O FF -D YNAMICS R EINFORCEMENT L EARNING : T RAINING FOR T RANSFER WITH D OMAIN C LASSIFIERS
Anonymous authors
Paper under double-blind review
A BSTRACT
We propose a simple, practical, and intuitive approach for domain adaptation in reinforcement learning.
Our approach stems from the idea that the agent’s experience in the source domain should look similar to its experience in the target domain.
Building off of a probabilistic view of RL, we achieve this goal by compensating for the difference in dynamics by modifying the reward function.
This modiﬁed reward function is simple to estimate by learning auxiliary classiﬁers that distinguish source-domain transitions from target-domain transitions.
Intuitively, the agent is penalized for transitions that would indicate that the agent is interacting with the source domain, rather than the target domain.
Formally, we prove that applying our method in the source domain is guaranteed to obtain a near-optimal policy for the target domain, provided that the source and target domains satisfy a lightweight assumption.
Our approach is applicable to domains with continuous states and actions and does not require learning an explicit model of the dynamics.
On discrete and continuous control tasks, we illustrate the mechanics of our approach and demonstrate its scalability to high-dimensional tasks.
I NTRODUCTION
Reinforcement learning (RL) can automate the acquisition of complex behavioral policies through real-world trial-and-error experimentation.
However, many domains where we would like to learn policies are not amenable to such trial-and-error learning, because the errors are too costly: from autonomous driving to ﬂying airplanes to devising medical treatment plans, safety-critical RL problems necessitate some type of transfer learning , where a safer source domain, such as a simulator, is used to train a policy that can then function effectively in a target domain.
In this paper, we examine a speciﬁc transfer learning scenario that we call domain adaptation, by analogy to domain adaptation problems in computer vision (Csurka, 2017), where the training process in a source domain can be modiﬁed so that the resulting policy is effective in a given target domain.
RL algorithms today require a large amount of experience in the target domain .
However, for many tasks we may have access to a different but structurally similar source domain .
While the source domain has different dynamics than the target domain, experience in the source domain is much cheaper to collect.
However, transferring policies from one domain to another is challenging because strategies which are effective in the source domain may not be effective in the target domain.
For example, aggressive driving may work well on a dry racetrack but fail catastrophically on an icy road.
While prior work has studied the domain adaptation of observations in RL (Bousmalis et al., 2018; Ganin et al., 2016; Higgins et al., 2017), it ignores the domain adaptation of the dynamics .
This paper presents a simple approach for domain adaptation in RL, illustrated in Fig.
Our main idea is that the agent’s experience in the source domain should look similar to its experience in the target domain.
Building off of a probabilistic view of RL, we formally show that we can achieve this goal by compensating for the difference in dynamics by modifying the reward function .
This modiﬁed reward function is simple to estimate by learning auxiliary classiﬁers that distinguish source- domain transitions from target-domain transitions.
Because our method learns a classiﬁer, rather than a dynamics model, we expect it to handle high-dimensional tasks better than model-based methods, a conjecture supported by experiments on the 111-dimensional Ant task.
Unlike prior work based on similar intuition (Koos et al., 2012; Wulfmeier et al., 2017b), a key contribution of our work is a formal guarantee that our method yields a near-optimal policy for the target domain.
The main contribution of this work is an algorithm for domain adaptation to dynamics changes in RL, based on the idea of compensating for differences in dynamics by modifying the reward function.
We call this algorithm Domain Adaptation with Rewards from Classiﬁers, or DARC for short.
DARC does not estimate transition probabilities, but rather modiﬁes the reward function using a pair of classiﬁers.
We formally analyze the conditions under which our method produces near-optimal policies for the target domain.
On a range of discrete and continuous control tasks, we both illustrate the mechanics of our approach and demonstrate its scalability to higher-dimensional tasks.
R ELATED W ORK
While our work will focus on domain adaptation applied to RL, we start by reviewing more general ideas in domain adaptation, and defer to Kouw & Loog (2019) for a recent review of the ﬁeld.
Two common approaches to domain adaptation are importance weighting and domain-agnostic features.
Importance-weighting methods (e.g., (Zadrozny, 2004; Cortes & Mohri, 2014; Lipton et al., 2018)) estimate the likelihood ratio of examples under the target domain versus the source domain, and use this ratio to re-weight examples sampled from the source domain.
Similar to prior work on importance weighting (Bickel et al., 2007; Sønderby et al., 2016; Mohamed & Lakshminarayanan, 2016; Uehara et al., 2016), our method will use a classiﬁer to estimate a probability ratio.
Since we will need to estimate the density ratio of conditional distributions (transition probabilities), we will learn two classiﬁers.
Importantly, we will use the logarithm of the density ratio to modify the reward function instead of weighting samples by the density ratio, which is often numerically unstable (see, e.g., Schulman et al.
and led to poor performance in our experiments.
Prior methods for applying domain adaptation to RL include approaches based on system identiﬁcation, domain randomization, and observation adaptation.
Perhaps the most established approach, system identiﬁcation (Ljung, 1999), uses observed data to tune the parameters of a simulator (Feldbaum, 1960; Werbos, 1989; Wittenmark, 1995; Ross & Bagnell, 2012; Tan et al., 2016;
Zhu et al., 2017b; Farchy et al., 2013)
More recent work has successfully used this strategy to bridge the sim2real gap (Chebotar et al., 2019; Rajeswaran et al., 2016).
