1 Introduction
Imitation Learning has become an incredibly convenient scheme to teach robots skills for specific tasks [Wang et al.2017, Pathak et al.2018, Yu et al.2018, Stadie, Abbeel, and Sutskever2017, Sermanet et al.2018, Edmonds et al.2017]. It is often achieved by showing the robot various expert trajectories of stateaction pairs. Existing imitation methods like MAML [Finn, Abbeel, and Levine2017] and OneShot Imitation Learning [Duan et al.2017] requires perfect demonstrations in the sense that the experts should perform the same as they expect the robot would do. However, this requirement may not always hold since collecting exactlyrelevant demonstrations is resourceconsuming.
One possible relaxation is assuming the expert to perform a basic task that is related but not necessary the same as the target task (sharing some common features, parts, etc). This relaxation, at the very least, can reduce the human effort on demonstration collecting and enrich the diversity of the demonstrations for task transfer. For example in Figure 1, the expert demonstrations contain the agent movements along an arbitrary direction, while the desired target is to move along only one specified direction.
Clearly, it does not come for free to learn target action policy from the relaxed expert demonstrations. More advanced strategies are required to transfer the action policy from the demonstrations to the target task. The work by [De Gemmis et al.2009] suggests that using experts’ preference as a supervised signal can achieve nearly optimal learning result. Here, the preference refers to the highlyabstract evaluation rules or choice tendency of a human for making comparison and selection among data samples. Indeed, the preference mechanism has been applied in many other scenarios, such as complex tasks learning [Wirth et al.2017], policy updating [Christiano et al.2017], and policy optimization combing with Inverse Reinforcement Learning (IRL) [Wirth and Fürnkranz2013] to name a few.
However, previous preferencebased methods mainly focus on learning the utility function behind each comparison, where the distribution of trajectories is never studied. However, this would be inadequate for task transfer. The importance of modeling distribution comes from two aspects: 1. Learning the trajectory distribution takes a critical role in preferencebased selection, which will be discussed lately; 2. With the distribution, it is more convenient to provide a theoretical analysis of the efficiency and stability of the task transfer algorithm (See Section 3.4).
In this work, we approach the task transfer by utilizing the expert preference in a principled way. We first model the preference selection as a rejection sampling where a hidden cost is proposed to compute the acceptance probability. After selection, we then learn the distribution of the target trajectories based on the preferred demonstrations. Since the candidate demonstrations would usually be insufficient after selection, we augment the demonstrations with the samples of the current learned trajectory distribution and perform the preference selection and distribution learning iteratively. The distribution here acts as the knowledge which we make the transfer on. The theoretical derivations prove that it can improve the preference after each iteration and the target distribution will eventually converge.
As the core of our framework, the trajectory distribution and cost learning are based on but has advanced the Maximum Entropy Inverse Reinforcement Learning (MaxEnt IRL) [Ziebart et al.2008] and its adversary version [Finn et al.2016]. The MaxEnt IRL framework models the trajectory distribution as the exponential of the explicitlycoded cost function. Nevertheless in MaxEnt IRL, computing the partition function requires MCMC or iterative optimization, which is timeconsuming and numerically unstable. Hence in adversary MaxEnt IRL, it avoids the computation of the partition function by casting the whole IRL problem into optimization of a generator and a discriminator. Although the adversary MaxEnt IRL is more flexible, it never delivers any form of the cost function, which is crucial for downstream applications and policy learning. Our method enhances the original adversary MaxEnt IRL by redefining the samples from the trajectory level to the stateaction level and devise the cost function using the outputs of the discriminator and generator. With the cost function, we can optimize the generator by any offtheshelf reinforcement learning method and then the optimal generator could be used as a policy on the target task.
To summarize, our key contributions are as follow.

We propose to perform imitation learning from related but not exactlyrelevant demonstrations by making use of the expert preferencebased selection.

We enhance the Adversarial MaxEnt IRL framework for learning the trajectory distribution and cost function simultaneously.

