CS 285 Deep Reinforcement Learning, Decision Making, and Control

Assignment 2: Policy Gradients

1 Introduction

The goal of this assignment is to experiment with policy gradient and its variants, including variance reduction

tricks such as implementing reward-to-go and neural network baselines. The startercode can be found at

https://github.com/berkeleydeeprlcourse/homework_fall2021/tree/master/hw2

2 Review

2.1 Policy gradient

Recall that the reinforcement learning objective is to learn a θ

∗

that maximizes the objective function:

J(θ) = Eτ∼πθ(τ)

[r(τ )] (1)

where each rollout τ is of length T, as follows:

πθ(τ ) = p(s0, a0, …, sT −1, aT −1) = p(s0)πθ(a0|s0)

T

Y−1

t=1

p(st|st−1, at−1)πθ(at|st)

and

r(τ ) = r(s0, a0, …, sT −1, aT −1) =

T

X−1

t=0

r(st, at).

The policy gradient approach is to directly take the gradient of this objective:

∇θJ(θ) = ∇θ

Z

πθ(τ )r(τ )dτ (2)

=

Z

πθ(τ )∇θ log πθ(τ )r(τ )dτ. (3)

= Eτ∼πθ(τ)

[∇θ log πθ(τ )r(τ )] (4)

(5)

In practice, the expectation over trajectories τ can be approximated from a batch of N sampled trajectories:

∇θJ(θ) ≈

1

N

X

N

i=1

∇θ log πθ(τi)r(τi) (6)

=

1

N

X

N

i=1 T

X−1

t=0

∇θ log πθ(ait|sit)

! T

X−1

t=0

r(sit, ait)

!

. (7)

Here we see that the policy πθ is a probability distribution over the action space, conditioned on the state. In

the agent-environment loop, the agent samples an action at from πθ(·|st) and the environment responds with

a reward r(st, at).

2.2 Variance Reduction

2.2.1 Reward-to-go

One way to reduce the variance of the policy gradient is to exploit causality: the notion that the policy cannot

affect rewards in the past. This yields the following modified objective, where the sum of rewards here does

Berkeley CS 285 Deep Reinforcement Learning, Decision Making, and Control Fall 2021

not include the rewards achieved prior to the time step at which the policy is being queried. This sum of

rewards is a sample estimate of the Q function, and is referred to as the “reward-to-go.”

∇θJ(θ) ≈

1

N

X

N

i=1

T

X−1

t=0

∇θ log πθ(ait|sit)

T

X−1

t

0=t

r(sit0 , ait0 )

!

. (8)

2.2.2 Discounting

Multiplying a discount factor γ to the rewards can be interpreted as encouraging the agent to focus more on

the rewards that are closer in time, and less on the rewards that are further in the future. This can also be

thought of as a means for reducing variance (because there is more variance possible when considering futures

that are further into the future). We saw in lecture that the discount factor can be incorporated in two ways,

as shown below.

The first way applies the discount on the rewards from full trajectory:

∇θJ(θ) ≈

1

N

X

N

i=1 T

X−1

t=0

∇θ log πθ(ait|sit)

! T

X−1

t

0=0

γ

t

0−1

r(sit0 , ait0 )

!

(9)

and the second way applies the discount on the “reward-to-go:”

∇θJ(θ) ≈

1

N

X

N

i=1

T

X−1

t=0

∇θ log πθ(ait|sit)

T

X−1

t

0=t

γ

t

0−t

r(sit0 , ait0 )

!

. (10)

.

2.2.3 Baseline

Another variance reduction method is to subtract a baseline (that is a constant with respect to τ ) from the

sum of rewards:

∇θJ(θ) = ∇θEτ∼πθ(τ)

[r(τ ) − b] . (11)

This leaves the policy gradient unbiased because

∇θEτ∼πθ(τ)

[b] = Eτ∼πθ(τ)

[∇θ log πθ(τ ) · b] = 0.