Closely related is work on online system identiﬁcation and meta-learning, which directly uses the inferred system parameters to update the policy (Yu et al., 2017; Clavera et al., 2018;
Tanaskovic et al., 2013; Sastry &
Isidori, 1989).
However, these approaches typically require either a model of the environment or a manually-speciﬁed distribution over potential test-time dynamics, requirements that our method will lift.
Another approach, domain randomization , randomly samples the parameters of the source domain and then ﬁnds the best policy for this randomized environment (Sadeghi & Levine, 2016;
Tobin et al., 2017; Peng et al., 2018; Cutler et al., 2014).
While often effective, this method is sensitive to the choice of which parameters are randomized, and the distributions from which these simulator parameters are sampled.
A third approach, observation adaptation , modiﬁes the observations of the source domain to appear similar to those in the target domain (Fernando et al., 2013; Hoffman et al., 2016; Wulfmeier et al., 2017a).
While this approach has been successfully applied to video games (Gamrian & Goldberg, 2018) and robot manipulation (Bousmalis et al., 2018), it ignores the fact that the source and target domains may have differing dynamics.
Finally, our work is similar to prior work on transfer learning (Taylor & Stone, 2009) and metalearning in RL, but makes less strict assumptions than most prior work.
For example, most work on meta-RL (Killian et al., 2017; Duan et al., 2016; Mishra et al., 2017; Rakelly et al., 2019) and some work on transfer learning (Perkins et al., 1999; Tanaka & Yamamura, 2003; Sunmola & Wyatt, 2006) assume that the agent has access to many source tasks, all drawn from the same distribution as the target task.
Selfridge et al.
; Madden & Howley (2004) assume a manually-speciﬁed curriculum of tasks, Ravindran & Barto (2004) assume that the source and target domains have the same dynamics locally, and Sherstov & Stone (2005) assume that the set of actions that are useful in the source domain is the same as the set of actions that will be useful in the target domain.
Our method does not require these assumptions, allowing it to successfully learn in settings where these prior works would fail.
For example, the assumption of Sherstov & Stone (2005) is violated in our experiments with broken robots: actions which move a joint are useful in the source domain (where the robot is fully-function) but not useful in the target domain (where that joint is disabled).
Our method will signiﬁcantly outperform an importance weighting baseline (Lazaric, 2008).
Unlike Vemula et al.
, our method does not require learning a dynamics model and is applicable to stochastic environments and those with continuous states and actions.
Our algorithm bears a resemblance to that in Wulfmeier et al.
(2017b),
but a crucial algorithmic difference allows us to prove that our method acquires a near-optimal policy in the target domain, and also leads to improved performance empirically.
The theoretical derivation of our method is inspired by prior work which formulates control as a problem of probabilistic inference (e.g., (Toussaint, 2009; Rawlik et al., 2013; Levine et al., 2018)).
Algorithms for model-based RL (e.g., (Deisenroth & Rasmussen, 2011; Hafner et al., 2018; Janner et al., 2019)) and off-policy RL (e.g., (Munos et al., 2016; Fujimoto et al., 2018; Dann et al., 2014; Dud´ık et al., 2011) similarly aim to improve the sample efﬁciency of RL, but do use the source domain to accelerate learning.
Our method is applicable to any maximum entropy RL algorithm, including on-policy (Song et al., 2019), off-policy (Abdolmaleki et al., 2018; Haarnoja et al., 2018), and model-based (Janner et al., 2019; Williams et al., 2015) algorithms.
We will use the
SAC (Haarnoja et al., 2018) in our experiments and compare against model-based baselines.
P RELIMINARIES
In this section, we introduce notation and formally deﬁne domain adaptation for RL.
Our problem setting will consider two MDPs: M source represents the source domain (e.g., a practice facility, simulator, or learned approximate model of the target domain) while M target represents a the target domain.
We assume that the two domains have the same state space S , action space A , reward function r , and initially state distribution p 1 ( s 1 ) ; the only difference between the domains is the dynamics, p source ( s t +1 | s t , a t ) and p target ( s t +1 | s t , a t ) .
We will learn a Markovian policy π θ ( a | s ) , parametrized by θ .
Our objective is to learn a policy π that maximizes the expected discounted sum of rewards on M target , E π, M target [ (cid:80) t γ t r ( s t , a t )] .
We now formally deﬁne our problem setting:
Deﬁnition 1.
Domain Adaptation for RL is the problem of using interactions in the source MDP
M source together with a small number of interactions in the target MDP M target to acquire a policy that achieves high reward in the target MDP, M target .
We will assume every transition with non-zero probability in the target domain will have non-zero probability in the source domain:
[Equation]
This assumption is common in work on importance sampling (Koller & Friedman, 2009, § 12.2.2), and the converse need not hold: transitions that are possible in the source domain need not be possible in the target domain.
If this assumption did not hold, then the optimal policy for the target domain might involve behaviors that are not possible in the source domain, so it is unclear how one could learn a near-optimal policy by practicing in the source domain.
A V ARIATIONAL P ERSPECTIVE ON D OMAIN A DAPTATION IN RL
The probabilistic inference interpretation of RL (Kappen, 2005; Todorov, 2007; Toussaint, 2009;
Ziebart, 2010; Rawlik et al., 2013; Levine, 2018) treats the reward function as deﬁning a desired distribution over trajectories.
