How can we evaluate a portfolio of algorithms to extract meaningful interpretations about them? Suppose we have a set of algorithms. These can be classification, regression, clustering or any other type of algorithm. And suppose we have a set of problems that these algorithms can work on. We can evaluate these algorithms on the problems and get the results. From these results, can we explain the algorithms in a meaningful way? The easy option is to find which algorithm performs best for each problem and find the algorithm that performs best on the greatest number of problems. But, there is a limitation with this approach. We are only looking at the overall best! Suppose a certain algorithm gives the best performance on hard problems, but not on easy problems. We would miss this algorithm by using the “overall best” approach. How do we obtain a salient set of algorithm features?

1. Explainable
algorithm
evaluation from
lessons in education
Sevvandi Kandanaarachchi
Work with Kate Smith-Miles
MODSIM 2023
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2. What is an explanation?
• “To explain an event is to provide some information about its
causal history.” – Lewis , 1986 (Causal Explanation)
• “A statement or an account that makes something clear” –
Google
• “It is important to note that the solution to explainable AI is not
just ‘more AI’ “ - Miller, 2019
2

3. Explanation in the social sciences
• Miller (2019) argues for Social Science + Computer Science in XAI
“In the fields of philosophy, cognitive psychology/science, and social
psychology, there is a vast and mature body of work that studies these
exact topics. For millennia, philosophers have asked the questions
about what constitutes an explanation, what is the function of
explanations, and what are their structure. For over 50 years, cognitive
and social psychologists have analysed how people attribute and
evaluate the social behaviour of others in physical environments. For
over two decades, cognitive psychologists and scientists have
investigated how people generate explanations and how they evaluate
their quality.
I argue here that there is considerable scope to infuse this valuable
body of research into explainable AI.”
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4. Our contribution: bring methods
from social sciences for algorithm
evaluation
Evaluate algorithm performance in a way that we understand
algorithms and problems better!
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5. What is algorithm evaluation?
• Performance of many algorithms to many problems
• How do you explain the algorithm performance?
• Standard statistical analysis misses many things
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Algo 1 Algo 2 Algo 3 Algo 4
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5

6. Item Response Theory (IRT)
• Models used in social sciences/psychometrics
• Unobservable characteristics and observed outcomes
• Verbal or mathematical ability
• Racial prejudice or stress proneness
• Political inclinations
• Intrinsic “quality” that cannot be measured directly
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This Photo by Unknown Author is licensed under CC BY-SA

7. IRT in education
• Finds the discrimination and difficulty of test
questions
• And the ability of the test participants
• By fitting an IRT model
• In education – questions that can discriminate
between students with different ability is
preferred to “very difficult” questions.
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8. How it works
8
Questions
Students Q 1 Q 2 Q 3 Q 4
Stu 1 0.95 0.87 0.67 0.84
Stu 2 0.57 0.49 0.78 0.77
Stu n 0.75 0.86 0.57 0.45
IRT Model
Discrimination of questions 𝛼𝑗
Difficulty of questions 𝛽𝑗
Ability of students 𝜃𝑖 (latent trait)
Matrix 𝑌𝑁×𝑛
Finds the discrimination and
difficulty of test questions

9. The causal understanding in traditional IRT
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𝛼𝑗 𝛽𝑗 𝜃𝑖
𝑥𝑖𝑗
Discrimination of Q j Difficulty of Q j Ability of student i
Marks of student i for Question j
Student
Question
Marks

10. The mapping to algorithm evaluation
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𝛼𝑗 𝛽𝑗 𝜃𝑖
𝑥𝑖𝑗
Discrimination of Algo j Difficulty of Algo j Ability of Dataset i
Marks of dataset i for algorithm j
Dataset
Algorithm
Performance

11. After fitting the model, we get . . .
• Algorithm metrics
• Discrimination parameter -> Consistency, anomalousness indicator
• Difficulty parameter -> Difficulty limit of algorithm
• Dataset metrics
• Ability scores -> Difficulty of datasets
• AIRT – Algorithmic IRT
• Scottish word – to guide
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12. What can we say . . .
• Consistent algorithms give similar performance for easy or hard datasets
• Algorithms with higher difficulty limits can handle harder problems
• Anomalous algorithms give bad performance for easy problems and good
performance for difficult problems
• Difficult datasets give poorer performances to algorithms compared to
easier datasets
• General algorithm evaluation has best algorithm, or best 5 algorithms
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13. Anomaly detection
algorithms 13
Algorithm Consistency Difficulty Limit Anomalous
Ensemble 55.08 -66.55 FALSE
LOF 4.50 5.10 FALSE
KNN 1.72 2.30 FALSE
FAST_ABOD 9.39 10.23 FALSE
Isolation Forest 2.35 3.05 FALSE
KDEOS_1 0.86 -0.31 TRUE
KDEOS-2 1.16 -0.51 TRUE
LDF 2.01 2.08 FALSE