In this assignment, we will implement a value function V

π

φ which acts as a state-dependent baseline. This

value function will be trained to approximate the sum of future rewards starting from a particular state:

V

π

φ

(st) ≈

T

X−1

t

0=t

Eπθ

[r(st

0 , at

0 )|st] , (12)

so the approximate policy gradient now looks like this:

∇θJ(θ) ≈

1

N

X

N

i=1

T

X−1

t=0

∇θ log πθ(ait|sit)

T

X−1

t

0=t

γ

t

0−t

r(sit0 , ait0 )

!

− V

π

φ

(sit)

!

. (13)

2.2.4 Generalized Advantage Estimation

The quantity PT −1

t

0=t

γ

t

0−t

r(st

0 , at

0 )

− V

π

φ

(st) from the previous policy gradient expression (removing the i

index for clarity) can be interpreted as an estimate of the advantage function:

A

π

(st, at) = Q

π

(st, at) − V

π

(st), (14)

Berkeley CS 285 Deep Reinforcement Learning, Decision Making, and Control Fall 2021

where Qπ

(st, at) is estimated using Monte Carlo returns and V

π

(st) is estimated using the learned value

function V

π

φ

. We can further reduce variance by also using V

π

φ

in place of the Monte Carlo returns to estimate

the advantage function as:

A

π

(st, at) ≈ δt = r(st, at) + γV π

φ

(st+1) − V

π

φ

(st), (15)

with the edge case δT −1 = r(sT −1, aT −1) − V

π

φ

(sT −1). However, this comes at the cost of introducing bias

to our policy gradient estimate, due to modeling errors in V

π

φ

. We can instead use a combination of n-step

Monte Carlo returns and V

π

φ

to estimate the advantage function as:

A

π

n

(st, at) = Xt+n

t

0=t

γ

t

0−t

r(st

0 , at

0 ) + γ

nV

π

φ

(st+n+1) − V

π

φ

(st). (16)

Increasing n incorporates the Monte Carlo returns more heavily in the advantage estimate, which lowers bias

and increases variance, while decreasing n does the opposite. Note that n = T − t − 1 recovers the unbiased

but higher variance Monte Carlo advantage estimate used in (13), while n = 0 recovers the lower variance but

higher bias advantage estimate δt.

We can combine multiple n-step advantage estimates as an exponentially weighted sum, which is known as

the generalized advantage estimator (GAE). Let λ ∈ [0, 1]. Then we define:

A

π

GAE(st, at) = 1 − λ

T −t−1

1 − λ

T

X−t−1

n=1

λ

n−1A

π

n

(st, at), (17)

where 1−λ

T−t−1

1−λ

is a normalizing constant. Note that a higher λ emphasizes advantage estimates with higher

values of n, and a lower λ does the opposite. Thus, λ serves as a control for the bias-variance tradeoff, where

increasing λ decreases bias and increases variance. In the infinite horizon case (T = ∞), we can show:

A

π

GAE(st, at) = 1

1 − λ

X∞

n=1

λ

n−1A

π

n

(st, at) (18)

=

X∞

t

0=t

(γλ)

t

0−t

δt

0 , (19)

where we have omitted the derivation for brevity (see the GAE paper https://arxiv.org/pdf/1506.02438.pdf

for details). In the finite horizon case, we can write:

A

π

GAE(st, at) =

T

X−1

t

0=t

(γλ)

t

0−t

δt

0 , (20)

which serves as a way we can efficiently implement the generalized advantage estimator, since we can recursively

compute:

A

π

GAE(st, at) = δt + γλAπ

GAE(st+1, at+1) (21)

Berkeley CS 285 Deep Reinforcement Learning, Decision Making, and Control Fall 2021

3 Overview of Implementation

3.1 Files

To implement policy gradients, we will be building up the code that we started in homework 1. All files needed

to run your code are in the hw2 folder, but there will be some blanks you will fill with your solutions from

homework 1. These locations are marked with # TODO: get this from hw1 and are found in the following

files:

• infrastructure/rl trainer.py

• infrastructure/utils.py

• policies/MLP policy.py

After bringing in the required components from the previous homework, you can begin work on the new policy

gradient code. These placeholders are marked with TODO, located in the following files:

• agents/pg agent.py

• policies/MLP policy.py

The script to run the experiments is found in scripts/run hw2.py (for the local option) or scripts/run hw2.ipynb

(for the Colab option).

