Description
1 Theory
Question 1. [15 points] Multi-Arm Bandits
Consider the pure exploration setting where a learner is confronted with a multi-arm
bandit problem defined by K arms with reward distributions {Ri}
K
i=1. Each Ri has
bounded support in [0, 1] and mean µi
. The learner explores the arms for a number
of steps and has to pick the best one after T steps: µ
∗ = maxi=[K] µi
. Imagine the
simple setting where the agent chooses each arm uniformly for a fixed number of
trials T and then picks the one with the highest mean (each arm will be pulled T/K
times). Derive the value of T, i.e. the number of arm pulls – needed to guarantee that
µ
∗ − µbi 6 ε, ∀ε > 0 with probability 1 − δ, δ ∈ (0, 1), where bi is the arm the agent
chooses after T steps of interaction. T should be in big O notation as a function
of ε and δ. (Tip: use union bound and a concentration inequality, e.g. Hoeffding’s
inequality).
Question 2. [10 points] Markov Decision Processes
In this question we consider two MDPs which differ slightly. You will analyze how
the value function differs for these MDPs.
i Consider two MDPs M = (S, A, P, R, γ) and M¯ = (S, A, P, R, γ ¯ ), which only
differ in their reward functions. We have for any s ∈ S, a ∈ A, R¯(s, a) =
R(s, a) + N (µ, σ2
), where N is a Gaussian with constant mean µ, and variance
σ
2
that does not depend on s or a. For any policy π, let V
π
M denote its value
function in M and V
π
M¯ denote its value function in M¯ . For any s ∈ S, express
V
π
M(s) in terms of µ and V
π
M¯ (s).
ii In the previous part, we saw how the values differs if the reward function is changed.
In this question we explore the effects if the transition matrix is changed. Consider
two MDPs M = (S, A, P, R, γ) and M¯ = (S, A, P, R, γ ¯ ) where P¯ = (αP + βQ),
α ∈ [0, 1], β ∈ [0, 1] are constants such that α + β = 1, Q is another transition
matrix with the same dimension as P. Let V
π
M denote the value function in M
and V
π
M¯ denote the value function in M¯ . For any s ∈ S, express V
π
M¯ (s) in terms
of V
π
M(s).
Question 3. [15 points] Policy Evaluation and Improvement
Consider a discrete, finite state MDP whose optimal value is V
?
. Let γ ∈ [0, 1) be
1
Assignment 1
January 9, 2020 Reinforcement Learning
COMP 767
Due: Jan 29, 2020
the discount factor. Consider any value function Vb such that |V
?
(s) − Vb(s)| 6 ε for
all states s ∈ S. Prove that LVb (s) 6
2γε
1−γ
for all s, where LVb (s) = V
?
(s) − VVb (s), VVb
is the value function obtained after evaluating the greedy policy with respect to Vb.
2 Coding
In addition to the instructions provided for each question, please make sure you abide
by the following general rules of reproducible and meaningful research:
• When applicable please do a thorough hyperparameter search. A detailed report
of the hyperparameter search applicable to the algorithm has to be included in
the submission (Justify the choice of hyperparameters with supporting plots).
• Each experiment has to be run on different seeds as specified in each individual
question. Mean curve along with variance should be highlighted in the plots.
For more information regarding reproducibility and reporting of results, please refer
to the instructions file posted along with the assignment.
Question 1. [30 points] Explore-Exploit in Bandits (Choose one of the following
two tracks)
A. Implement Boltzmann (softmax) exploration, the UCB algorithm and Thompson
sampling algorithm. Compare the performance of these algorithms on the 10-armed
problem described in the course textbook (which we also discussed in class). Please
plot three graphs: average training return (averaged over 10 independent runs),
average regret per step (which you can compute in this case, as we know apriori the
best arm) and testing plots (showing the reward of the policy, computed after every
10 pulls, and averaged over 5 pulls). Explain what you expected to see and what you
actually saw.
B. Implement UCB and Thompson sampling for a simple 2-arm bandit setting
described in the following parts.
i Consider the setting where the reward distributions of the bandit arms are
independent Bernoulli (rewards for each arm i is 1 with probability pi and 0 with
probability 1−pi), where the parameter pi
for each arm are sampled independently
and uniformly over [0, 1]. Test UCB and Thompson sampling algorithms in this
setting.
