = 0
Loop N = 50 times
H = 0
Loop M = 30 times
;; stocks that increase in this fund
;; M is number of stocks in this fund
Toss a coin with bias p to decide whether this stock increases in value and if so increment H
Push H on a list ;; We get N values of H
B := maximum(H)
;; number of stocks that increase in
;; the best of N funds
;; N is number of funds
Push B on a list
;; The number of increasing stocks in
;; the best fund
;; We get K values of B
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 The probability that the best of 50 funds reports 28 of 30 stocks increase in price is roughly 0.4
 Why? The probability that an arbitrary fund would report this increase is Pr(28 successes | pr(success)=.75)≈.01, but the probability that the best of 50 funds would report this is much higher.
 Machine learning algorithms use critical values based on arbitrary elements, when they are actually testing the best element; they think elements are more unusual than they really are. This is why
ML algorithms overfit.
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 Monte Carlo estimation of sampling distributions assume you know the parameters of the population from which samples are drawn.
 What if you don’t?
 Use the sample as an estimate of the population.
 Draw samples from the sample!
 With or without replacement?
 Example: Sampling distribution of the mean; check the results against the central limit theorem.
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 S is a sample of size N:
Loop K = 1000 times
Draw a pseudosample S* of size N from S by sampling with replacement
Calculate the mean of S* and push it on a list L
 L is the bootstrapped sampling distribution of the mean**
 This procedure works for any statistic, not just the mean.
* Recall we can get the sampling distribution of the mean via the central limit theorem – this example is just for illustration.
** This distribution is not a null hypothesis distribution and so is not directly used for hypothesis testing, but can easily be transformed into a null hypothesis distribution (see Cohen, 1995).
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 Used to test hypotheses that involve association between elements of two or more groups; very general.
 Example: Paul tosses H H H H, Carole tosses T T T T is outcome independent of tosser?
 Example: 4 women score 54 66 64 61, six men score 23 28 27 31
51 32. Is score independent of gender?
 Basic procedure: Calculate a statistic f for your sample; randomize one factor relative to the other and calculate your pseudostatistic f*. Compare f to the sampling distribution for f*.
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 Four women score 54 66 64 61, six men score 23 28 27 31 51 32. Is score independent of gender?
 f = difference of means of men’s and women’s scores: 29.25
 Under the null hypothesis of no association between gender and score, the score 54 might equally well have been achieved by a male or a female.
 Toss all scores in a hopper, draw out four at random and without replacement, call them female*, call the rest male*, and calculate f*, the difference of means of female* and male*. Repeat to get a distribution of f*. This is an estimate of the sampling distribution of f under H0: no difference between male and female scores.
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 Those ignorant of history are doomed to repeat it
 we have committed many howlers
 We hope to help others avoid similar ones …
… and illustrate how easy it is to screw up!
 “How Not to Do It”
I Gent, S A Grant, E. MacIntyre, P Prosser, P Shaw,
B M Smith, and T Walsh
University of Leeds Research Report, May 1997
 Every howler we report committed by at least one of the above authors!
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 Do measure with many instruments
 in exploring hard problems, we used our best algorithms
 missed very poor performance of less good algorithms better algorithms will be bitten by same effect on larger instances than we considered
 Do measure CPU time
 in exploratory code, CPU time often misleading
 but can also be very informative e.g. heuristic needed more search but was faster
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 Do vary all relevant factors
 Don’t change two things at once
 ascribed effects of heuristic to the algorithm
 changed heuristic and algorithm at the same time
 didn’t perform factorial experiment
 but it’s not always easy/possible to do the “right” experiments if there are many factors
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 Do Collect All Data Possible ….
( within reason)
 one year Santa Claus had to repeat all our experiments
 ECAI/AAAI/IJCAI deadlines just after new year!
 we had collected number of branches in search tree
 performance scaled with backtracks, not branches
 all experiments had to be rerun
 Don’t Kill Your Machines
 we have got into trouble with sysadmins
… over experimental data we never used
 often the vital experiment is small and quick
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 Do It All Again … (or at least be able to)
 e.g. storing random seeds used in experiments
 we didn’t do that and might have lost important result
 Do Be Paranoid
 “identical” implementations in C, Scheme gave different results
 Do Use The Same Problems
 reproducibility is a key to science (c.f. cold fusion)
 can reduce variance
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 We’ve seen the possible problem of over-fitting
 remember machine learning benchmarks?
 Two common approaches
 benchmark libraries
 random problems
 Both have potential pitfalls
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 +ve
 can be based on real problems
 lots of structure
 -ve
 library of fixed size possible to over-fit algorithms to library
 problems have fixed size so can’t measure scaling
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 +ve
 problems can have any size so can measure scaling
 can generate any number of problems hard to over-fit?
 -ve
 may not be representative of real problems
 lack structure
 easy to generate “flawed” problems
 CSP, QSAT, …
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 Constraint satisfaction example
 40+ papers over 5 years by many authors used Models A,
B, C, and D
 all four models are “flawed” [Achlioptas et al. 1997]
 asymptotically almost all problems are trivial
 brings into doubt many experimental results
• some experiments at typical sizes affected
• fortunately not many
 How should we generate problems in future?