3.2 Overview

As in the previous homework, the main training loop is implemented in

infrastructure/rl trainer.py.

The policy gradient algorithm uses the following 3 steps:

1. Sample trajectories by generating rollouts under your current policy.

2. Estimate returns and compute advantages. This is executed in the train function of pg agent.py

3. Train/Update parameters. The computational graph for the policy and the baseline, as well as the

update functions, are implemented in policies/MLP policy.py.

4 Implementing Policy Gradients

You will be implementing two different return estimators within pg agent.py. The first (“Case 1” within

calculate q vals) uses the discounted cumulative return of the full trajectory and corresponds to the

“vanilla” form of the policy gradient (Equation 9):

r(τi) =

T

X−1

t

0=0

γ

t

0

r(sit0 , ait0 ). (22)

The second (“Case 2”) uses the “reward-to-go” formulation from Equation 10:

r(τi) =

T

X−1

t

0=t

γ

t

0−t

r(sit0 , ait0 ). (23)

Note that these differ only by the starting point of the summation.

Implement these return estimators as well as the remaining sections marked TODO in the code. For the smallscale experiments, you may skip those sections that are run only if nn baseline is True; we will return to

baselines in Section 6. (These sections are in MLPPolicyPG:update and PGAgent:estimate advantage.)

Berkeley CS 285 Deep Reinforcement Learning, Decision Making, and Control Fall 2021

5 Small-Scale Experiments

After you have implemented all non-baseline code from Section 4, you will run two small-scale experiments to

get a feel for how different settings impact the performance of policy gradient methods.

Experiment 1 (CartPole). Run multiple experiments with the PG algorithm on the discrete CartPole-v0

environment, using the following commands:

python cs285/scripts/run_hw2.py –env_name CartPole-v0 -n 100 -b 1000 \

-dsa –exp_name q1_sb_no_rtg_dsa

python cs285/scripts/run_hw2.py –env_name CartPole-v0 -n 100 -b 1000 \

-rtg -dsa –exp_name q1_sb_rtg_dsa

python cs285/scripts/run_hw2.py –env_name CartPole-v0 -n 100 -b 1000 \

-rtg –exp_name q1_sb_rtg_na

python cs285/scripts/run_hw2.py –env_name CartPole-v0 -n 100 -b 5000 \

-dsa –exp_name q1_lb_no_rtg_dsa

python cs285/scripts/run_hw2.py –env_name CartPole-v0 -n 100 -b 5000 \

-rtg -dsa –exp_name q1_lb_rtg_dsa

python cs285/scripts/run_hw2.py –env_name CartPole-v0 -n 100 -b 5000 \

-rtg –exp_name q1_lb_rtg_na

What’s happening here:

• -n : Number of iterations.

• -b : Batch size (number of state-action pairs sampled while acting according to the current policy at

each iteration).

• -dsa : Flag: if present, sets standardize_advantages to False. Otherwise, by default, standardizes

advantages to have a mean of zero and standard deviation of one.

• -rtg : Flag: if present, sets reward_to_go=True. Otherwise, reward_to_go=False by default.

• –exp_name : Name for experiment, which goes into the name for the data logging directory.

Various other command line arguments will allow you to set batch size, learning rate, network architecture,

and more. You can change these as well, but keep them fixed between the 6 experiments mentioned above.

Deliverables for report:

• Create two graphs:

– In the first graph, compare the learning curves (average return at each iteration) for the experiments

prefixed with q1_sb_. (The small batch experiments.)

– In the second graph, compare the learning curves for the experiments prefixed with q1_lb_. (The

large batch experiments.)

• Answer the following questions briefly:

– Which value estimator has better performance without advantage-standardization: the trajectorycentric one, or the one using reward-to-go?

– Did advantage standardization help?

– Did the batch size make an impact?

Berkeley CS 285 Deep Reinforcement Learning, Decision Making, and Control Fall 2021

• Provide the exact command line configurations (or #@params settings in Colab) you used to run your

experiments, including any parameters changed from their defaults.

