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Foundations of Machine learning, HT 2022

创建日期：2024-04-27 16:58:09

所属分类：计算机computer science

标签： HT 2022

Foundations of Machine learning

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Problemset (3),

Foundations of Machine learning,
HT 2022
Maximilian Kasy
In this problemset you are asked to implement some simulations and
estimators in R. Please make sure that your solutions satisfy the following
conditions:
• The code has to run from start to end on the grader’s machine, pro-
ducing all the output.
• Output and discussion of findings have to be integrated in a report
generated in R-Markdown.
• Figures and tables have to be clearly labeled and interpretable.
• The findings need to be discussed in the context of the theoretical
results that we derived in class.
1. In this problem, you are asked to simulate data for a Bernoulli bandit
problem, where
Dt ∈ {1, . . . , k}, Yt = Y Dt , Y dt ∼ Ber(θd).
and treatment is assigned using Thompson sampling with a uniform
prior, (θ1, . . . , θk) ∼ U([0, 1]k).
(a) Set up a function which accepts a sample size T and a k-vector
(θ1, . . . , θk) as its arguments, and returns a history (Dt, Yt)
T
t=1
generated based on the Bernoulli bandit model and Thompson
sampling.
1
(b) Write a second function which takes the same arguments, plus a
number of replications R, and evaluates the first function R times
(using parallel computing; for instance the future package).
This function should return 4 vectors of length T : The averages of
Yt, θ
Dt , 1(Dt = max θ
d), and max θd− θDt , for each time periord
t.
(c) Pick a fixed vector of parameters (θ1, . . . , θk) and a time hori-
zon T and use the second function to plot cumulative average
regret as a function of t, using a large number of replications R
(such as R = 10.000). Repeat this for several different choices of
(θ1, . . . , θk).
How does the result relate to the theoretical regret rate bound
discussed in class, and to Agrawal and Goyal (2012)?
(d) Now let k = 2, fix θ1 = .5 and T = 200. Plot cumulative average
regret for T as a function of θ2, for θ2 ∈ [0, 1]. Do the same for
the share of observations assigned to the optimal treatment.
How does the result relate to the local-to-zero asymptotics dis-
cussed in class, and to Figure 3 in Wager and Xu (2021)?
2. In this problem, we will again consider the Bernoulli bandit, and com-
pare Thompson sampling to exploration sampling, as discussed in Kasy
and Sautmann (2021).
(a) Create a modified version of the first function from problem 1,
where instead of Thompson sampling treatment is assigned using
exploration sampling.
Let this function additionally return the treatment d∗T with the
highest posterior mean.
(b) Create a modified version of the second function from problem 1,
again replacing Thompson sampling by exploration sampling.
Let this function additionally return the average policy regret,
and the probability of choosing the best arm. Edit the second
function from problem 1 to do the same for Thompson sampling.
(c) Pick a fixed vector of parameters (θ1, . . . , θk) and a time horizon T
and calculate cumulative average regret as well as average policy
regret, for both Thompson sampling and exploration sampling.
Do so using a large number of replicationsR (such asR = 10.000).
2
How does the result line up with the theoretical characterization
and simulations of Kasy and Sautmann (2021)?
(d) Repeat this exercise for several different parameter vectors (θ1, . . . , θk)
and sample sizes T . Discuss any patterns you might find.
3. In this problem, we repeat the exercise of problem 1, but replace the
discrete treatment Dt by a continuous treatment Xt, and replace the
uniform prior for θ with a Gaussian process prior for the response
function m(·):
Xt ∈ [0, 1], Yt = m(Xt) + t, t|Xt ∼ N(0, 1).
Assume that Thompson sampling uses a Gaussian process prior of the
form
m(·) ∼ GP (0, C), C(x, x′) = τ2 exp
(
− (x1−x2)22λ
)
.
for τ2 = 4 and λ = 1/4.
(a) Write a function which takes a history of observations Xt, Yt as
its argument, and returns a draw from the posterior for m(·),
evaluated at a set of grid points (0, .01, .02, . . . , 1).
Let the function also return the maximizer of this posterior draw
(over the set of grid points).
(b) Write a second function which accepts a sample size T and a mean
function m(·) as its arguments1, and returns a history (Xt, Yt)Tt=1
generated based on the normal sampling model and Thompson
sampling.
(c) Repeat the task of items (b) and (c) from problem 1 for this
version of Thompson sampling, for different functions m(·).
You might for instance try polynomials, or trigonometric func-
tions.
(d) Vary the hyper-parameters λ and τ2, to see how the performance
of Thompson sampling is affected.
1Recall that you can pass functions as variables in R.
3
References
Agrawal, S. and Goyal, N. (2012). Analysis of thompson sampling for the
multi-armed bandit problem. In Conference on Learning Theory, pages
39–1.
Kasy, M. and Sautmann, A. (2021). Adaptive treatment assignment in
experiments for policy choice. Econometrica, 89(1):113–132.
Wager, S. and Xu, K. (2021). Diffusion asymptotics for sequential experi-
ments. arXiv preprint arXiv:2101.09855.

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