1 Introduction
Teaching new actions to robot actors through demonstration is one of the most attractive methods for behavior learning. While robots can learn new behaviors using reinforcement learning with a prespecified reward function (Sutton & Barto, 1998), significant exploration is often required to extract the behavior from the reward. In some cases, denser reward functions can help speed up the exploration process, but designing them requires a certain level of skill and understanding of the reinforcement learning process, and can often result in unexpected behaviors when the reward function doesn’t precisely guide the action. Instead, teaching a robot a behavior simply by demonstrating it removes the requirement of explicitly specifying a reward function altogether. Anyone who knows how to perform the task can demonstrate it without understanding the learning process, and the learning process requires much less exploration. This process–learning from demonstration (LfD)–aims to take a series of observed states (e.g. joint angles, position in space) and actions (e.g. decisions to move a joint at some speed) and extract a policy that approximates the demonstrated behavior (Argall et al., 2009).
While being able to imitate a behavior after observing the state and actions of a demonstrator is useful, there are many situations where the actions of the demonstrator are unknown. Common approaches to LfD require both the states and actions of the demonstrator to be recorded (Argall et al., 2009). In imitation from external observation (IfO) (Liu et al., 2018; Torabi et al., 2019c), on the other hand, just the observable states of the demonstrator are known–no action information is available. Imitating behaviors solely from observable data greatly expands the set of possible demonstrators: behaviors could be learned from inperson human demonstrators or even the vast collection of videos available online.
While imitation from external observation has been studied and performed with some success for two decades (Ijspeert et al., 2001)
, recent advances in deep neural networks have widened the set of behaviors that can be imitated and the ways that demonstration data can be collected. One way deep learning has been applied to IfO is through generative adversarial networks
(Torabi et al., 2018b; Ho & Ermon, 2016; Chen et al., 2016). In this approach–generative adversarial imitation from observation (GAIfO)–one network learns a control policy for imitating the demonstrator while the other learns to discriminate between the demonstrator’s behavior and that of the imitator. While GAIfO advanced the state of the art in imitation from observation, it comes with its own set of challenges. First, in comparison with simpler regressed models, deep networks are notorious for requiring orders of magnitude more training data, and GAIfO is no exception. Second, this algorithm uses modelfree reinforcement algorithms which are usually very data inefficient. Some of the possible benefits of the applications of IfO break down when a high sample size is required. Therefore, in practice, this algorithm has been largely limited to being studied in simulation. In simulation, many experiences and large demonstration sets can be collected quickly. Physical demonstrations are more costly to perform, and realtime constraints limit the speed at which control policies can be evaluated and thus behavior learned. For imitation from observation to work on a physical robot, a higher degree of sample efficiency is required.Deep reinforcement learning has faced similar obstacles with learning with limited samples, especially in the context of robotic control policies with complex dynamics. However, recently, trajectory centric reinforcement learning algorithms are being used to guide neural network policy search which has been shown that is very sampleefficient (Levine & Koltun, 2013; Levine & Abbeel, 2014; Levine et al., 2015, 2016). These algorithms achieve this sample efficiency in part by gaining insight into dynamics through the iterative training of linear quadratic regulators (iLQR’s) (Tassa et al., 2012) on a set of trajectory controllers.
In this paper, we propose an imitation from observation algorithm, LQR+GAIfO, that takes advantage of both (1) the high performance of the adversarial learning algorithms, and (2) the sample efficiency of trajectory centric reinforcement learning algorithms. We apply the proposed algorithm to a 6degreeoffreedom robot arm to learn to imitate behaviors from a set of lowlevel state trajectories. We find that this new method results in successful imitation learning with fewer samples than the previous algorithms.
In Section 2 of this paper, we discuss previous work related to this topic. In Section 3, we cover the techniques involved in GAIfO and LQR. Section 4, describes our approach to combining LQR and GAIfO into one functional algorithm. In Section 5, we share our experimental setup and results, and we discuss results in Section 6. Finally, in Section 7, we summarize and discuss potential future work.
2 Related Work
Our approach to sampleefficient imitation learning is built upon previous works in the field of imitation learning and trajectorycentric reinforcement learning. In the following we discuss previous works on both topics.