What to Expect:

• The best configuration of CartPole in both the large and small batch cases should converge to a maximum

score of 200.

Experiment 2 (InvertedPendulum). Run experiments on the InvertedPendulum-v2 continuous control

environment as follows:

python cs285/scripts/run_hw2.py –env_name InvertedPendulum-v2 \

–ep_len 1000 –discount 0.9 -n 100 -l 2 -s 64 -b <b*> -lr <r*> -rtg \

–exp_name q2_b<b*>_r<r*>

where your task is to find the smallest batch size b* and largest learning rate r* that gets to optimum

(maximum score of 1000) in less than 100 iterations. The policy performance may fluctuate around 1000; this

is fine. The precision of b* and r* need only be one significant digit.

Deliverables:

• Given the b* and r* you found, provide a learning curve where the policy gets to optimum (maximum

score of 1000) in less than 100 iterations. (This may be for a single random seed, or averaged over

multiple.)

• Provide the exact command line configurations you used to run your experiments.

6 Implementing Neural Network Baselines

You will now implement a value function as a state-dependent neural network baseline. This will require filling

in some TODO sections skipped in Section 4. In particular:

• This neural network will be trained in the update method of MLPPolicyPG along with the policy gradient

update.

• In pg agent.py:estimate advantage, the predictions of this network will be subtracted from the

reward-to-go to yield an estimate of the advantage. This implements PT −1

t

0=t

γ

t

0−t

r(sit0 , ait0 )

−V

π

φ

(sit).

7 More Complex Experiments

Note: The following tasks take quite a bit of time to train. Please start early! For all remaining experiments,

use the reward-to-go estimator.

Experiment 3 (LunarLander). You will now use your policy gradient implementation to learn a controller

for LunarLanderContinuous-v2. The purpose of this problem is to test and help you debug your baseline

implementation from Section 6.

Run the following command:

python cs285/scripts/run_hw2.py \

–env_name LunarLanderContinuous-v2 –ep_len 1000

–discount 0.99 -n 100 -l 2 -s 64 -b 40000 -lr 0.005 \

–reward_to_go –nn_baseline –exp_name q3_b40000_r0.005

Deliverables:

• Plot a learning curve for the above command. You should expect to achieve an average return of around

180 by the end of training.

Berkeley CS 285 Deep Reinforcement Learning, Decision Making, and Control Fall 2021

Experiment 4 (HalfCheetah). You will be using your policy gradient implementation to learn a controller

for the HalfCheetah-v2 benchmark environment with an episode length of 150. This is shorter than the default

episode length (1000), which speeds up training significantly. Search over batch sizes b ∈ [10000, 30000, 50000]

and learning rates r ∈ [0.005, 0.01, 0.02] to replace <b> and <r> below.

python cs285/scripts/run_hw2.py –env_name HalfCheetah-v2 –ep_len 150 \

–discount 0.95 -n 100 -l 2 -s 32 -b <b> -lr <r> -rtg –nn_baseline \

–exp_name q4_search_b<b>_lr<r>_rtg_nnbaseline

Deliverables:

• Provide a single plot with the learning curves for the HalfCheetah experiments that you tried. Describe

in words how the batch size and learning rate affected task performance.

Once you’ve found optimal values b* and r*, use them to run the following commands (replace the terms in

angle brackets):

python cs285/scripts/run_hw2.py –env_name HalfCheetah-v2 –ep_len 150 \

–discount 0.95 -n 100 -l 2 -s 32 -b <b*> -lr <r*> \

–exp_name q4_b<b*>_r<r*>

python cs285/scripts/run_hw2.py –env_name HalfCheetah-v2 –ep_len 150 \

–discount 0.95 -n 100 -l 2 -s 32 -b <b*> -lr <r*> -rtg \

–exp_name q4_b<b*>_r<r*>_rtg

python cs285/scripts/run_hw2.py –env_name HalfCheetah-v2 –ep_len 150 \

–discount 0.95 -n 100 -l 2 -s 32 -b <b*> -lr <r*> –nn_baseline \

–exp_name q4_b<b*>_r<r*>_nnbaseline

python cs285/scripts/run_hw2.py –env_name HalfCheetah-v2 –ep_len 150 \

–discount 0.95 -n 100 -l 2 -s 32 -b <b*> -lr <r*> -rtg –nn_baseline \

–exp_name q4_b<b*>_r<r*>_rtg_nnbaseline

Deliverables: Provide a single plot with the learning curves for these four runs. The run with both rewardto-go and the baseline should achieve an average score close to 200.

