COMP90051 Statistical Machine Learning

Project 2 Description

Due date: 4:00pm Thursday, 17th October 2019 Weight: 25%1

Multi-armed bandits (MABs) are a powerful tool in statistical machine learning: they bridge decision

making, control, optimisation and learning; they address practical problems of sequential decision making while

backed by elegant theoretical guarantees; they are relatively easily implemented, efficient to run, and are used

in many industrial applications. They are neither fully supervised nor unsupervised, being partial supervised by

indirect rewards—as a subroutine they employ supervised learning for predicting future rewards. Exploitation

behaviour in MABs optimises short-term rewards by acting greedily based on current knowledge; but this must

be balanced against imprecision in knowledge by exploration; and when effectively balanced, MABs optimise

for long-term cumulative reward. In this project, you will work individually (not in teams) to implement several

MAB learners. Some will be directly from class, while others will be more advanced and come out of papers

that you will have to read and understand yourself.

By the end of the project you should have developed

ILO1. A deeper understanding of the MAB setting and common MAB approaches, and an appreciation of how

MABs are applied;

ILO2. Better understanding of how the Bayesian paradigm can support machine learning;

ILO3. Demonstrable ability to implement ML approaches in code; and

ILO4. An ability to pick up recent machine learning publications in the literature, understand their focus,

contributions, and algorithms enough to be able to implement and apply them. (And being able to

ignore other presented details not needed for your task.)

Overview

Through the 2000s Yahoo! Research led the way in applying MABs to problems in online advertising, information retrieval, and media recommendation. One of their many applications was to Yahoo! News, in deciding

what news items to recommend to users based on article content, user profile, and the historical engagement

of the user with articles. Given decision making in this setting is sequential—what do we show next—and

feedback is only available for articles shown, Yahoo! researchers observed a perfect formulation for MABs like

those (-Greedy and UCB) learned about in class. Going further, however, they realised that incorporating

some element of user-article state requires contextual bandits: articles are arms; context per round incorporates information about both user and article (arm); and {0, 1}-valued rewards represent clicks. Therefore the

per round cumulative reward represents click-through-rate (CTR) which is exactly what services like Yahoo!

News want to maximise to drive user engagement and advertising revenue. You will be implementing these

approaches, noting that you need not necessarily complete the entire project.

Required Resources The LMS page for project 2 comprises

• project2.pdf this spec;

• proj2.ipynb Jupyter notebook: skeleton in Python; and

• dataset.txt A text-file dataset (see below for details).

1Forming a combined hurdle with project 1.

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You will implement code in Python Jupyter notebooks, which after running on your machine you will submit

via LMS. Further detailed rules about what is expected with code are available towards the end of this spec. We

appreciate that while some have entered COMP90051 with little/no prior Python experience, many workshops

so far have exercised and built up basic Python and Jupyter knowledge.

Part 1: Implementing -Greedy and UCB [3 marks total]

Implement Python classes EpsGreedy and UCB for both -Greedy and UCB learners as covered in class. You

should use inheritance: make your classes sub-classes of the abstract MAB base class. Include components:

• All necessary properties for storing MAB state

• __init__ constructor methods for initialising MAB state with respective signatures:

– def __init__(self, narms, epsilon, Q0) for positive integer narms, floating-point probability

epsilon, real-valued Q0 taking by default numpy.inf; and

– def __init__(self, narms, rho, Q0) for positive integer narms, positive real rho, real-valued Q0

taking by default numpy.inf.

• Additional methods (where in your implementations context will go unused)

– def play(self, tround, context) for positive integer tround, and unused (for now) context.

This should return an arm integer in {1, . . . , self.narms}; and

– def update(self, arm, reward, context) for positive integer arm no larger than property self.narms,

floating-point reward, and unused (for now) context. This method should not return anything.

Tie-breaking in play() should be completed uniformly-at-random among value-maximising arms.

Part 2: The Basic Thompson Bandit [5 marks total]

Your next task is to implement a third bandit learner, one that you haven’t seen in class. Thompson sampling

is named after Thompson who discovered the idea in 1933 (before the advent of machine learning), and went

unnoticed by the machine learning community until relatively recently. It is now regarded as a leading MAB

technique, that uses a Bayesian model of rewards. The simplest Thompson sampler models rewards in {0, 1}

as Bernoulli draws with different parameters per arm each starting with a common Beta prior.

