Deep Deterministic Policy Gradient¶
Table of Contents
Background¶
(Previously: Introduction to RL Part 1: The Optimal QFunction and the Optimal Action)
Deep Deterministic Policy Gradient (DDPG) is an algorithm which concurrently learns a Qfunction and a policy. It uses offpolicy data and the Bellman equation to learn the Qfunction, and uses the Qfunction to learn the policy.
This approach is closely connected to Qlearning, and is motivated the same way: if you know the optimal actionvalue function , then in any given state, the optimal action can be found by solving
DDPG interleaves learning an approximator to with learning an approximator to , and it does so in a way which is specifically adapted for environments with continuous action spaces. But what does it mean that DDPG is adapted specifically for environments with continuous action spaces? It relates to how we compute the max over actions in .
When there are a finite number of discrete actions, the max poses no problem, because we can just compute the Qvalues for each action separately and directly compare them. (This also immediately gives us the action which maximizes the Qvalue.) But when the action space is continuous, we can’t exhaustively evaluate the space, and solving the optimization problem is highly nontrivial. Using a normal optimization algorithm would make calculating a painfully expensive subroutine. And since it would need to be run every time the agent wants to take an action in the environment, this is unacceptable.
Because the action space is continuous, the function is presumed to be differentiable with respect to the action argument. This allows us to set up an efficient, gradientbased learning rule for a policy which exploits that fact. Then, instead of running an expensive optimization subroutine each time we wish to compute , we can approximate it with . See the Key Equations section details.
Quick Facts¶
 DDPG is an offpolicy algorithm.
 DDPG can only be used for environments with continuous action spaces.
 DDPG can be thought of as being deep Qlearning for continuous action spaces.
 The Spinning Up implementation of DDPG does not support parallelization.
Key Equations¶
Here, we’ll explain the math behind the two parts of DDPG: learning a Q function, and learning a policy.
The QLearning Side of DDPG¶
First, let’s recap the Bellman equation describing the optimal actionvalue function, . It’s given by
where is shorthand for saying that the next state, , is sampled by the environment from a distribution .
This Bellman equation is the starting point for learning an approximator to . Suppose the approximator is a neural network , with parameters , and that we have collected a set of transitions (where indicates whether state is terminal). We can set up a meansquared Bellman error (MSBE) function, which tells us roughly how closely comes to satisfying the Bellman equation:
Here, in evaluating , we’ve used a Python convention of evaluating True
to 1 and False
to zero. Thus, when d==True
—which is to say, when is a terminal state—the Qfunction should show that the agent gets no additional rewards after the current state. (This choice of notation corresponds to what we later implement in code.)
Qlearning algorithms for function approximators, such as DQN (and all its variants) and DDPG, are largely based on minimizing this MSBE loss function. There are two main tricks employed by all of them which are worth describing, and then a specific detail for DDPG.
Trick One: Replay Buffers. All standard algorithms for training a deep neural network to approximate make use of an experience replay buffer. This is the set of previous experiences. In order for the algorithm to have stable behavior, the replay buffer should be large enough to contain a wide range of experiences, but it may not always be good to keep everything. If you only use the verymost recent data, you will overfit to that and things will break; if you use too much experience, you may slow down your learning. This may take some tuning to get right.
You Should Know
We’ve mentioned that DDPG is an offpolicy algorithm: this is as good a point as any to highlight why and how. Observe that the replay buffer should contain old experiences, even though they might have been obtained using an outdated policy. Why are we able to use these at all? The reason is that the Bellman equation doesn’t care which transition tuples are used, or how the actions were selected, or what happens after a given transition, because the optimal Qfunction should satisfy the Bellman equation for all possible transitions. So any transitions that we’ve ever experienced are fair game when trying to fit a Qfunction approximator via MSBE minimization.
Trick Two: Target Networks. Qlearning algorithms make use of target networks. The term
is called the target, because when we minimize the MSBE loss, we are trying to make the Qfunction be more like this target. Problematically, the target depends on the same parameters we are trying to train: . This makes MSBE minimization unstable. The solution is to use a set of parameters which comes close to , but with a time delay—that is to say, a second network, called the target network, which lags the first. The parameters of the target network are denoted .
In DQNbased algorithms, the target network is just copied over from the main network every somefixednumber of steps. In DDPGstyle algorithms, the target network is updated once per main network update by polyak averaging:
where is a hyperparameter between 0 and 1 (usually close to 1). (This hyperparameter is called polyak
in our code).