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 [Gent et al. 1998] fix flaw ….
 introduce “flawless” problem generation
 defined in two equivalent ways
 though no proof that problems are truly flawless
 Undergraduate student at Strathclyde found new bug
 two definitions of flawless not equivalent
 Eventually settled on final definition of flawless
 gave proof of asymptotic non-triviality
 so we think that we just about understand the problem generator now
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 Often need to implement an algorithm
 usually novel algorithm, or variant of existing one
 e.g. new heuristic in existing search algorithm
 novelty of algorithm should imply extra care
 more often, encourages lax implementation
 it’s only a preliminary version
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 Don’t Trust Yourself
 bug in innermost loop found by chance
 all experiments re-run with urgent deadline
 curiously, sometimes bugged version was better!
 Do Preserve Your Code
 Or end up fixing the same error twice
Do use version control!
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 Do Make it Fast Enough
 emphasis on enough
 it’s often not necessary to have optimal code
 in lifecycle of experiment, extra coding time not won back
 e.g. we have published many papers with inefficient code
 compared to state of the art
• first GSAT version O(N 2 ), but this really was too slow!
• Do Report Important Implementation Details
 Intermediate versions produced good results
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 Do Look at the Raw Data
 Summaries obscure important aspects of behaviour
 Many statistical measures explicitly designed to minimise effect of outliers
 Sometimes outliers are vital
 “exceptionally hard problems” dominate mean
 we missed them until they hit us on the head when experiments “crashed” overnight old data on smaller problems showed clear behaviour
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 Do face up to the consequences of your results
 e.g. preprocessing on 450 problems
 should “obviously” reduce search
 reduced search 448 times
 increased search 2 times
 Forget algorithm, it’s useless?
 Or study in detail the two exceptional cases
 and achieve new understanding of an important algorithm
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 Outline some of the basic issues
 exploration, experimental design, data analysis, ...
 Encourage you to consider some of the pitfalls
 we have fallen into all of them!
 Raise standards
 encouraging debate
 identifying “best practice”
 Learn from your experiences
 experimenters get better as they get older!
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 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!
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 Bernard Moret, U. New Mexico
“Towards a Discipline of Experimental Algorithmics”
 David Johnson, AT&T Labs
“A Theoretician’s Guide to the Experimental Analysis of
Algorithms”
Both linked to from www.cs.york.ac.uk/~tw/empirical.html
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 Useful types of empirical results:
 accuracy/correctness of theoretical results
 real-world performance
 heuristic quality
 impact of data structures
 ...
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 Hallmarks of a good experimental paper
 clearly defined goals
 large scale tests both in number and size of instances
 mixture of problems real-world, random, standard benchmarks, ...
 statistical analysis of results
 reproducibility publicly available instances, code, data files, ...
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 Pitfalls for experimental papers
 simpler experiment would have given same result
 result predictable by (back of the envelope) calculation
 bad experimental setup e.g. insufficient sample size, no consideration of scaling,
…
 poor presentation of data e.g. lack of statistics, discarding of outliers, ...
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 Ideal experimental procedure
 define clear set of objectives which questions are you asking?
 design experiments to meet these objectives
 collect data do not change experiments until all data is collected to prevent drift/bias
 analyse data consider new experiments in light of these results
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 3 types of paper describe the implementation of an algorithm
 application paper
“Here’s a good algorithm for this problem”
 sales-pitch paper
“Here’s an interesting new algorithm”
 experimental paper
“Here’s how this algorithm behaves in practice”
 These lessons apply to all 3
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 Perform “newsworthy” experiments
 standards higher than for theoretical papers!
 run experiments on real problems theoreticians can get away with idealized distributions but experimentalists have no excuse!
 don’t use algorithms that theory can already dismiss
 look for generality and relevance don’t just report algorithm A dominates algorithm B, identify why it does!
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 Place work in context
 compare against previous work in literature
 ideally, obtain their code and test sets verify their results, and compare with your new algorithm
 less ideally, re-implement their code report any differences in performance
 least ideally, simply report their old results try to make some ball-park comparisons of machine speeds
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 Use efficient implementations
 “somewhat” controversial
 efficient implementation supports claims of practicality tells us what is achievable in practice
 can run more experiments on larger instances can do our research quicker!
 don’t have to go over-board on this
 exceptions can also be made e.g. not studying CPU time, comparing against a previously newsworthy algorithm, programming time more valuable than processing time, ...
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 Use testbeds that support general conclusions
 ideally one (or more) random class, & real world instances predict performance on real world problems based on random class, evaluate quality of predictions
 structured random generators parameters to control structure as well as size
 don’t just study real world instances hard to justify generality unless you have a very broad class of real world problems!
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 Provide explanations and back them up with experiment
 adds to credibility of experimental results
 improves our understanding of algorithms leading to better theory and algorithms
 can “weed” out bugs in your implementation!
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 Ensure reproducibilty
 easily achieved via the Web
 adds support to a paper if others can (and do) reproduce the results
 requires you to use large samples and wide range of problems otherwise results will not be reproducible!
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 Ensure comparability (and give the full picture)
 make it easy for those who come after to reproduce your results
 provide meaningful summaries give sample sizes, report standard deviations, plot graphs but report data in tables in the appendix
 do not hide anomalous results
 report running times even if this is not the main focus readers may want to know before studying your results in detail
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 Failing to report key implementation details
 Extrapolating from tiny samples
 Using irreproducible benchmarks
 Using running time as a stopping criterion
 Ignoring hidden costs (e.g. preprocessing)
 Misusing statistical tools
 Failing to use graphs
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 Obscuring raw data by using hard-to-read charts
 Comparing apples and oranges
 Drawing conclusions not supported by the data
 Leaving obvious anomalies unnoted/unexplained
 Failing to back up explanations with further experiments
 Ignoring the literature the self-referential study!
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