In this part you are to implement a Python class for the Beta-Bernoulli Thompson MAB as described in

Algorithm 1 “Thompson Sampling for Bernoulli bandits” from the paper:

Shipra Agrawal and Navin Goyal, ‘Analysis of Thompson sampling for the multi-armed bandit

problem’, in Proceedings of the Conference on Learning Theory (COLT 2012), 2012.

http://proceedings.mlr.press/v23/agrawal12/agrawal12.pdf

While the COLT’2012 Algorithm 1 only considers a uniform Beta(1, 1) prior, you are to implement a more

flexible Beta(α0, β0) prior for any given α0, β0 > 0. A point that may be missed on first reading, is that while

each arm begins with the same prior, each arm updates its own posterior.

Your class BetaThompson should sub-class abstract MAB base class with components similar to above:

• All necessary properties for storing MAB state

• Constructor for initialising MAB state with signature __init__(self, narms, alpha0, beta0) for positive integer narms, positive reals alpha0, beta0 taking by default 1;

• Additional methods (where again context will go unused) of play() and update() as above.

Tie-breaking in play() should again be completed uniformly-at-random among value-maximising arms.

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Part 3: Off-Policy Evaluation [3 marks total]

A major practical challenge for industry deployments of MAB learners has been the requirement to let the

learner loose on real data. Inevitably bandits begin with little knowledge about arm reward structure, and so

a bandit must necessarily suffer poor rewards in beginning rounds. For a company trying out and evaluating

dozens of bandits in their data science groups, this is potentially very expensive.

A breakthrough was made when it was realised that MABs can be evaluated offline or off policy. The

idea being that you collect just once a dataset of uniformly-random arm pulls and resulting rewards. Then you

evaluate any possible future bandit learner of interest on that one historical data—there is no need to run bandits

online in order to evaluate them! In this part you are to implement a Python function for offline/off-policy

evaluation.

You must implement the algorithm first described as Algorithm 3 “Policy Evaluator” from the paper:

Lihong Li, Wei Chu, John Langford, Robert E. Schapire, ‘A Contextual-Bandit Approach to Personalized News Article Recommendation’, in Proceedings of the Nineteenth International Conference

on World Wide Web (WWW 2010), Raleigh, NC, USA, 2010.

https://arxiv.org/pdf/1003.0146.pdf

You should begin by reading Section 4 of the WWW2010 paper which describes the algorithm. You may

find it helpful to read the rest of the paper up to this point for background (skipping Sec 3.2) as this also relates

to Part 4. If you require further detail of the algorithm you may find the follow-up paper useful (particularly

Sec 3.1):

Lihong Li, Wei Chu, John Langford, and Xuanhui Wang. ‘Unbiased offline evaluation of contextualbandit-based news article recommendation algorithms.’ In Proceedings of the Fourth ACM International Conference on Web Search and Data Mining (WSDM’2011), pp. 297-306. ACM, 2011.

https://arxiv.org/pdf/1003.5956.pdf

Note that what is not made clear in the pseudo-code of Algorithm 3, is that after the bandit plays (written

as function or policy π) an arm that matches a given log, you should not only note down the reward as if the

bandit really received this reward, but you should also update the bandit with the played arm a, reward ra, and

later in the project the context x1, . . . , xK over the K arms. Bandits that do not make use of context—such as

your Part 1 and 2 bandits—can still take context as an argument even if unused.

A second point that is implied in the pseudo-code, but may be missed, is that when asking the bandit to

play an arm, the supplied round number should not be the current round in the log file, but instead the length

of history recorded so far, plus one. That is, after playing a matching arm on the first logged event, a bandit

may play different arms for events 2, 3, and 4, and on the 5th event may for a second time play a matching

arm. For the function calls to play() for each of events 2, 3, 4, 5 you would pass as the tround argument the

value 2. You would then increment to 3 for tround from the 6th event.

Implement your function (nominally outside any Python class) with signature

def offlineEvaluate(mab, arms, rewards, contexts, nrounds=None)

for a MAB class object mab such as EpsGreedy, UCB, BetaThompson (and the classes implemented

in later project Parts), a (numpy) array arms of values in {1, . . . , mab.narms}, an array of scalar

numeric rewards of the same length as arms, a numeric 2D array contexts with number of rows

equal to the length of arms and number of columns equal to a positive multiple of mab.narms, a

positive integer nrounds with default value None.