The agent’s task is to sample from this distribution by picking trajectories with probability proportional to their exponentiated reward.
This section will reinterpret this model in the context of domain transfer, showing that domain adaptation of dynamics can be done by modifying the rewards .
To apply this model to domain adaptation, deﬁne p ( τ ) as the desired distribution over trajectories in the target domain,
[Equation]
and q ( τ ) as our agent’s distribution over trajectories in the source domain,
[Equation]
As noted in Section 3, we assume both trajectory distributions have the same initial state distribution.
Our aim is to learn a policy whose behavior in the source domain both receives high reward and has high likelihood under the target domain dynamics.
We codify this objective by minimizing the reverse KL divergence between these two distributions:
[Equation]
where
[Equation]
The constant c is the partition function of p ( τ ) , which is independent of the policy and dynamics.
While ∆ r is deﬁned in terms of transition probabilities, in Sec.
5 we show how to estimate ∆ r by learning a classiﬁer.
We therefore call our method domain adaptation with rewards from classiﬁers (DARC), and will use π ∗ DARC to refer to the policy that maximizes the objective in Eq. 2.
Where the source and target dynamics are equal, the correction term ∆ r is zero and we recover maximum entropy RL (Ziebart, 2010; Todorov, 2007).
The reward correction is different from prior work that adds log β ( a | s ) to the reward to regularize the policy to be close to the behavior policy
β (e.g., Jaques et al.
; Abdolmaleki et al.
In the case where the source dynamics are not equal to the true dynamics, this objective is not the same as maximum entropy RL on trajectories sampled from the source domain.
Instead, this objective suggests a corrective term ∆ r that should be added to the reward function to account for the discrepancy between the source and target dynamics.
The correction term, ∆ r , is quite intuitive.
If a transition ( s t , a t , s t +1 ) has equal probability in the source and target domains, then ∆ r ( s t , a t ) =
so no correction is applied.
For transitions that are likely in the source but are unlikely in the target domain, ∆ r < 0 , the agent is penalized for “exploiting” inaccuracies or discrepancies in the source domain by taking these transitions.
For the example environment in Figure 1, transitions through the center of the environment are blocked in the target domain but not in the source domain.
For these transitions, ∆ r would serve as a large penalty, discouraging the agent from taking these transitions and instead learning to navigate around the wall.
Appendix A presents additional interpretations of ∆ r in terms of coding theory, mutual information, and a constraint on the discrepancy between the source and target dynamics.
Appendix C discusses how prior work on domain agnostic feature learning can be viewed as a special case of our framework.
4. T HEORETICAL G UARANTEES
We now analyze when maximizing the modiﬁed reward r + ∆ r in the source domain yields a nearoptimal policy for the target domain.
Our proof relies on the following lightweight assumption:
Assumption 1.
Let π ∗ =
arg max π E p
[ (cid:80) r ( s t , a t )] be the reward-maximizing policy in the target domain.
Then the expected reward in the source and target domains differs by at most 2 R max (cid:112) (cid:15)/ 2 :
[Equation]
The variable R max refers to the maximum entropy-regularized return of any trajectory.
This assumption says that the optimal policy in the target domain is still a good policy for the source domain, and its expected reward is similar in both domains.
We do not require that the opposite be true: the optimal policy in the source domain does not need to receive high reward in the target domain.
If there are multiple optimal policies, we only require that this assumption hold for one of them.
We now state our main result:
Theorem 4.1.
Let π ∗ DARC be the policy that maximizes the modiﬁed (entropy-regularized) reward in the source domain, let π ∗ be the policy that maximizes the (unmodiﬁed, entropy-regularized) reward in the target domain, and assume that π ∗ satisﬁes Assumption 1.
Then the following holds:
[Equation]
See Appendix B for the proof and deﬁnition of (cid:15) .
This result says that π ∗ DARC attains near-optimal (entropy-regularized) reward on the target domain.
Thus, we can expect that modifying the reward function should allow us to adapt to different dynamics.
The next section will present a practical algorithm for acquiring π ∗ DARC by estimating and effectively maximizing the modiﬁed reward in the source domain.
D OMAIN A DAPTATION IN RL WITH A L EARNED R EWARD
[Figure]
The variational perspective on model-based RL in the previous section suggests that we should modify the reward in the source domain by adding ∆ r .
In this section we develop a practical algorithm for off-dynamics RL by showing how ∆ r can be estimated without learning an explicit dynamics model.
To estimate ∆ r , we will use a pair of (learned) binary classiﬁers, which will infer whether transitions came from the source or target domain.
The key idea is that the transition probabilities are related to the classiﬁer probabilities via Bayes’ rule:
[Equation]
We estimate the term p ( s t , a t | target ) on the RHS via another classiﬁer, p ( target | s t , a t ) :
[Equation]
Substituting these expression into our deﬁnition for ∆ r and simplifying, we obtain an estimate for ∆ r that depends solely on the predictions of these two classiﬁers:
[Equation]
The ........orange terms are the difference in logits from the classiﬁer conditioned on s t , a t , s t +1 , while the blue terms are the difference in logits from the classiﬁer conditioned on just s t , a t .