14. Dataset difficulty spectrum
IRT Student Ability - > Dataset Difficulty
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15. Focusing on algorithm performance
• We have algorithm performance (y axis)
and problem difficulty (x axis)
• We can fit a model and find how each
algorithm performs
• We use smoothing splines
• Can visualize them
• No parameters to specify
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16. 16
Strengths and
weaknesses

17. AIRT framework
17
Performance
data
IRT Model
Algorithm and
dataset metrics
Fitting
smoothing
splines
Algorithm
strengths and
weaknesses
Airt portfolio
Evaluate
portfolios
General block
AIRT output

18. 18
AIRT
IRT
Question and
student
characteristics
Algorithm and
dataset
characteristics
Visualize strengths
and weaknesses
Portfolio
construction

19. AIRT performs well, when . . .
• The set of algorithms is diverse.
• Ties back to IRT basics
• IRT in education – If all the questions are equally discriminative and
difficult, IRT doesn’t add much
• IRT useful when we have a diverse set of questions and we want to
know
• Which questions are more discriminative
• Which questions are difficult
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20. Summary
• Understand more about algorithms
• Anomalousness, consistency, difficulty limit
• Visualise the strengths and weaknesses of algorithms
• Select a portfolio of good algorithms
• Includes diagnostics
• R package airt on CRAN (uses EstCRM)
• https://sevvandi.github.io/airt/
• Pre-print: http://bit.ly/algorithmirt
• Comprehensive Algorithm Portfolio Evaluation using Item Response Theory
• More applications included
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21. We are hiring! – FairML Research
• CSIRO Postdoctoral Fellowship in Fairness
Research in Machine Learning
• Salary Range: AU$92,624 to AU$101,459 pa
• plus up to 15.4% superannuation
• 3-year contract
• https://jobs.csiro.au/go/CERC-Postdoctoral-and-
Engineering-Fellowships/7829300/
• Job will be advertised in August

22. 22

23. IRT in Data Science/Machine Learning
• Relatively new area of research
• Seminal paper
• 2019 - Item response theory in AI: Analysing machine learning
classifiers at the instance level – F. Martínez-Plumed et al.
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24. Dichotomous IRT
• Multiple choice
• True or false
• 𝜙 𝑥𝑖𝑗 = 1 𝜃𝑖, 𝛼𝑗, 𝑑𝑗, 𝛾𝑗 = 𝛾𝑗 +
1 −𝛾𝑗
1+exp(−𝛼𝑗(𝜃𝑖−𝑑𝑗))
• 𝑥𝑖𝑗 - outcome/score of examinee 𝑖 for item 𝑗
• 𝜃𝑖 - examinee’s (𝑖) ability
• 𝛾𝑗 - guessing parameter for item 𝑗
• 𝑑𝑗 - difficulty parameter
• 𝛼𝑗 - discrimination
This Photo by Unknown Author is licensed under CC BY-NC
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25. Continuous IRT
• Grades out of 100
• A 2D surface of probabilities
• 𝑃(𝑥𝑖𝑗|𝜃𝑖, 𝑑𝑗, 𝛼𝑗)
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𝑃(𝑥𝑖𝑗|𝜃𝑖, 𝑑𝑗, 𝛼𝑗)

26. Continuous IRT
• 𝑃(𝑥𝑖𝑗|𝜃𝑖, 𝑑𝑗, 𝛼𝑗)
• At a fixed 𝜃
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𝑃(𝑥𝑖𝑗|𝜃𝑖, 𝑑𝑗, 𝛼𝑗)
𝜃 = 2
𝜃 = −3,
x axis is the normalized score

27. Fitting the IRT model
• Maximising the expectation
• 𝐸 = 𝑁 𝑗(ln 𝛼𝑗 + ln |𝛾𝑗|) − 1/ 2 𝑖 𝑗 𝛼𝑗
2
𝛽𝑗 + 𝛾𝑗𝑧𝑖𝑗 − 𝜇𝑖
𝑡
2
+

28. Mapping algorithm evaluation to IRT
IRT Model
Students
Test questions
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Problems/datasets
Algorithms