Theoretical analyses have been provided to guarantee the convergence of the proposed task transfer frameworks. Considerable experimental evaluations demonstrate that our method obtains comparable results with other algorithms that require accurate demonstrations or costs.
2 Preliminaries
This section reviews fundamental conceptions and introduces related works to our method. Before further introduction, we first provide key notations used in this paper.
Notations.
For modeling the action decision procedure of an agent, The Markov Decision Processes (MDP) without reward
is used, where denotes a set of states which can be acquired from environment; denotes a set of actions controlled by the agent; denotes the transition probability from state to by action ; is a discount factor; is the distribution of the initial state ; defines the policy. A trajectory is given by the sequence of stateaction pairs . We define the cost function parameterized by over a sa pair as , and define the cost over a trajectory as where is time step. A trajectory set is formulated by expert demonstrations, i.e. .2.1 MaxEnt IRL
Given a demonstration set , the Inverse Reinforcement Learning (IRL) method [Ng, Russell, and others2000] seeks to learn optimal parameters of the cost function . The solution could be multiple when using insufficient demonstrations. The MaxEnt IRL [Ziebart2010, Boularias, Kober, and Peters2011] handles this ambiguity by training the parameters to maximize the entropy over trajectories, leading to the optimization problem as:
(1) 
Here is the distribution of trajectories; is the probability of the expert trajectory; computes the expectation. The optimal is derived to be the Boltzmann distribution associated with the cost , namely,
(2) 
Here is the partition function given by the integral of over all trajectories.
2.2 Generative Adversarial Networks
Generative Adversarial Networks (GANs) provides a framework to model a generator and a discriminator simultaneously. generates sample from noise , while takes as input, and outputs the likelihood value indicates whether is sampled from underlying data distribution or from generator [Goodfellow et al.2014]
(3) 
Generator loss , discriminator loss and optimization goals are defined as (3). Here is modified as the sum among logarithm confusion and opposite loss of
for keeping training signal in case generated sample is easily classified by the discriminator.
3 Methodology
Our preferencebased task transfer framework consists of 2 iterative subprocedures: 1) querying expert preference and construct a selected trajectory samples set; 2) learning the trajectory distribution and cost function from this samples set for regenerating more samples for next episode. Starting from the demonstrations of the basic task, the trajectory distribution and cost function we learned are improved continuously. Finally, with the learned cost function, we can derive a policy of the target task.
The following sections will cover the modeling and analysis for all the two steps mentioned above. In Section 3.1, we will introduce the hiddencost model for modeling the expert preferencebased selection. Then in Section 3.2, our enhanced Adversarial MaxEnt IRL for distribution and cost learning will be presented. We will combine the above two components to develop a preferencebased task transfer framework and provide the theoretical analysis on it.
3.1 Preferencebased Sampling and Hidden Cost Model
The main idea of our task transfer framework is transferring trajectory distribution with sample selection. Different from other transfer learning algorithms, the selection in our method only depends on preference provided by experts instead of any quantities. The preference of expert here could be abstract conceptions or rules on the performance of agents in target task, which are hard to directly be formalized as cost functions or provided numerically by the expert. In our preferencebased cost learning framework, however, we only require experts to choose their most preferred samples among the given set generated on the last step, and try to use the selection result as the guidance on migrating the distribution from current policy to the target task policy.
We migrate the distribution by preferencebased selection of samples in current set, the agent should be able to generate feasible trajectories on target task, which requires the probability of a trajectory on current task to be nonzero whenever the probability of that trajectory on target task is nonzero, and there should exist one finite value (which indicates the expected rejections made before a sample is accepted) that
(4) 
where and are feasible trajectory sets of current task and target task respectively. In previous section, we have shown that under MaxEnt IRL, the expert trajectories are assumed to be sampled from a Boltzmann distribution with negative cost function as energy. For an arbitrary trajectory , there will be
(5) 
where and are ground truth costs over a trajectory of current and target task, while and are corresponding cost functions. During selection, we suppose that the expert intends to keep the trajectory which have lower cost value on target task, which means the preference selection procedure could be seen as a rejection sampling over set with acceptance probability
(6) 
We define the gap between target cost and current cost as hidden cost and for trajectory . Thus we can view as a latent factor, or formally, a negative utility function [Wirth et al.2016] that indicates the preference and at the same while indicates the gap between target distribution and current distribution. Lower expectation of over the set of samples indicates greater acceptance possibilities and indicated current distribution to be more similar as target one. After each step, by reintroducing the accept rate, the probability of a sample presenting in the set after selection should be
(7) 
With preferencebased sample selection, the trajectory distribution is expected to approach to the one under the target task finally. The convergence analysis will be provided in Section 3.4.
3.2 Enhanced Adversarial MaxEnt IRL for Distribution and Cost Learning
In the previous section, we introduce how the preferencebased sample selection works in our task transfer framework. However, since the task transfer is an iterative process, we need to generate more samples with the same distribution as the selected samples set to keep it selectable by experts. Additionally, a cost function needs to be extracted from the selected demonstrations to optimize policies. With our enhanced Adversarial MaxEnt IRL, we can tackle these problems by learning the trajectory distribution and unbiased cost function simultaneously.
Adversarial MaxEnt IRL [Finn et al.2016] is a recently proposed GANbased IRL algorithm that explicitly recovers the trajectory distribution from demonstrations. We enhance it to meet the requirements in our task transfer framework. Our enhancement is twofold:

Redefining the GAN from trajectory level to stateaction pair level to extract a cost function that can be directly used for policy optimization.