Techniques for imitation learning differ in the way they approach the problem. Two popular approaches to imitation learning have been behavioral cloning (Pomerleau, 1991) and inverse reinforcement learning (IRL) (Ng et al., 2000; Russell, 1998)
. Behavioral cloning views the imitation learning problem as a supervised learning problem that attempts to learn a direct mapping from states to actions. On the other hand, inverse reinforcement learning works to find a cost function under which the expert demonstrator is optimal. One approach of this type is guided cost learning
(Finn et al., 2016) which builds on maximum entropy IRL (Ziebart et al., 2008) and guided policy search algorithm (Levine & Abbeel, 2014) and achieves impressive results on physical robots. Later, in 2016, Ho & Ermon used generative adversarial networks to imitate policies when both states and actions are available using a technique called generative adversarial imitation learning (GAIL) (Ho & Ermon, 2016). One imitator network attempts to imitate the policy while another attempts to discriminate between the imitation and provided demonstration data(Goodfellow et al., 2014). Several followup works have improved upon this approach on different aspects (Fu et al., 2018; Song et al., 2018) and recently, there has been efforts to address sample efficiency of this algorithm by proposing approaches for unbiasing rewards and deriving an offpolicy formulation of adversarial imitation learning algorithms (Kostrikov et al., 2019).These approaches however, require access to the demonstrator’s actions. Recently, on the other hand, imitation learning from observation (Torabi et al., 2018a, 2019c) is becoming more popular in which the agent only has access to state demonstrations of the expert. An algorithm of this type is generative adversarial learning from observation (GAIfO) (Torabi et al., 2018b, 2019a, 2019b; Stadie et al., 2017) which uses a GANs like architecture to bring the statetransition distribution of the imitator closer to that of the demonstrator. While this technique has been shown to discover accurate imitation policies, to date, they have only been evaluated in a simulated experimental domain. Because experiments consist of thousands of iterations in which each iteration includes executing a policy several times, the time required for monitoring experiments is prohibitive.
On the other hand, in reinforcement learning–policy learning through environmentprovided reward functions only–direct policy search in a large stateaction space requires numerous samples and often can fall into poor local optima. Guided policy search (GPS) is a method to improve the sample efficiency of direct policy search and guide learning in a large space away from poor local optima (Levine & Koltun, 2013). The basis of GPS is to use trajectory optimization to focus policy learning on highreward actions.
In guided policy search under unknown dynamics, timevarying linear Gaussian models of the dynamics for a small set of specific tasks are first trained to fit a small set of sample data through LQR (Levine & Abbeel, 2014). These Gaussian controllers are then sampled to generate samples to optimize a general policy for a model with thousands of parameters that would typically require much more training data. Specifically, samples in regions of trajectories that have been found to lead to higher reward are generated, guiding the policy learning.
GPS has had success in learning policies in reinforcement learning situations with complex dynamics and highdimensional inputs, including training a policy that directly controls the torque on motors in a robot arm to perform a task like screwing a cap on a bottle solely from raw images of the system (Levine et al., 2016). Current applications of GPS have focused on reinforcement learning, and the technique’s applications to IfO have not been adequately explored.
In this work, our goal is to resolve GAIfO’s sample inefficiency with the help of linear quadratic regulators to the extent that it can be applied to learning a behavior on a real robot.
3 Preliminaries and Overview
In this section, we describe the notation considered throughout the paper, and the two methods that our proposed algorithm are based on, (1) adversarial imitation from observation, and (2) trajectory centric reinforcement learning.
3.1 Notation
We consider agents acting within the broad framework of Markov decision processes (MDPs). We denote a MDP using the 5tuple
, where is the agent’s state space, is its action space,is a function denoting the probability of the agent transitioning from state
to after taking action , is a function specifying the immediate reward that the agent receives for taking a specific action in a given state, and is a discount factor. In this framework, agent behavior can be specified by a policy, , which specifies the action (or distribution over actions) that the agent should use when in a particular state.In reinforcement Learning the goal is to learn a policy, , by maximizing the accumulated reward, , through interaction with the environment. However, imitation learning considers the setting of M\r, i.e. the reward function is excluded. Instead the agent has access to some demonstrated trajectories. The problem that we are interested in this paper is imitation from observation where these demonstrations only include statetrajectories of the expert .
3.2 Adversarial Imitation from Observation
Generative adversarial imitation from observation (Torabi et al., 2018b) is an algorithm of this type in which attempts to learn tasks by bringing the state transition distribution of the imitator closer to that of the demonstrator. The algorithm works as follows. There is an imitator policy network, , that is initialized randomly. This policy is then executed in the environment to generate trajectories where each trajectory is a set of states . There is also a discriminator network parameterized by weights and maps input trajectories to a score between and : , The discriminator is trained in a way to output values close to zero for the data coming from the expert and close to one for the data coming from the imitator. To do so, is updated by taking an iteration towards solving the following optimization problem.
(1) 
From a reinforcement learning point of view, the discriminator network provides a cost function that could change to move the distribution of trajectories created by towards the distribution of the demonstrated trajectories .