DDPG Detail: Calculating the Max Over Actions in the Target. As mentioned earlier: computing the maximum over actions in the target is a challenge in continuous action spaces. DDPG deals with this by using a target policy network to compute an action which approximately maximizes . The target policy network is found the same way as the target Qfunction: by polyak averaging the policy parameters over the course of training.
Putting it all together, Qlearning in DDPG is performed by minimizing the following MSBE loss with stochastic gradient descent:
where is the target policy.
The Policy Learning Side of DDPG¶
Policy learning in DDPG is fairly simple. We want to learn a deterministic policy which gives the action that maximizes . Because the action space is continuous, and we assume the Qfunction is differentiable with respect to action, we can just perform gradient ascent (with respect to policy parameters only) to solve
Note that the Qfunction parameters are treated as constants here.
Exploration vs. Exploitation¶
DDPG trains a deterministic policy in an offpolicy way. Because the policy is deterministic, if the agent were to explore onpolicy, in the beginning it would probably not try a wide enough variety of actions to find useful learning signals. To make DDPG policies explore better, we add noise to their actions at training time. The authors of the original DDPG paper recommended timecorrelated OU noise, but more recent results suggest that uncorrelated, meanzero Gaussian noise works perfectly well. Since the latter is simpler, it is preferred. To facilitate getting higherquality training data, you may reduce the scale of the noise over the course of training. (We do not do this in our implementation, and keep noise scale fixed throughout.)
At test time, to see how well the policy exploits what it has learned, we do not add noise to the actions.
You Should Know
Our DDPG implementation uses a trick to improve exploration at the start of training. For a fixed number of steps at the beginning (set with the start_steps
keyword argument), the agent takes actions which are sampled from a uniform random distribution over valid actions. After that, it returns to normal DDPG exploration.
Documentation¶
You Should Know
In what follows, we give documentation for the PyTorch and Tensorflow implementations of DDPG in Spinning Up. They have nearly identical function calls and docstrings, except for details relating to model construction. However, we include both full docstrings for completeness.
Documentation: PyTorch Version¶

spinup.
ddpg_pytorch
(env_fn, actor_critic=<MagicMock spec='str' id='140554320181456'>, ac_kwargs={}, seed=0, steps_per_epoch=4000, epochs=100, replay_size=1000000, gamma=0.99, polyak=0.995, pi_lr=0.001, q_lr=0.001, batch_size=100, start_steps=10000, update_after=1000, update_every=50, act_noise=0.1, num_test_episodes=10, max_ep_len=1000, logger_kwargs={}, save_freq=1)¶ Deep Deterministic Policy Gradient (DDPG)
Parameters:  env_fn – A function which creates a copy of the environment. The environment must satisfy the OpenAI Gym API.
 actor_critic –
The constructor method for a PyTorch Module with an
act
method, api
module, and aq
module. Theact
method andpi
module should accept batches of observations as inputs, andq
should accept a batch of observations and a batch of actions as inputs. When called, these should return:Call Output Shape Description act
(batch, act_dim) Numpy array of actions for eachobservation.pi
(batch, act_dim) Tensor containing actions from policygiven observations.q
(batch,) Tensor containing the current estimateof Q* for the provided observationsand actions. (Critical: make sure toflatten this!)  ac_kwargs (dict) – Any kwargs appropriate for the ActorCritic object you provided to DDPG.
 seed (int) – Seed for random number generators.
 steps_per_epoch (int) – Number of steps of interaction (stateaction pairs) for the agent and the environment in each epoch.
 epochs (int) – Number of epochs to run and train agent.
 replay_size (int) – Maximum length of replay buffer.
 gamma (float) – Discount factor. (Always between 0 and 1.)
 polyak (float) –
Interpolation factor in polyak averaging for target networks. Target networks are updated towards main networks according to:
where is polyak. (Always between 0 and 1, usually close to 1.)
 pi_lr (float) – Learning rate for policy.
 q_lr (float) – Learning rate for Qnetworks.
 batch_size (int) – Minibatch size for SGD.
 start_steps (int) – Number of steps for uniformrandom action selection, before running real policy. Helps exploration.
 update_after (int) – Number of env interactions to collect before starting to do gradient descent updates. Ensures replay buffer is full enough for useful updates.
 update_every (int) – Number of env interactions that should elapse between gradient descent updates. Note: Regardless of how long you wait between updates, the ratio of env steps to gradient steps is locked to 1.
 act_noise (float) – Stddev for Gaussian exploration noise added to policy at training time. (At test time, no noise is added.)
 num_test_episodes (int) – Number of episodes to test the deterministic policy at the end of each epoch.
 max_ep_len (int) – Maximum length of trajectory / episode / rollout.
 logger_kwargs (dict) – Keyword args for EpochLogger.
 save_freq (int) – How often (in terms of gap between epochs) to save the current policy and value function.