Although the GAN does not directly work on trajectory anymore, we prove that the generator can still be a sampler to the trajectory distribution of demonstrations.
We first briefly review the main ideas of Adversarial MaxEnt IRL. In this algorithm, demonstrations are supposed to be drawn from a Boltzmann distribution (2
), and the optimizing target can be regarded as Maximum Likelihood Estimation(MLE) over trajectory set
(8) 
The optimization in (8) can be cast into an optimization problem of a GAN [Goodfellow et al.2014, Finn et al.2016], where the discriminator takes the form as followed
(9) 
CONN_GAN_IRL showed that, when the model is trained to optimal, the generator will be an optimal sampler of the trajectory distribution . However, we still cannot extract a closedform cost function from the model. As a result, we enhanced it to meet our requirements.
Since the cost function should be defined on each stateaction pair, we first modified the input of the model in (9) from a trajectory to a stateaction pair
(10) 
The connection between the accurate cost and outputs , of GANs can be established
(11) 
Here we define as a cost estimator, while is the accurate cost function. Since the partition function is a constant while cost function is fixed, it will not affect the policy optimization, which means that can be directly integrated in common policy optimization algorithms as a unbiased cost function.
Notice that, after this modification, there will be several issues we need to address. Firstly, since the GAN is not defined on trajectory anymore, the equivalence between Guided Cost Learning and GAN training need to be reverified. We will discuss it in Section 3.4. Moreover, it is not straightforward whether is a sampler to the distribution of demonstrations.
We now show that when is trained to optimal, the distribution of trajectories sampled from it is exactly the distribution of demonstrations:
Assumption 1.
The environment is stationary.
Lemma 1.
Suppose that we have an expert policy to produce demonstrations , a trajectory is sampled from . Then will have the same probability as drawn from if Assumption 1 holds ( is the trajectory distribution of ).
Proof.
We first introduce the environment model and the state distribution . In Reinforcement Learning, environment is basically a condition distribution over state transitions . Thus the probability of a given trajectory will be
(12) 
Now we sample a trajectory with by executing rollouts. Under Assumption 1, the environment model for sampling from will be the same as sampling the demonstrations , while is also identical. Therefore, the probability of sampling can be derived as
(13) 
It’s obvious that . ∎
Lemma 2.
[Goodfellow et al.2014] The global minimum of the discriminator objective (3) is achieved when .
For a GAN defined on stateaction level, with Lemma 2, , is the expert policy for producing demonstrations. Then with Lemma 1, it’s obviously that the trajectory sampled with will have the same density as , which means that can still be a sampler to the trajectory distribution of demonstrations.
We formulate the minimization of generator loss as a policy optimization problem. We regard the unbiased cost estimator as the cost function instead of in (3), and as a policy . Thus the policy objective will be
(14) 
This is quite similar to the generator objective used by GAIL [Ho and Ermon2016] but with an extra entropy penalty. We’ll compare the performances of cost learning between our method and GAIL in Section 4.
3.3 Preferencebased Task Transfer
The entire task transfer framework is demonstrated in Algorithm 1, which combines the hidden cost model for preferencebased selection and enhanced Adversarial MaxEnt IRL for distribution and cost learning. With this framework, a welltrained policy on the basic task can be transferred to target task without accurate demonstrations or cost.
Comparing to Section 3.2, we adopt a stop condition with and which indicates the termination of the loop, and an extra selection constraint which is observed to be helpful for stability in preliminary experiments. In practice, the parameters of and can be directly inherited from and when . Compare to initialize from scratch, this will converge faster in each iteration, while the results remain the same.
3.4 Theoretical Analysis
In this section, we will discuss how can our framework learn the distribution from trajectories in each episode and finally transfer the cost function to target task. Remember the core part in our framework: Transferring the trajectory distribution from to . There is a finite loop in this process, during which we query for preference as and improve the distribution for each episode . If the distribution improves monotonically and the improvement can be maintained, we can guarantee the convergence of our method, which means that can be learned. Then the cost function we learned together with will also approach to the cost for target task . This intuition is shown as following:
Proposition 1.
Given a finite set of trajectories sampled from distribution and an expert with select probability (6), the hidden cost over a trajectory is improved monotonically after each selection.
This proposition can be proved with some elementary derivations. Here we only provide the proof sketch. Since all the trajectories in are sampled from corresponding distribution , the expect cost can be estimated. Notice that we use a normalized select probability . Thus the estimations of expectation before and after the selection will be
(15) 
Obviously, trajectories after selection can not be seen as samples drawn from , here we use , which can be regarded as an improved . Under linear expansion of cost, can be proved. Thus the expect cost over a trajectory is improved monotonically.
Then we need to reverify that whether the proposed stateaction level GAN in our enhanced Adversarial MaxEnt IRL is still equivalent to Guided Cost Learning [Finn, Levine, and Abbeel2016]:
Theorem 1.
Suppose we have demonstrations , a GAN with generator , discriminator . Then when the generator loss is minimized, the sampler loss in Guided Cost Learning [Finn, Levine, and Abbeel2016] is also minimized. is the learned trajectory distribution, and is corresponding sampler.
Since the is minimized along with , when the adversarial training ends, an optimal sampler of can be obtained. Now we need to prove is drawn from :
Theorem 2.
In Theorem 2, MaxEnt IRL is regarded as a MLE of (2), while the unknown partition function needs to be estimated. Therefore, training a stateaction level GAN is still equivalent to maximizing the likelihood of trajectory distribution. Thus we can learn the optimal cost function and distribution under the current trajectory set at the same time.
With Proposition 1, we can start from an arbitrary trajectory distribution and trajectory set drawn from it. Then we can define a trajectory distribution iteration as [Haarnoja et al.2017]. Then expected hidden cost over a trajectory improves monotonically in each episode. With Theorem 1, 2, by strictly recovering the distribution as from trajectory set (after selection), our algorithm can guarantee to maintain the improvement of expect cost over a trajectory to next episode.
Under certain regularity conditions [Haarnoja et al.2017], converges to . For trajectories that sampled from the target distribution , their corresponding select probability (6) will approach to . Thus can be a fixed point of this iteration when the iteration starts from . Since all the nonoptimal distribution can be improved this way, the learned distribution will converge to at infinity. As we have showed before, with a limited demonstrated trajectories sampled from arbitrary trajectory distribution , an optimal cost can be extracted through our enhanced Adversarial MaxEnt IRL proposed in Section 3.2. Therefore, the target cost can also be learned from transferred distribution .
4 Experiments
We evaluate our algorithm on several control tasks in MuJoCo [Todorov, Erez, and Tassa2012] physical simulator with predefined groundtruth cost function on basic tasks and on target tasks in each experiments, and are accumulated costs over trajectory for basic and target task respectively. All the initial demonstrations are generated by a welltrained PPO using , and during the transfer process, preference is given by emulator with negative utility function (or hidden cost over a trajectory) . The select probability follows the definition in (6). For performance evaluation, we use averaged return with respect to as the criterion.
4.1 Overview
In experiments, we mainly want to answer three questions:

During the task transfer procedure, can our method recover the trajectory distribution from demonstrations in each episode?

Starting from a basic task, can our method finally transfer to the target task and learn the cost function of it?

Under the same task transfer problem, can our method (based on preference only) obtain a policy with comparable performance, compared to other task transfer algorithms (based on accurate cost or demonstrations)?
To answer the first question, we need to verify the distribution learning part in our method functionally. Since our enhanced Adversarial MaxEnt IRL is built upon MaxEnt IRL, the recovered trajectory distribution can be reflected as a cost function, and the trajectories we learn from being generated by the optimal policy under that cost. Intuitively, given the expert trajectories generated by PPO and its corresponding cost , if we can train a policy which can generate with similar average , we believe that the trajectory distribution can be recovered.
To answer the second question, we evaluate the complete preferencebased task transfer algorithm under some customized environments and tasks. In each environment, we transfer current policy under basic task to the target one. During the transfer process, expert preference (emulated by computer) is given as a selection result only, while any information of cost or selecting rule is unknown to the agent. We also train an expert policy with PPO and for comparison. In each episode, we generate using our learned policy and record . If the average finally approaches to , we can verify that our method can learn the cost function of target task.
To answer the third question, we compare our method with MAML [Finn, Abbeel, and Levine2017], a task transfer algorithm requiring accurate . We use averaged cost on target task in each episode (we consider gradient step in MAML the same as episode in our method) for evaluation, to see whether the result of our method is comparable.
4.2 Environments and Tasks
Here we outline some specifications of the environments and tasks in our experiments:

Hopper, Walker2d, Humanoid and Reacher: These environments and tasks are directly picked from OpenAI Gym [Brockman et al.2016] without customization. Since they are only used for functionally verifying our distribution learning part and comparing with the original GAIL algorithm, there are no transfer settings.