Here arms corresponds to the arms played by a uniformly-random policy recorded in a dataset of say M

events. While rewards corresponds to the resulting observed M rewards. In the next Part we will consider

contextual bandits, in which each arm may have a feature vector representing its state/context (and potentially

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factoring in the context of the user also). So that if each of the K arms have d features, each row of contexts

will have these feature vectors flattened as the d features of arm 1 followed by the d features of arm 2, all the

way up to arm K so that we have d × K features (a multiple of K).

Finally nrounds is the desired number of matching events we would like to evaluate bandit mab on. Once

your function finds this many matching arm plays, it should stop and return the per round rewards—and not

their sum as in the WWW’2010 Algorithm 3. If it reaches the end of the logged dataset without reaching the

required number (or in the case of default None) then it should return the per round rewards recorded.

Dataset: The LMS page for project 2 contains a 2 MB dataset.txt suitable for validating MAB implementations. You may download this file and familiarise yourself with its format:

• 10,000 lines (i.e., rows) corresponding to distinct site visits by users—events in the language of this part;

• Each row comprises 102 space-delimited columns of integers:

– Column 1: The arm played by a uniformly-random policy out of 10 arms (news articles);

– Column 2: The reward received from the arm played—1 if the user clicked 0 otherwise; and

– Columns 3–102: The 100-dim flattened context: 10 features per arm (incorporating the content of

the article and its match with the visiting user), first the features for arm 1, then arm 2, etc. up to

arm 10.

Your function should be able to run on this file where column 1 forms arms, column 2 forms rewards, and

columns 3–102 form contexts. On both classes you’ve implemented thus far. You should output the result of

running

mab = EpsGreedy(10, 0.05)

results_EpsGreedy = offlineEvaluate(mab, arms, rewards, contexts, 800)

print(“EpsGreedy average reward “, np.mean(results_EpsGreedy))

mab = UCB(10, 1.0)

results_UCB = offlineEvaluate(mab, arms, rewards, contexts, 800)

print(“UCB average reward “, np.mean(results_UCB))

mab = BetaThompson(10, 1.0, 1.0)

results_BetaThompson = offlineEvaluate(mab, arms, rewards, contexts, 800)

print(“BetaThompson average reward “, np.mean(results_BetaThompson))

Part 4: Contextual Bandits—LinUCB [5 marks total]

In this part you are to implement a fourth MAB learner as a fourth Python class. This time you are to read up

to and including Section 3.1 of the WWW’2010 paper to understand and then implement the LinUCB learner

with disjoint linear models (Algorithm 1). This is a contextual bandit—likely the first you’ve seen—however

it’s workings are a direct mashup of UCB and ridge regression both of which you’ve seen in class. Practicing

reading and implementing papers is the best way to turbo-charge your ML skills. Your class LinUCB should

have methods

• def __init__(self, narms, ndims, alpha) constructor for positive integer narms the number of arms,

positive integer ndims the number of dimensions for each arm’s context, positive real-valued alpha a

hyperparameter balancing exploration-exploitation

• def play(self, tround, context) as for your other classes. For positive integer tround, and context

being a numeric array of length self.ndims * self.narms; and

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• def update(self, arm, reward, context) as for your other classes. For positive integer arm no larger

than property self.narms, floating-point reward, and context as previous.

While the idea of understanding LinUCB enough to implement it correctly may seem daunting, the WWW’2010

paper is written for a non-ML audience and is complete in its description. The pseudo-code is detailed. There

is one unfortunate typo however: pg. 3, column. 2, line 3 of the linked arXiv version should read ca rather

than ba. The pseudo-code uses the latter (correctly) as shorthand for the former times the contexts.

Note also one piece of language you may not have encountered: “design matrix” means a feature matrix in

the statistics literature.