29. What happens to the IRT parameters?
• IRT - 𝜃𝑖 - ability of student 𝑖
• As 𝜃 increases probability of a
higher score increases
• What is 𝜃𝑖, in terms of a
dataset?
• 𝜃𝑖 easiness of the dataset
• 𝛿𝑖 = −𝜃𝑖
• 𝛿𝑖 Dataset difficulty score
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30. Discrimination parameter
• Discrimination of item 𝑗 = 𝛼𝑗
• 𝛼𝑗increases → slope of curve
increases
• What is 𝛼𝑗, in terms of an
algorithm?
• 𝛼𝑗- lack of stability/robustness
of algo
• (1/|𝛼𝑗|) Consistency of algo
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31. Consistent algorithms
• Education – such a question
doesn’t give any information
• Algorithms – these algorithms
are really stable or consistent
• Consistency = 1/|𝛼𝑗|
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32. Anomalous algorithms
• Algorithms that perform poorly
on easy datasets and well on
difficult datasets
• Negative discrimination
• In education – such items are
discarded or revised
• If an algorithm anomalous, it is
interesting
• Anomalousness = sign(𝛼𝑗)
This Photo by Unknown Author is licensed under CC BY-NC-ND
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33. Example – fitting IRT model
• Graph Colouring algorithms (Smith-Miles et al 2014)
• 8 graph colouring algorithms
• RandomGreedy, DSATUR, Bktr, HillClimber, HEA, PartialCol, TabuCol, AntCol
• How many colours did each algorithm use to colour 6712 graphs
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34. Graph colouring
algorithms
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Algorithm Consistency Difficulty Limit Anomalous
Random Greedy 1.73 1.96 FALSE
DSATUR 0.57 1.79 FALSE
Bktr 0.32 1.97 FALSE
Hill Climber 0.68 3.18 FALSE
HEA 7.76 87.53 FALSE
PartialCol 2.25 19.21 FALSE
TabuCol 4.68 65.87 FALSE
AntCol 3.09 11.61 FALSE

35. Dataset difficulty spectrum
35

36. Focusing on algorithm performance
• We have algorithm performance (y axis)
and problem difficulty (x axis)
• We can fit a model and find how each
algorithm performs
• We use smoothing splines
• Can visualize them
• No parameters to specify
36

37. The smoothing splines
37

38. Strengths and weaknesses of algorithms
• If the curve is on top – it is strong in that region
• If the curve is at bottom – weak in that region
• ℎ𝑗∗ 𝛿 = max
𝑗
ℎ𝑗(𝛿) - this is the best performance for a given
difficulty
• 𝑗 ∗ is the best algorithm for that difficulty
• 𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ𝑠 𝑗, 𝜖 = { 𝛿: ℎ𝑗 𝛿 − ℎ𝑗∗ 𝛿 ≤ 𝜖}
• We give 𝜖 leeway
• Weaknesses are similar
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39. Strengths and
weaknesses
39

40. Algorithm portfolio selection
• Can use algorithm strengths to select a good portfolio of algorithms
• We call this portfolio airt portfolio
• airt – Algorithm IRT (old Scottish word – to guide)
• 𝑎𝑖𝑟𝑡 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝜖 = 𝑠𝑒𝑡 𝑜𝑓 𝑎𝑙𝑔𝑜𝑟𝑖𝑡ℎ𝑚𝑠 𝑤𝑖𝑡ℎ 𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ𝑠 (𝜖)
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41. Evaluating this portfolio
• Let 𝑖 denote a problem, 𝑃 a portfolio of algorithms, 𝐹 the full set of
algorithms
• 𝑃𝑒𝑟𝑓𝑜𝑟𝑚𝑎𝑛𝑐𝑒. 𝑑𝑒𝑡𝑒𝑟𝑖𝑜𝑟𝑎𝑡𝑖𝑜𝑛 𝑖, 𝑃 = 𝑏𝑒𝑠𝑡. 𝑝𝑒𝑟𝑓𝑜𝑟𝑚𝑎𝑛𝑐𝑒 𝑖, 𝐹 −
𝑏𝑒𝑠𝑡. 𝑝𝑒𝑟𝑓𝑜𝑟𝑚𝑎𝑛𝑐𝑒(𝑖, 𝑃)
Graph colouring example
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42. An example from ASlib data repository
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43. • SAT11-individual
example
• Example where
airt doesn’t do
so well.
• The curves are
quite similar to
each other
• Reason to
believe SAT11
has pre-selected
algorithms
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