Intuitively, ∆ r answers the following question: for the task of predicting whether a transition came from the source or target domain, how much better can you perform after observing s t
+1 ?
We make this connection precise in Appendix A.2 by relating ∆ r to mutual information.
Ablation experiments (Fig. 11) conﬁrm that both classiﬁers are important to the success of our method.
The use of transition classiﬁers makes our method look somewhat similar to adversarial imitation learning (Ho & Ermon, 2016; Fu et al., 2017).
While our method is not solving an imitation learning problem (we do not assume access to any expert experience), our method can be interpretted as learning a policy such that the dynamics observed by that policy in the source domain imitate the dynamics of the target domain.
Algorithm Summary
Our algorithm modiﬁes an existing MaxEnt RL algorithm to additionally learn two classiﬁers, q θ SAS ( target | s t , a t , s t +1 ) and q θ SA ( target | s t , a t ) , parametrized by θ SAS and θ SA respectively, to minimize the standard cross-entropy loss.
(cid:96) SAS ( θ SAS ) (cid:44) − E D target [log q θ SAS ( target | s t , a t , s t +1 )] − E D source [log q θ SA ( source | s t , a t , s t +1 )] (cid:96) SA ( θ SA ) (cid:44) − E D target [log q θ SA ( target | s t , a t )] − E D source [log q θ SA ( source | s t , a t )] .
Our algorithm, Domain Adaptation with Rewards from Classiﬁers (DARC), is presented in Alg.and illustrated in Fig.
To simplify notation, we deﬁne θ (cid:44) ( θ SAS , θ SA ) and (cid:96) ( θ ) (cid:44) (cid:96) SAS ( θ SAS ) + (cid:96) SA ( θ SA ) .
At each iteration, we collect transitions from the source and (less frequently) target domain, storing the transitions in separate replay buffers.
We then sample a batch of experience from both buffers to update the classiﬁers.
We use the classiﬁers to modify the rewards from the source domain, and apply MaxEnt RL to this experience.
We use SAC (Haarnoja et al., 2018) as our
MaxEnt RL algorithm, but emphasize that DARC is applicable to any MaxEnt RL
algorithm (e.g., on-policy, off-policy, and model-based).
When training the classiﬁers, we add Gaussian input noise to prevent overﬁtting to the small number of target-domain transitions (see Fig.
11 for an ablation).
E XPERIMENTS
We start with a didactic experiment to build intuition for the mechanics of our method, and then evaluate on more complex tasks.
Our experiments will show that DARC outperforms alternative approaches, such as directly applying RL to the target domain or learning importance weights.
We will also show that our method can account for domain shift in the termination condition, and conﬁrm the importance of learning two classiﬁers.
[Figure]
Illustrative example.
We start with a simple gridworld example, shown on the right, where we can apply our method without function approximation.
The goal is to navigate from the top left to the bottom left.
The real environment contains an obstacle (shown in red), which is not present in the source domain.
If we simply apply RL on the source domain, we obtain a policy that navigates directly to the goal (blue arrows), and will fail when used in the target domain.
We then apply our method: we collect trajectories from the source domain and real world to ﬁt the two tabular classiﬁers.
These classiﬁers give us a modiﬁed reward, which we use to learn a policy in the source domain.
The modiﬁed reward causes our learned policy to navigate around the obstacle, which succeeds in the target environment.
Visualizing the reward modiﬁcation in stochastic domains.
In our next experiment, we use an “archery” task to visualize how the modiﬁed reward accounts for differences in dynamics.
The task, shown in Fig. 4, requires choosing an angle at which to shoot an arrow.
The practice range (i.e., the source domain) is outdoors, with wind that usually blows from left to right.
The competition range (i.e., the target domain) is indoors with no wind.
The reward is the negative distance to the target.
We plot the reward as a function of the angle in both domains in Fig.
The optimal strategy for the outdoor range is to compensate for the wind by shooting slightly to the left ( θ = − 0 . 8 ), while the optimal strategy for the indoor range is to shoot straight ahead ( θ = 0 ).
We estimate the modiﬁed reward function with DARC, and plot the modiﬁed reward in the windy outdoor range and indoor range.
We aggregate across episodes using J ( θ ) =
log E p ( s (cid:48) | θ ) [exp( r ( s (cid:48) ))] ; see Appendix E.4 for details.
We observe that maximizing the modiﬁed reward in the windy range does not yield high reward in the windy range, but does yield a policy that performs well in the indoor range.
Scaling to more complex tasks.
We now apply DARC to the more complex tasks shown in
Fig.
We deﬁne three tasks by crippling one of the joints of each robot in the target domain, but using the fully-functional robot in the source domain.
We use three simulated robots taken from OpenAI Gym (Brockman et al., 2016):
DOF reacher, half cheetah, and ant.
The broken reacher is based on the task described by Vemula et al.
We also include a task where the shift in dynamics is external to the robot, by modifying the cheetah task to reward the agent for running both forward and backwards.
It is easier to learn to run backwards, an obstacle in the target domain prevents the agent from running backwards.
This “half cheetah obstacle” task does not entirely satisfy the assumption in Eq.
1 because transitions such as bouncing off the obstacle are only possible in the target domain, not the source domain.
Nonetheless, the success of our method on this task illustrates that DARC can excel even in settings that do not satisfy the assumption in Eq. 1.