Saved Model Contents: PyTorch Version¶
The PyTorch saved model can be loaded with ac = torch.load('path/to/model.pt')
, yielding an actorcritic object (ac
) that has the properties described in the docstring for ddpg_pytorch
.
You can get actions from this model with
actions = ac.act(torch.as_tensor(obs, dtype=torch.float32))
Documentation: Tensorflow Version¶

spinup.
ddpg_tf1
(env_fn, actor_critic=<function mlp_actor_critic>, ac_kwargs={}, seed=0, steps_per_epoch=4000, epochs=100, replay_size=1000000, gamma=0.99, polyak=0.995, pi_lr=0.001, q_lr=0.001, batch_size=100, start_steps=10000, update_after=1000, update_every=50, act_noise=0.1, num_test_episodes=10, max_ep_len=1000, logger_kwargs={}, save_freq=1)¶ Deep Deterministic Policy Gradient (DDPG)
Parameters:  env_fn – A function which creates a copy of the environment. The environment must satisfy the OpenAI Gym API.
 actor_critic –
A function which takes in placeholder symbols for state,
x_ph
, and action,a_ph
, and returns the main outputs from the agent’s Tensorflow computation graph:Symbol Shape Description pi
(batch, act_dim) Deterministically computes actionsfrom policy given states.q
(batch,) Gives the current estimate of Q* forstates inx_ph
and actions ina_ph
.q_pi
(batch,) Gives the composition ofq
andpi
for states inx_ph
:q(x, pi(x)).  ac_kwargs (dict) – Any kwargs appropriate for the actor_critic function you provided to DDPG.
 seed (int) – Seed for random number generators.
 steps_per_epoch (int) – Number of steps of interaction (stateaction pairs) for the agent and the environment in each epoch.
 epochs (int) – Number of epochs to run and train agent.
 replay_size (int) – Maximum length of replay buffer.
 gamma (float) – Discount factor. (Always between 0 and 1.)
 polyak (float) –
Interpolation factor in polyak averaging for target networks. Target networks are updated towards main networks according to:
where is polyak. (Always between 0 and 1, usually close to 1.)
 pi_lr (float) – Learning rate for policy.
 q_lr (float) – Learning rate for Qnetworks.
 batch_size (int) – Minibatch size for SGD.
 start_steps (int) – Number of steps for uniformrandom action selection, before running real policy. Helps exploration.
 update_after (int) – Number of env interactions to collect before starting to do gradient descent updates. Ensures replay buffer is full enough for useful updates.
 update_every (int) – Number of env interactions that should elapse between gradient descent updates. Note: Regardless of how long you wait between updates, the ratio of env steps to gradient steps is locked to 1.
 act_noise (float) – Stddev for Gaussian exploration noise added to policy at training time. (At test time, no noise is added.)
 num_test_episodes (int) – Number of episodes to test the deterministic policy at the end of each epoch.
 max_ep_len (int) – Maximum length of trajectory / episode / rollout.
 logger_kwargs (dict) – Keyword args for EpochLogger.
 save_freq (int) – How often (in terms of gap between epochs) to save the current policy and value function.
Saved Model Contents: Tensorflow Version¶
The computation graph saved by the logger includes:
Key  Value 

x 
Tensorflow placeholder for state input. 
a 
Tensorflow placeholder for action input. 
pi 
Deterministically computes an action from the agent, conditioned
on states in
x . 
q 
Gives actionvalue estimate for states in x and actions in a . 
This saved model can be accessed either by
 running the trained policy with the test_policy.py tool,
 or loading the whole saved graph into a program with restore_tf_graph.
References¶
Relevant Papers¶
 Deterministic Policy Gradient Algorithms, Silver et al. 2014
 Continuous Control With Deep Reinforcement Learning, Lillicrap et al. 2016
Why These Papers?¶
Silver 2014 is included because it establishes the theory underlying deterministic policy gradients (DPG). Lillicrap 2016 is included because it adapts the theoreticallygrounded DPG algorithm to the deep RL setting, giving DDPG.