2
Assignment 1
January 9, 2020 Reinforcement Learning
COMP 767
Due: Jan 29, 2020
ii In this part, consider Bernoulli distributions where the parameters p1 and p2 of the
two arms are correlated. Specifically, p2 = 1 − p1. Consider the following settings
for selecting the parameter p1: (a) uniform – means that p1 ∼ U([0, 1]) (uniform
distribution over the unit interval), (b) easy – means that p1 ∼ U(0.1, 0.9) (uniform
distribution over those two possible values) (c) medium – p1 ∼ U(0.25, 0.75) and
(d) hard – p1 ∼ U(0.4, 0.6). For each setting, please plot the following graphs:
average training return (averaged over 10 independent runs), average regret per
step (which you can compute in this case, as we know a priori the best arm) and
a testing plots (showing the reward of the optimal policy, computed after every
10 pulls, and averaged over 5 pulls). Explain what you expect to see and what
you actually saw.
Question 2. [30 points] Dynamic Programming (Choose one of the following two
tracks) Pick two tabular environments of your choice where the transition probabilities
and the reward function are known. Some examples of tabular environment include
Frozen lake (Brockman et al., 2016), Taxi environment (Dietterich, 1998), grid world,
MiniGrid, towers of Hanoi (Abel, 2019), etc. You can either use gym environments
or design your own environment in python.
A. Implement tabular policy iteration and value iteration to find the optimal value
function and optimal policy.
i Report the training performance of these methods with the following plots: (i)
cumulative reward per episode obtained by the agent, and (ii) the number of
timesteps to solve the task per episode of experience, both reported over 5 different
seeds (as specified in the beginning of the coding section).
ii For the test performance, after every 10 episodes of training (for every seed), run
the estimated policy in the environment for 5 test episodes and report the mean
over (i) cumulative reward per episode obtained by the agent, and (ii) the number
of timesteps required to solve the task per episode of experience. You will extend
the reporting scheme over seeds from training.
iii In both training and testing plots, highlight the optimal performance (maximum
return in an episode / minimum number of timesteps per episode) in a dotted
line.
iv Please provide your own observations with respect to the results obtained. Also,
provide the time complexity, space complexity of these algorithms.
3
Assignment 1
January 9, 2020 Reinforcement Learning
COMP 767
Due: Jan 29, 2020
B. Implement the following variants of value iteration and policy iteration to find
the optimal value function and optimal policy.
i Modified policy iteration: instead of fully (to convergence) evaluating a policy,
just compute a few Bellman backups and then improve the policy (See section
6.5 of (Puterman, 1994)).
ii Asynchronous value iteration, Gauss-Seidel and Jacobi variants of value iteration
(See section 6.3.3 of (Puterman, 1994)).
For both the parts,
• Report the training performance of these methods with the following plots: (i)
cumulative reward per episode obtained by the agent, and (ii) the number of
timesteps required to solve the task per episode of experience, reported over 5
different seeds (as specified in the beginning of the coding section).
• For the test performance, after every 10 episodes of training (for every seed),
run the estimated policy in the environment for 5 test episodes and report the
mean over (i) cumulative reward per episode obtained by the agent, and (ii)
the number of timesteps required to solve the task per episode of experience.
You will extend the reporting scheme over seeds from training.
• In both training and testing plots, highlight the optimal performance (maximum
return in an episode/minimum number of timesteps per episode) in a dotted
line.
• Please provide your own observations with respect to the results obtained,
specifically in terms of speed of convergence to the right optimal solutions.
References
Abel, David (2019). “simple rl: Reproducible Reinforcement Learning in Python”. In:
Brockman, Greg et al. (2016). OpenAI Gym. eprint: arXiv:1606.01540.
Dietterich, Thomas G. (1998). “The MAXQ Method for Hierarchical Reinforcement
Learning”. In: Proceedings of the Fifteenth International Conference on Machine
Learning. ICML ’98. San Francisco, CA, USA: Morgan Kaufmann Publishers Inc.,
pp. 118–126. isbn: 1558605568.
Puterman, Martin L. (1994). Markov Decision Processes: Discrete Stochastic Dynamic
Programming. 1st. USA: John Wiley and Sons, Inc. isbn: 0471619779.
4