We compare our method to eight baselines.
RL on Source and RL on Target directly perform RL on the source and target domains, respectively.
The Finetuning baseline takes the result of running
RL on the source domain, and further ﬁnetunes the agent on the target domain.
The Importance
Weighting baseline performs RL on importance-weighted samples from the source domain; the importance weights are exp(∆ r ) .
Recall that DARC collects many ( r = 10 ) transitions in the source domain and performs many gradient updates for each single transition collected in the target domain (Alg. 1 Line 5). We therefore compared against a RL on Target (10x) baseline that likewise performs many ( r = 10 ) gradient updates per transition in the target domain.
Next, we compared against two recent model-based RL methods: MBPO (Janner et al., 2019) and PETS (Chua et al., 2018).
Finally, we also compared against MATL (Wulfmeier et al., 2017b), which is similar in spirit to our method but uses a different modiﬁed reward.
We show the results of this experiment in Fig. 6, plotting the reward on the target domain as a function of the number of transitions in the target domain.
In this ﬁgure, the transparent lines correspond to different random seeds, and the darker lines are the average of these random seeds.
On all tasks, the RL on source baseline (shown as a dashed line because it observes no target transitions) performs considerably worse than the optimal policy from RL on the target domain, suggesting that good policies for the source domain are suboptimal for the target domain.
Nonetheless, on three of the four tasks our method matches (or even surpasses) the asymptotic performance of doing RL on the target domain, despite never doing RL on experience from the target domain, and despite observing 5 - 10x less experience from the target domain.
On the broken reacher and broken half cheetah tasks, ﬁnetuning on the target domain performs on par with our method.
On the simpler broken reacher task, just doing RL on the target domain with a large number of gradient steps works quite well (we did not tune this parameter for our method).
While the model-based baselines (PETS and MBPO) also performed well on for low-dimensional tasks (broken reacher, broken half cheetah), they perform quite poorly on more challenging tasks like broken ant, supporting our intuition that classiﬁcation is easier than learning a dynamics model in high dimensional tasks.
Finally, DARC outperforms MATL on all tasks.
We include additional ablation experiments in Appendix D.1.
To gain more intuition for our method, we recorded the reward correction ∆ r throughout training on the broken reacher environment.
In this experiment, we ran RL on the source domain for 100k steps before switching to our method.
Said another way, we ignored ∆ r for the ﬁrst 100k steps of training.
As shown in
Fig. 7, ∆ r steadily decreases during these ﬁrst 100k steps, suggesting that the agent is learn- ing a strategy that takes transitions where the source domain and target domain have different dynamics: the agent is making use of its broken joint.
After 100k steps, when we maximize the combination of task reward and ∆ r , we observe that ∆ r increases, so the agent’s transitions in the source domain are increasingly consistent with target domain dynamics.
After around 1e6 training steps ∆ r is zero: the agent has learned a strategy that uses transitions that are indistinguishable between the source and target domains.
Safety emerges from domain adaptation to the termination condition.
In many safetycritical applications, the real-world and simulator have different safeguards, which kick in to stop the agent and terminate the episode.
For an agent to effectively transfer from the simulator to the real world, it cannot rely on safeguards which are present in one domain but not the other.
Since this termination condition is part of the dynamics (White, 2017), we can readily apply DARC to this setting.
We use the humanoid shown in Fig.
8 for this experiment and set the task reward to 0.
In the source domain episodes have a ﬁxed length of 300 steps; in the target domain the episode terminates when the robot falls.
The scenario mimics the real-world setting where robots have freedom to practice in a safe, cushioned, practice facility, but are preemptively stopped when they try to take unsafe actions in the real world.
Our aim is for the agent to learn to avoid unsafe transitions in the source domain that would result in episode termination in the target domain.
As shown in Fig.
8, our method learns to remain standing for nearly the entire episode.
As expected, baselines that maximize the zero reward on the source and target domains fall immediately.
While DARC was not designed as a method for safe RL (Tamar et al., 2013; Achiam et al., 2017; Eysenbach et al., 2017; Berkenkamp et al., 2017), this experiment suggests that safety may emerge automatically from DARC, without any manual reward function design.
with MATL Comparison with Prior Transfer Learning
Methods We are not the ﬁrst work that modiﬁes the reward function to perform transfer in RL (Koos et al., 2012), nor the ﬁrst work to learn how to modify the reward function (Wulfmeier et al., 2017a).
However, these prior works lack theoretical justiﬁcation.
In contrast, our approach maximizes a welldeﬁned variational objective and our analysis guarantees that agents learned with our method will achieve similar rewards in the source and target domains.
Our formal guarantees (Sec. 4) do not apply to MATL (Wulfmeier et al., 2017b) because their classiﬁer is not conditioned on the action.
Indeed, our results on the four tasks in Fig.
6 indicate that DARC ourperforms MATL on all tasks.
To highlight this difference, we compared DARC and MATL on a gridworld (right), where the source and target domains differed by assigning opposite effects to the “up” and “down” in the purple state in the source and target domains.
We collected data from a uniform random policy, so the marginal distribution p ( s t +1 | s t ) was the same in the source and target domains, even though the dynamics p ( s t +1 | s t , a t ) where different.
In this domain, MATL fails to recognize that the source and target domains are different.