MountainCar, Two Peaks One Peak: In this environment, there are two peaks for the agent to climb. The basic task is to make the vehicle higher, while the target task is climbing to a specified one.

Reacher, Two Targets Center of Targets: In this environment, the agent needs to control a 2DOF manipulator to reach some specified targets. For the basic task, there will be two targets, and the agent can reach any of them, while in target task the agent is expected to reach the central position between the two targets.

HalfCheetah, Arbitrary Backward: In this environment, the agent needs to control a multijoint (6) robot to move forward or backward. The two directions are all acceptable in the basic task, while only moving backward is expected in the target task.

Ant, Arbitrary Single: This environment enhances the HalfCheetah environment in two aspects: First, there will be more joints (8) to control; Second, the robot can move to arbitrary directions. In the basic task, any directions are allowed, while only one specified direction is expected in the target task.
4.3 Distribution and Cost Learning
We first concern the question whether our method can recover the trajectory distribution from demonstrations during the task transfer procedure. Experiment results are shown in Figure 2. All the selected control tasks are equipped with highdimensional continuous state and action spaces, which can be challenging to common IRL algorithms. We find that our method achieves nearly the same final performance as the expert (PPO) that provides the demonstrations, indicating that our method can recover the trajectory distribution. Also, comparing with other stateoftheart IRL methods like GAIL, our method can learn a better trajectory distribution and a cost function more efficiently.
4.4 Preferencebased Task Transfer
In Figure 3, we demonstrate the transfer results on two environments. The transfer in Reacher environment is more difficult than MountainCar toy environment. The reason would be that the later one can be clustered easily since there are only two actual goals that a trajectory may reach, and the target goal (to reach one specified peak) is exactly one of them. In Reacher environment, although the demonstrations in the basic task still seem to be easily clustered, the target task cannot be directly derived from any of the clusters. In both two transfer experiments, the adapted policies produced by our algorithm show nearly the same performances as the experts that directly trained on these two target tasks. As the transferred policy is trained with the learned cost function, we can conclude that our algorithm can transfer to target task by learning the target cost function. In our experiments, we find that within less than 10 episodes and less than 100 querying number at each episode can sufficiently derive desired performance. Another potential improvement of our method is to apply some commutable rules to simulate the human selection and reduce the querying time.
4.5 Comparison with Other Methods
We compare our method with some stateoftheart task transfer algorithms including MAML [Finn, Abbeel, and Levine2017]. Results are shown in Figure 4. HalfCheetah environment is pretty similar to MountainCar for the limited moving directions. However, its state and action space dimensions are much higher, which increase the difficulties for trajectory distribution and cost learning. Ant is the most difficult one among all the environments. Due to its unrestricted moving directions, the demonstrations on the basic task are highly entangled. The results illustrate that our method achieves comparable performances to those methods that require the accurate cost of the target task on the testing environments. Notice that, for some hard environment like Ant, our method may run for more episodes than MAML, since our algorithm only depends on preference, the results can still be convincing and impressive.
5 Conclusion
In this paper, we present an algorithm that can transfer policies through learning the cost function on the target task with expertprovided preference selection results only. By modeling the preferencebased selection as rejection sampling and utilizing enhanced Adversarial MaxEnt IRL for directly recovering the trajectory distribution and cost function from selection results, our algorithm can efficiently transfer policies from a related but not exactlyrelevant basic task to the target one, while theoretical analysis on convergence can be provided at the same time. Comparing to other task transfer methods, our algorithm can handle the scenario in which acquiring the accurate demonstrations or cost functions from experts is inconvenient. Our results achieve comparable task transfer performances to other methods which depend on accurate costs or demonstrations. Future work could focus on the quantitative evaluation of the improvement on the transferred cost function. Also, the upper bound on the sum of total operating episodes could be analyzed.
Acknowledgment
This research work was jointly supported by the Natural Science Foundation great international cooperation project (Grant No:61621136008) and the National Natural Science Foundation of China (Grant No:61327809). Professor Fuchun Sun(fcsun@tsinghua.edu.cn) is the corresponding author of this paper and we would like to thank Tao Kong, Chao Yang and Professor Chongjie Zhang for their generous help and insightful advice.
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