8 Implementing Generalized Advantage Estimation

You will now use the value function you previously implemented to implement a simplified version of GAE-λ.

This will require filling in the remaining TODO section in pg agent.py:estimate advantage.

Experiment 5 (HopperV2). You will now use your implementation of policy gradient with generalized

advantage estimation to learn a controller for a version of Hopper-v2 with noisy actions. Search over λ ∈

[0, 0.95, 0.99, 1] to replace <λ> below. Note that with a correct implementation, λ = 1 is equivalent to the

vanilla neural network baseline estimator. Do not change any of the other hyperparameters (e.g. batch size,

learning rate).

python cs285/scripts/run_hw2.py \

–env_name Hopper-v2 –ep_len 1000

–discount 0.99 -n 300 -l 2 -s 32 -b 2000 -lr 0.001 \

–reward_to_go –nn_baseline –action_noise_std 0.5 –gae_lambda <λ> \

–exp_name q5_b2000_r0.001_lambda<λ>

Deliverables:

Berkeley CS 285 Deep Reinforcement Learning, Decision Making, and Control Fall 2021

• Provide a single plot with the learning curves for the Hopper-v2 experiments that you tried. Describe in

words how λ affected task performance. The run with the best performance should achieve an average

score close to 400.

9 Bonus!

Choose any (or all) of the following:

• A serious bottleneck in the learning, for more complex environments, is the sample collection time. In

infrastructure/rl trainer.py, we only collect trajectories in a single thread, but this process can be

fully parallelized across threads to get a useful speedup. Implement the parallelization and report on

the difference in training time.

• In PG, we collect a batch of data, estimate a single gradient, and then discard the data and move on.

Can we potentially accelerate PG by taking multiple gradient descent steps with the same batch of data?

Explore this option and report on your results. Set up a fair comparison between single-step PG and

multi-step PG on at least one MuJoCo gym environment.

10 Submission

10.1 Submitting the PDF

Your report should be a document containing

(a) All graphs and answers to short explanation questions requested for Experiments 1-4.

(b) All command-line expressions you used to run your experiments.

(c) (Optionally) Your bonus results (command-line expressions, graphs, and a few sentences that comment

on your findings).

10.2 Submitting the code and experiment runs

In order to turn in your code and experiment logs, create a folder that contains the following:

• A folder named run logs with all the experiment runs from this assignment. These folders can be

copied directly from the cs285/data folder. Do not change the names originally assigned to the

folders, as specified by exp name in the instructions. Video logging is disabled by default in the

code, but if you turned it on for debugging, you need to run those again with –video log freq -1, or

else the file size will be too large for submission.

• The cs285 folder with all the .py files, with the same names and directory structure as the original

homework repository (excluding the cs285/data folder). Also include any special instructions we need

to run in order to produce each of your figures or tables in the form of a README file.

As an example, the unzipped version of your submission should result in the following file structure. Make

sure that the submit.zip file is below 15MB.

Berkeley CS 285 Deep Reinforcement Learning, Decision Making, and Control Fall 2021

submit.zip

run logs

q1 lb rtg na CartPole-v0 12-09-2019 17-53-4

events.out.tfevents.1567529456.e3a096ac8ff4

q3 b40000 r0.005 LunarLanderContinuous-v2 12-09-2019 00-17-58

events.out.tfevents.1567529456.e3a096ac8ff4

…

cs285

agents

bc agent.py

…

policies

…

…

README.md

…

10.3 Turning it in

Turn in your assignment by the deadline on Gradescope. Uploade the zip file with your code to HW2 Code,

and upload the PDF of your report to HW2.