Therefore, following the update to , the imitator policy, , is updated using the technique of Trust Region Policy Optimization (Schulman et al., 2015) under the cost function
(2) 
where is the newly updated discriminator network. The whole process is repeated until convergence.
It is a quite wellknown fact that modelfree reinforcement learning algorithms (e.g. TRPO) often require a large number of environment interactions. Therefore, it is not practical to deploy these types of algorithms on physical robots. On the other hand, modelbased RL algorithms have shown promising performance in the real world (Levine et al., 2015, 2016).
3.3 Trajectory Centric Reinforcement Learning
Linear quadratic regulators (LQR’s) learn control policies under two assumptions (Bemporad et al., 2002):

The dynamics of the environment are linear. This means that the transition from a particular state given an action can be represented as the product of the state/action and a matrix
plus a constant vector
: 
The cost is quadratic. The cost is represented by a quadratic term and a linear vector :
The algorithm attempts to solve an optimization problem that returns the actions that have the highest return in the course of an episode. Solving this optimization problem, results in a linear controller:
(3) 
where the s and s are matrices and vectors which are combinations of s, s, s, and s that can be computed for each timestep.
In situations where the dynamics are assumed to be close to linear but are not completely known or are nondeterministic, the linear transition function is often replaced by a conditional probability specified under a normal Gaussian distribution, with a mean of the linear dynamics and a covariance:
When the covariance is constant (independent of the state and action), the optimal policy is identical to the nonstochastic LQR.
In nonlinear systems where the cost is not quadratic, the techniques of LQR can be used by approximating the dynamics with a firstorder Taylor expansion and approximating the cost with a secondorder Taylor expansion:
Iterative linear quadratic regulators (iLQR’s) can be used to find optimal controllers under nonlinear models by running LQR with the approximated dynamics, then updating the dynamics fit on each iteration (Li & Todorov, 2004). The resulting controller is:
Where and are the states and actions around which the Taylor expansion is computed.
LQR assumes that the dynamics of the environment are known. Learning dynamics for a given situation involves building a model to define from a set of observed state/action transitions
. A simple approach to this model building is to use linear regression to estimate the dynamics, finding some matrices
and that model the transition as , or in a stochastic environment, . Modelling dynamics with a Gaussian approximation of the linear regression (often called linear Gaussian models) has the advantage of being very sampleefficient.To avoid the erroneous pursuit of an incorrect global optimal, a set of local models can be used to replace a global model. The most expressive case of local models is a set of models with a single model for every timestep. In the linear regression approach, this amounts to fitting new and for every timestep, often called timevarying controllers. Because dynamics are often highly correlated between timesteps, this approach can be refined by using a global model as a prior for a Bayesian linear regression at each timestep. For a better approximation of the local models it is shown that linearGaussian controllers, , should be used for generating the training data (Levine et al., 2016). The covariance depends on the sensitivity of the total cost to the choice of action.
Because linear regression can overshoot optimals of nonlinear dynamics, policy adjustment can be bounded so that each iteration’s update to the model’s transition distribution (or trajectory distribution) is not too large. This can be achieved with a bound on the Kullback–Leibler (KL) divergence–a relative measure of divergence between distributions–between the previous trajectory distribution and the current trajectory distribution.
4 Proposed Algorithm
In this section, we propose an imitation from observation algorithm, LQR+GAIfO, to learn an imitation policy from state only demonstrations, . Our algorithm takes advantage of the high performance of adversarial imitation from observation algorithms and the sample efficiency of trajectorycentric reinforcement learning algorithms. To do so, we build upon the methods described in Section 3. For LQR to be useful in an imitation learning scenario, it can no longer depend on a prespecified reward function that defines the task. Instead, the trajectory optimization step in LQR should be based on the existing controller’s ability to imitate the expert demonstration. To achieve this capability, we train a discriminator network on each iteration and use an approximate version of its loss on the sampled trajectories to optimize the controllers.
Our algorithm begins by initializing the linear Gaussian controller and executing it inside the environment to collect stateaction trajectories . Then it randomly initializes a timevarying model to model the trajectory dynamics. is specified as . Given a set of stateaction trajectories , , , and
are fit to the sample data at each timestep using Bayesian linear regression with a normalinverseWishart prior. For this prior, it fits the entire trajectory sample to a Gaussian mixture model (GMM), which previous research has found to be effective
(Levine et al., 2016).Following the dynamics model update, a randomly initialized neural network is considered as the discriminator, , which takes statetransitions as input and outputs a value. Similar to Section 3.2, The goal is to train the discriminator to distinguish between the statetransitions coming from the controller and the demonstrator. However, in order to stabilize the learning, our algorithm uses Wasserstein loss (Arjovsky et al., 2017) and takes an iteration on the following optimization problem.