After you have implemented your class, include and run an evaluation on the given dataset with

mab = LinUCB(10, 10, 1.0)

results_LinUCB = offlineEvaluate(mab, arms, rewards, contexts, 800)

print(“LinUCB average reward “, np.mean(results_LinUCB))

Part 5: Contextual Bandits—LinThompson [6 marks total]

Just as LinUCB mashes up ridge regression (with confidential intervals) with UCB for the contextual MAB

problem, we may plug Bayesian linear regression into the Thompson sampling framework to tackle contextual

bandit learning. This idea is considered by the paper:

Shipra Agrawal and Navin Goyal, ‘Thompson sampling for contextual bandits with linear payoffs’,

in Proc. International Conference on Machine Learning (ICML 2013), pp. 127-135. 2013.

http://proceedings.mlr.press/v28/agrawal13.pdf

In this Part you are to implement as a fifth MAB class LinThompson, the ICML’2013 Algorithm 1 (described

in Section 2.2). Your class LinThompson should have methods

• def __init__(self, narms, ndims, v) constructor for positive integer narms the number of arms,

positive integer ndims the number of dimensions for each arm’s context, v a hyperparameter controlling

exploration vs. exploitation;

• def play(self, tround, context) as for your other classes. For positive integer tround, and context

being a numeric array of length self.ndims * self.narms; and

• def update(self, arm, reward, context) as for your other classes. For positive integer arm no larger

than property self.narms, floating-point reward, and context as previous.

While the ICML’2013 intro does re-introduce the Thompson sampling framework explored in Part 2, it can

be very much skimmed. Section 2.1 introduces the setting formally—information on regret and the assumptions2

are not important for simple experimentation. Section 2.2 and Algorithm 1 ‘Thompson Sampling for Contextual

bandits’ is the key place to find the described algorithm to be implemented. Note that the first 14 lines of

Section 2.3 explains the hyperparameters , δ further: the latter controls our confidence of regret being provably

low (and so we might imagine taking it to be 0.05 as in typical confidence intervals); while advice is given for

setting when you know the total number of rounds to be played. All that said, R, , δ only feature in the

expression v, while we wouldn’t really know R. And so you should just use v as a hyperparameter to control

exploration balance as you have in the previous part with α.

After you have implemented your class, include and run an evaluation on the given dataset with

2Sub-Gaussianity is a generalisation of Normally distributed rewards. I.e., while they don’t assume the rewards are Normal, they

assume something Normal-like in order to obtain theoretical guarantees. R takes the role of 1/σ and controls how fast likelihood of

extreme rewards decays. You can ignore all this–phew!

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mab = LinThompson(10, 10, 1.0)

results_LinThompson = offlineEvaluate(mab, arms, rewards, contexts, 800)

print(“LinThompson average reward “, np.mean(results_LinThompson))

Part 6: Evaluation [3 marks total]

In this part you are to delve deeper into the performance of your implemented bandit learners. This part’s first

sub-part does not necessarily require completion of Parts 4 and 5.

Part 6(a) [1 marks]: Run offlineEvaluate on each of your Python classes just with hyperparameters

as when you ran offlineEvaluate above. This time plot the running per-round cumulative reward i.e.

T

−1 PT

t=1 rt,a for T = 1..800 as a function of round T, all on one overlayed plot. Your plot will have up

to 5 curves, clearly labelled.

Part 6(b) [2 marks]: How can you optimise hyperparameters? Devise grid-search based strategies to select

the α and v hyperparameters in LinUCB and LinThompson, as Python code in your notebook. Output the

result of this strategy—which could be a graph, number, etc.

Project Submission

Preserving its structure, you must (1) rename proj2.ipynb as username.ipynb using your username3

(2) flesh

out with your project solutions with cells, (3) run on your local machine prior to submission so that outputs and

plots are preserved (you are strongly recommended to open your notebook again prior to upload to double check.

We may not run your notebook; given your environment might subtly differ to ours, it is your responsibility to

ensure results are contained), and then (4) submit in LMS.

Marks: graders will perform code reviews of your implementations. In general a portion will be available for

correctness, a portion will be available for code structure and style (primarily the former). Code should have

necessary commenting to understand interfaces, and major points of inner working, basic checks of well-formed

input, clear variable names and readable statements.

Further Rules. You may discuss the bandit learning deck or Python at a high-level with others, but do

not collaborate on solutions. You may consult resources to understand bandits conceptually, but do not make

any use of online code whatsoever. (We will run code comparisons against online partial implementations to

enforce these rules.) You must use the environment (Anaconda3, Python 3.6 or higher) as used in labs. (You

may use your own machine of course, but we strongly recommend you check code operation on lab machines

prior to submission. In case we run code.) You may only use the packages already imported in the provided

proj2.ipynb notebook. You should use matplotlib for plotting. Late submissions will be accepted to 4 days

with -3 penalty per day.

3LMS/UniMelb usernames look like brubinstein, not to be confused with email such as benjamin.rubinstein.

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