DARC succeeds in this task for 80% of trials while MATL succeeds for 0% of trials.
We conclude the conditioning on the action, as suggested by our analysis, is especially important when using experience collected from stochastic policies.
D ISCUSSION
In this paper, we proposed a simple, practical, and intuitive approach for domain adaptation to changing dynamics in RL.
We motivate this method from a novel variational perspective on domain adaptation in RL, which suggests that we can compensate for differences in dynamics via the reward function.
Moreover, we formally prove that, subject to a lightweight assumption, our method is guaranteed to yield a near-optimal policy for the target domain.
Experiments on a range of control tasks show that our method can leverage the source domain to learn policies that will work well in the target domain, despite observing only a handful of transitions from the target domain.
Limitations The main limitation of our method is that the source dynamics must be sufﬁciently stochastic, an assumption that can usually be satisﬁed by adding noise to the dynamics, or ensembling a collection of sources.
Empirically, we found that our method worked best on tasks that could be completed in many ways in the source domain but some of these strategies were not compatible with the target dynamics.
The main takeaway of this work is that inaccuracies in dynamics can be compensated for via the reward function.
In future work we aim to use the variation perspective on domain adaptation (Sec. 4) to learn the dynamics for the source domain.
A A DDITIONAL I NTERPRETATIONS OF THE R EWARD C ORRECTION
This section presents four additional interpretations of the reward correction, ∆ r .
A. C ODING T HEORY
The reward correction ∆ r can also be understood from the perspective of coding theory.
Suppose that we use a data-efﬁcient replay buffer that exploits that fact that the next state s t +1 is highly redundant with the current state and action, s t , a t .
If we assume that the replay buffer compression has been optimized to store transitions from the target environment, (negative)
∆ r is the number of additional bits (per transition) needed for our source replay buffer, as compared with our target replay buffer.
Thus, an agent which maximizes ∆ r will seek those transitions that can be encoded most efﬁciently, minimizing the size of the source replay buffer.
A. M UTUAL I NFORMATION
We can gain more intuition in the modiﬁed reward by writing the expected value of ∆ r from Eq.in terms of mutual information:
[Equation]
The mutual information I ( s t +1 ; target | s t , a t ) reﬂects how much better you can predict the next state if you know that you are interacting with the target domain, instead of the source domain.
Our approach does exactly this, rewarding the agent for taking transitions that provide information about the target domain while penalizing transitions that hint to the agent that it is interacting with a source domain rather than the target domain: we don’t want our are agent to ﬁnd bugs in the Matrix.
A. L OWER BOUND ON THE RISK - SENSITIVE REWARD OBJECTIVE .
While we derived DARC by minimizing a reverse KL divergence (Eq. 2), we can also show that DARC maximizes a lower bound on a risk-sensitive reward objective (Mihatsch & Neuneier, 2002):
[Equation]
The inequality on the last line is an application of Jensen’s inequality.
One interesting question is when it would be preferable to maximize Eq.
4 rather than Eq.
While Eq.
5 provides a loser bound on the risk sensitive objective, empirically it may avoid the risk-seeking behavior that can be induced by risk-sensitive objectives.
We leave the investigation of this trade-off as future work [Equation]
A. A C ONSTRAINT ON D YNAMICS D ISCREPANCY
Our method regularizes the policy to visit states where the transition dynamics are similar between the source domain and target domain:
[Equation]
This objective can equivalently be expressed as applying MaxEnt RL to only those policies which avoid exploiting the dynamics discrepancy.
More precisely, the KKT conditions guarantee that there exists a positive constant (cid:15) > 0 such that our objective is equivalent to the following constrained objective:
[Equation]
where Π DARC denotes the set of policies that do not exploit the dynamics discrepancy: Π DARC
[Equation]
One potential beneﬁt of considering our method as the unconstrained objective is that it provides a principled method for increasing or decreasing the weight on the ∆ r term, depending on how much the policy is currently exploiting the dynamics discrepancy.
We leave this investigation as future work.
B P ROOFS OF T HEORETICAL G UARANTEES
In this section we present our analysis showing that maximizing the modiﬁed reward r + ∆ r in the source domain yields a near-optimal policy for the target domain, subject to Assumption 1.
To start, we show that maximizing the modiﬁed reward in the source domain is equivalent to maximizing the unmodiﬁed reward, subject to the constraint that the policy not exploit the dynamics:
Lemma B.1.
Let a reward function r ( s, a ) , source dynamics p source ( s (cid:48) | s, a ) , and target dynamics p target ( s (cid:48) | s, a ) be given.
Then there exists (cid:15) >
such the optimization problem in Eq.
2 is equivalent to
[Equation]
where Π no exploit denotes the set of policies that do not exploit the dynamics:
[Equation]
The proof is a straightforward application of the KKT conditions.
This lemma says that maximizing the modiﬁed reward can be equivalently viewed as restricting the set of policies to those that do not exploit the dynamics.
Next, we will show that policies that do not exploit the dynamics have an expected (entropy-regularized) reward that is similar in the source and target domains:
Lemma B.2.
Let policy π ∈ Π no exploit be given, and let R max be the maximum (entropy-regularized) return of any trajectory.
Then the following inequality holds:
[Equation]
This Lemma proves that all policies in Π no exploit satisfy the same condition as the optimal policy (Assumption 1).