Gradient penalties are also used as the regularization for further stabilization of the learning process (Gulrajani et al., 2017). As discussed in Section 3, the discriminator—a function of statetransition —can be used as the cost function for training the controller. However, LQR requires the cost function to be a quadratic function of states and actions. Therefore, first, the discriminator is combined with the Gaussian dynamics models to create a composite cost function . This composite function is then quadratically approximated by taking the second order Taylor expansions of the cost:
Where and are the Hessian and gradient with respect to the concatenation of and vectors, respectively. Finally, an iteration of LQR uses this cost approximation to optimize the trajectory to form a new linearGaussian controller. The step size of this update is bounded by the KLDivergence compared to the previous iteration. The main components of this approach are depicted in Figure 1.
5 Experiments
To evaluate the performance of our algorithm, we studied its ability to imitate a reaching task on a robot arm–both on a physical arm and in a simulator.
5.1 Setup
For a testing platform, we used a Universal Robotics UR5, a 6degreeoffreedom robotic arm (Figure 2). The task that is demonstrated is a reaching task in which the arm begins in a consistent, retracted position and reaches towards a point in Cartesian space. When the end effector (the gripper at the end of the arm) reaches this point, the arm stops moving. This task is shown in Figure 4. The expert is trained by iterating between iLQR and dynamics learning with a specified reward function until convergence. This policy is then executed and recorded a number of times to create the demonstration data.
We modified the software to record the state of the arm and the action chosen at every timestep of the trajectory execution. For the initial experiments, the state consisted of:

Joint angles (3 dimensional)

Joint velocities (3 dimensional)

Cartesian distance to the goal position from the end effector (3 dimensional)

Cartesian velocity of the end effector (3 dimensional)
For testing in simulation, we used the Gazebo simulation environment (Figure 3) with a model of the UR5. Each trial lasts for 100 timesteps (10 seconds) and ends regardless of the end effector reaching the goal state. At each iteration, the policy being evaluated is executed five times to collect five sample trajectories. The policy is also evaluated once without noise per iteration, and the performance according to the cost function is logged.
The cost function used takes into account the distance from the end effector to the target position, weighted linearly as the trial progresses. With the distance from the goal position to the end effector at a given timestep , the cost of a trajectory with timesteps is calculated as:
The same cost function is used to train the expert through reinforcement learning as well as to evaluate the performance of the imitator. In this sense, the task of imitation learning can be seen as recovering the cost function that guided the expert (Torabi et al., 2018b). For a more complex task or more specific cost function than the one studied, it’s possible that the imitator could recover the task behavior correctly while not performing well in the eyes of the cost function, or vice versa. However, for the arm reaching task, the cost function is simple and directly related to the task, making it appropriate as an evaluator of imitation performance. For the imitation tasks, this cost function was used to evaluate each trajectory sample at a given iteration. The results were normalized on a range from zero to one, with zero mapping to the average cost of a random policy, and one mapping to the cost achieved by the expert. A policy that performs as well as the expert would achieve a score of one on this normalized performance scale.
We compare our algorithm with GAIfO which is instrumented to interface with the arm control and simulation platform. Trials for the GAIfO also involved taking five samples per iteration, in the same way as ours. The GAIfO policy network was updated using Proximal Policy Optimization (PPO).
5.2 Experimental Design
We conducted three main experiments to evaluate our algorithm. In the first experiment, the learning rate is compared to learning under GAIfO. In the second experiment, we test our algorithm’s ability to generalize to unseen target positions. Finally, we compare the performance of the algorithm in the simulated environment to the physical arm.
5.2.1 Comparison to GAIfO
To compare the learning rate of our algorithm to that of GAIfO, we ran trials for both algorithms for 100 iterations and tracked the policy’s performance at each iteration using the cost function described in Section 5.1. This process was repeated for both algorithms (n=30 for ours, n=55 for GAIfO) to collect average performance data. The algorithms’ performance along with the mean standard error is plotted in Figure 5. The performance of our algorithm quickly exceeds GAIfO and peaks around iteration 30.
5.2.2 Generalization
To test our algorithm’s ability to generalize a policy for a point that is not in the expert demonstration data, we collected expert demonstration trajectories for 8 points on the edge of a square (shown in Figure 7). For each point, we trained the expert and recorded five sample trajectories when the expert converged. Then, after choosing a subset of the points on the square as , we tasked the arm with moving to a point in the center of the square. Because the center point was not in , the control policy was required to generalize the expert trajectories to this unseen point. We varied the number of points included in , and tracked the normalized performance of our algorithm over 15 iterations. As shown in Figure 6, while performance was similar in the early iterations, our algorithm generally performed better in later iterations when more points were included in .
5.2.3 Performance on Physical Arm
Our algorithm was run on both the simulator and the physical arm to examine how closely simulated performance mapped to realworld performance. Over 25 iterations, the policy performance on the physical arm began to surpass the performance of the simulated arm, as shown in Figure 8.
6 Discussion
Our research began by asking if a combination of LQR and GAIfO could increase sample efficiency in imitation learning. The comparison of LQR+GAIfO to GAIfO suggests that LQR+GAIfO can indeed produce a policy that is better at imitating a behavior in a limited number of iterations, confirming our hypothesis. The steep initial learning curve of LQR+GAIfO indicates significantly higher sample efficiency compared to GAIfO alone. However, the performance of LQR+GAIfO seems to degrade around iteration 60. Without this performance degradation, LQR+GAIfO would outperform GAIfO past iteration 100. The reason for this degradation may be that in adversarial algorithms, improvement of the generator and the discriminator should occur at relatively similar rate. However, in our algorithm, since the controller’s representation complexity is limited, after some number of iteration, the controller does not improve as fast as the discriminator. In addition, even without this degradation, the GAIfO approach would eventually surpass the performance of LQR+GAIfO, likely due to the ability of the generator network in GAIfO to produce more complex policies than those that can be represented with linear Gaussian controllers in LQR.
Although most of the ability for a policy to perform a task that is different from the expert trajectories in GAIfO and GPS result from a complex model considered for the policy (neural network), the linear Gaussian controllers in LQR+GAIfO still have the ability to generalize to some degree. As expected, the ability to successfully generalize increases with demonstration trajectories, as shown in Figure 6. The reason may be that the discriminator learns a general cost function that could be applied to new target points and as a result LQR can learn a relatively good controller. Future work integrating the full GPS approach would likely lead to better generalization.
We studied the performance of LQR+GAIfO on the physical arm to validate the tractability of this technique on a real robot and to establish a sense of how directly the performance studied in the simulator would translate to the physical arm. Our results, as seen in Figure 8, show that the policy performance seen in the simulator can be trusted to model policy performance on the real arm. Surprisingly, the performance of LQR+GAIfO on the physical arm exceeds the simulator performance. It is possible that the noise introduced by the physical arm as a result of actuator noise or other physical effects lead to wider exploration and faster policy improvement. If this is the case, it could be possible to achieve similar performance in the simulator by introducing more policy noise.
7 Conclusion and Future Work
We have found that combining generative adversarial imitation from observation with Linear Quadratic Regulators leads to faster learning of imitation behavior over fewer samples than with GAIfO alone, confirming our hypothesis. While LQR+GAIfO doesn’t reach the absolute imitation performance of GAIfO over an extended training period with thousands of samples, achieving adequate imitation performance with limited samples opens the door to imitation research on physical robotic systems, for which imitation learning has posed logistical challenges in the past.
While LQR is a powerful technique by itself, a policy based solely on Gaussian controllers has limits in complexity. Work in GPS has already produced a method for combining sampleefficient Gaussian controllers with a deep network model that is trained through the controllers. Using a deep network as part of the policy offers increased performance in the long run and greatly increased generalization ability. Incorporating this deep network policy driven by importanceweighted samples of the linear Gaussian controllers is an obvious and promising next step for this work.
To validate the LQR+GAIfO technique, we represented the expert trajectories using lowlevel data like the Cartesian position of the arm’s end effector. GAIfO has had success in using higher level data–like a visual recording of the demonstrator–as the state in trajectories. Additionally, GPS has been used in learning neural network policies from visual observation (Levine et al., 2016). Pursuing imitation learning from visual data alone would greatly widen the situations in which demonstration data could be collected. Adding a convolutional layer to the discriminator so that it can accept visual data is a natural next step for extending this research.
Acknowledgements
This work has taken place in the Learning Agents Research Group (LARG) at the Artificial Intelligence Laboratory, The University of Texas at Austin. LARG research is supported in part by grants from the National Science Foundation (IIS1637736, IIS1651089, IIS1724157), the Office of Naval Research (N00014182243), Future of Life Institute (RFP2000), Army Research Lab, DARPA, Intel, Raytheon, and Lockheed Martin. Peter Stone serves on the Board of Directors of Cogitai, Inc. The terms of this arrangement have been reviewed and approved by the University of Texas at Austin in accordance with its policy on objectivity in research.
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