Proof.
To simplify notation, deﬁne ˜
r ( s, a ) =
r ( s, a ) − log π ( a | s ) .
We then apply Holder’s inequality and Pinsker’s inequality to obtain the desired result:
[Equation]
We restate our main result:
Theorem 4.1 (Repeated from main text) .
Let π ∗ DARC be the policy that maximizes the modiﬁed (entropy-regularized) reward in the source domain, let π ∗ be the policy that maximizes the (unmodiﬁed, entropy-regularized) reward in the target domain [Equation] and assume that π ∗ satisﬁes Assumption 1.
Then π ∗ DARC receives near-optimal (entropy-regularized) reward on the target domain:
[Equation]
Proof.
Assumption 1 guarantees that the optimal policy for the target domain, π ∗ , lies within Π no exploit .
Among all policies in Π no exploit , π ∗ DARC is (by deﬁnition) the one that receives highest reward on the source dynamics, so
[Equation]
Since the both π ∗ DARC and π ∗ lie inside the constraint set [Equation] Lemma B.2 dictates that their rewards on the target domain differ by at most 2 R max (cid:112) (cid:15)/ 2 from their source domain rewards.
In the worst case, the reward for π ∗ DARC decreases by this amount and the reward for π ∗ increases by this amount:
[Equation]
Substituting these inequalities on the LHS and RHS of Eq.
B and rearranging terms [Equation] we obtain the desired result.
C T HE S PECIAL C ASE OF AN O BSERVATION M ODEL
To highlight the relationship between domain adaptation of dynamics versus observations [Equation] we now consider a special case.
In this subsection
[Equation] we will assume that the state s t (cid:44) ( z t , o t ) is a combination of the system latent state z t (e [Equation] g., the poses of all objects in a scene) and an observation o t (e
[Equation]
g., a camera observation).
We will deﬁne q ( o t | z t ) and p ( o t | z t ) as the observation models for the source and target domains.
In this special case
[Equation]
we can decompose the KL objective (Eq. 2) into three terms:
[Equation]
[Figure] (cid:21) . Prior methods that perform observation adaptation (Bousmalis et al., 2018; Gamrian & Goldberg, 2018) effectively minimize the observation adaptation term, 1 but ignore the effect of dynamics.
In contrast, the ∆ r reward correction in our method provides one method to address both dynamics and observations.
These approaches could be combined; we leave this as future work.
D A DDITIONAL E XPERIMENTS
Figure 10 is a copy of Fig.
7 from the main text, modiﬁed to also show the agent’s reward on the target domain.
We observe that the reward does not start increasing until we start using DARC.
D. A BLATION EXPERIMENTS .
Our next experiment examines the importance of using two classiﬁers to estimate ∆ r .
We compared our method to an ablation that does not learn the SA classiﬁer, effectively ignoring the blue terms in Eq.
As shown in Fig.
11 (left), this ablation performs considerably worse than our method.
Intuitively, this makes sense: we might predict that a transition came from the source domain not because the next state had higher likelihood under the source dynamics, but rather because the state or action was visited more frequently in the source domain.
The second classiﬁer used in our method corrects for this distribution shift.
Finally, we examine the importance of input noise regularization in classiﬁers.
As we observe only a handful of transitions from the target domain, we hypothesized that regularization would be important to prevent overﬁtting.
We test this hypothesis in Fig.
11 (right) by training our method on the broken reacher environment with varying amounts of input noise.
With no noise or little noise our method performs poorly (likely due to overﬁtting); too much noise also performs poorly (likely due to underﬁtting).
We used a value of 1 in all our experiments, and did not tune this value.
See
Appendix D.1 for more plots of both ablation experiments.
E E XPERIMENT D ETAILS AND H YPERPARAMETERS
Our implementation of DARC is built on top of the implementation of SAC from Guadarrama et al.
Unless otherwise speciﬁed, all hyperparameters are taken from Guadarrama et al.
All neural networks (actor, critics, and classiﬁers) have two hidden layers with 256-units each and ReLU activations.
Since we ultimately will use the difference in the predictions of the two classiﬁers, we use a residual parametrization for the SAS classiﬁer q ( target | s t , a t , s t +1 ) .
Using f SAS ( s t , a t , s t +1 ) , f SA ( s t , a t ) ∈ R 2 to denote the outputs of the two classiﬁer networks, we compute the classiﬁer predictions as follows:
[Equation]
For the SAS classiﬁer we propagate gradients back through both networks parameters, θ SAS and θ SA .
Both classiﬁers use Gaussian input noise with σ = 1 .
Optimization of all networks is done with
Adam (Kingma & Ba, 2014) with a learning rate of 3e-4 and batch size of 128.
Most experiments with DARC collected 1 step in the target domain every 10 steps in the source domain (i.e., r = 10 ).
The one exception is the half cheetah obstacle domain, where we tried increasing r beyond 10 to 30, 100, 300, and 1000.
We found a large beneﬁt from increasing r to 30 and 100, but did not run the other experiments long enough to draw any conclusions.
Fig.
6 uses r = 30 for half cheetah obstacle.
We did not tune this parameter, and expect that tuning it would result in signiﬁcant improvements in sample efﬁciency.
We found that DARC was slightly more stable if we warm-started the method by applying RL on the source task without ∆ r for the ﬁrst t warmup iterations.
We used t warmup = 1 e 5 for all tasks except the broken reacher, where we used t warmup = 2 e 5 .
This discrepancy was caused by a typo in an experiment, and subsequent experiments found that DARC is relatively robust to different values of t warmup ; we did not tune this parameter.
E. B ASELINES
The RL on Source and RL on Target baselines are implemented identically to our method, with the exception that ∆ r is not added to the reward function.
The RL on Target (10x) is identical to RL on Target, with the exception that we take 10 gradient steps per environment interaction (instead of 1).
The Importance Weighting baseline estimates the importance weights as p target ( s t +1 | s t , a t )
/p source ( s t +1 | s t , a t ) ≈ exp(∆ r ) .
The importance weight is used to weight transitions in the SAC actor and critic losses.
PETS (Chua et al., 2018)
The PETS baseline is implemented using the default conﬁgurations used by (Chua et al., 2018) for the environments evaluated.
The broken-half-cheetah environment uses the hyperparameters as used by the half-cheetah environment in (Chua et al., 2018).
The broken-ant environment uses the same set of hyperparameters, namely: task horizon = 1000, number of training iterations = 300, number of planning (real) steps per iteration = 30, number of particles to be used in particle propagation methods = 20.
The PETS codebase can be found at https://github.com/kchua/handful-of-trials .
MBPO (Janner et al., 2019)
We used the authors implementation with the default hyperparameters: https://github.com/JannerM/mbpo .
We kept the environment conﬁgurations the same as their default unmodiﬁed MuJoCo environments, except for the domain and task name.
We added our custom environment xmls in mbpo/env/assets/ folder, and their corresponding environment python ﬁles in the mbpo/env/ folder .
Their static ﬁles were added under mbpo/static/ .
These environments can be registered as gym environments in the init ﬁle under mbpo odrl/mbpo/env/ or can be initialized directly in softlearning/environments/adapters/gym adapter.py .
We set the time limit to max episode steps=1000 for the Broken Half Cheetah, Broken Ant and Half Cheetah Obstacle environments and to 100 for the Broken Reacher environment.
E. E NVIRONMENTS
Broken Reacher
This environment uses the 7DOF robot arm from the Pusher environment in
OpenAI Gym.
The observation space is the position and velocities of all joints and the goal.
The reward function is
[Equation]
and episodes are 100 steps long.
In the target domain the 2nd joint (0-indexed) is broken: zero torque is applied to this joint, regardless of the commanded torque.
Broken Half Cheetah This environment is based on the HalfCheetah environment in OpenAI
Gym.
We modiﬁed the observation to include the agent’s global X coordinate so the agent can infer its relative position to the obstacle.
Episodes are 1000 steps long.
In the target domain the 0th joint (0-indexed) is broken: zero torque is applied to this joint, regardless of the commanded torque.
Broken Ant
This environment is based on the Ant environment in OpenAI Gym.
We use the standard termination condition and cap the maximum episode length at 1000 steps.
In the target domain the 3rd joint (0-indexed) is broken: zero torque is applied to this joint, regardless of the commanded torque.
In all the broken joint environments, we choose which joint to break to computing which joint caused the “RL on Source” baseline to perform worst on the target domain, as compared with the “RL on
Target” baseline.
Half Cheetah Obstacle
This environment is based on the HalfCheetah environment in OpenAI
Gym.
Episodes are 1000 steps long.
We modiﬁed the standard reward function to use the absolute value in place of the velocity, resulting the following reward function:
[Equation]
where s x vel is the velocity of the agent along the forward-aft axis and ∆ t = 0 .
01 is the time step of the simulator.
In the target domain, we added a wall at x = − 3 m , roughly 3 meters behind the agent.
Humanoid Used for the experiment in Fig.
8, we used a modiﬁed version of Humanoid from
OpenAI Gym.
The source domain modiﬁed this environment to ignore the default termination condition and instead terminate after exactly 300 time steps.
The target domain uses the unmodiﬁed environment, which terminates when the agent falls.
E. F IGURES
Unless otherwise noted, all experiments were run with three random seeds.
Figures showing learning curves (Figures 6, 11, 7, 12, and 13) plot the mean over the three random seeds, and also plot the results for each individual random seed with semi-transparent lines.
E. A RCHERY E XPERIMENT
We used a simple physics model for the archery experiment.
The target was located 70m North of the agent, and wind was applied along the East-West axis.
The system dynamics:
[Equation]
We trained the classiﬁer by sampling θ ∼ U [ − 2 , 2] (measured in degrees) for 10k episodes in the source domain and 10k episodes in the target domain.
The classiﬁer was a neural network with 1 hidden layer with 32 hidden units and ReLU activation.
We optimized the classiﬁer using the Adam optimizer with a learning rate of 3e-3 and a batch size of 1024.
We trained until the validation loss increased for 3 consecutive epochs, which took 16 epochs in our experiment.
We generated Fig. 4 by sampling 10k episodes for each value of θ and aggregating the rewards using
J ( θ ) = log E p ( s (cid:48) | θ ) [exp( r ( s (cid:48) ))] .
We found that aggregating rewards by taking the mean did not yield meaningful results, perhaps because the mean corresponds to a (possibly loose) lower bound on J (see Appendix A.3).