rad_openaigym
RAD: Reinforcement Learning with Augmented Data (code for state augmentation)
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Modelbased reinforcement learning (MBRL) is widely seen as having the potential to be significantly more sample efficient than modelfree RL. However, research in modelbased RL has not been very standardized. It is fairly common for authors to experiment with selfdesigned environments, and there are several separate lines of research, which are sometimes closedsourced or not reproducible. Accordingly, it is an open question how these various existing MBRL algorithms perform relative to each other. To facilitate research in MBRL, in this paper we gather a wide collection of MBRL algorithms and propose over 18 benchmarking environments specially designed for MBRL. We benchmark these algorithms with unified problem settings, including noisy environments. Beyond cataloguing performance, we explore and unify the underlying algorithmic differences across MBRL algorithms. We characterize three key research challenges for future MBRL research: the dynamics bottleneck, the planning horizon dilemma, and the earlytermination dilemma. Finally, to maximally facilitate future research on MBRL, we opensource our benchmark in http://www.cs.toronto.edu/ tingwuwang/mbrl.html.
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Reinforcement learning (RL) algorithms are most commonly classified in two categories: modelfree RL (MFRL), which directly learns a value function or a policy by interacting with the environment, and modelbased RL (MBRL), which uses interactions with the environment to learn a model of it. While modelfree algorithms have achieved success in areas including robotics
lillicrap2015continuous ; ppo ; deepmindppo ; andrychowicz2018learning , videogames dqn ; mnih2016asynchronous , and motion animation peng2018deepmimic , their high sample complexity limits largely their application to simulated domains. By learning a model of the environment, modelbased methods learn with significantly lower sample complexity. However, learning an accurate model of the environment has proven to be a challenging problem in certain domains. Modelling errors cripple the effectiveness of these algorithms, resulting in policies that exploit the deficiencies of the models, which is known as modelbias deisenroth2011pilco . Recent approaches have been able to alleviate the modelbias problem by characterizing the uncertainty of the learned models by the means of probabilistic models and ensembles. This has enabled modelbased methods to match modelfree asymptotic performance in challenging domains while using much fewer samples kurutach2018model ; chua2018deep ; clavera2018mbmpo .These recent advances have led to a great deal of excitement in the field of modelbased reinforcement learning. Despite the impressive results achieved, how these methods compare against each other and against standard baselines remains unclear. Reproducibility and lack of opensource code are persistent problems in RL henderson2018deep ; islam2017reproducibility , which makes it difficult to compare novel algorithms against prior lines of research. In MBRL, this problem is exacerbated by the modifications made to the environments: preprocessing of the observations, modification of the reward functions, or using different episode horizons. Such lack of standardized implementations and environments in MBRL makes it difficult to quantify scientific progress.
Systematic evaluation and comparison will not only further our understanding of the strengths and weaknesses of the existing algorithms, but also reveal their limitations and suggest directions for future research. Benchmarks have played a crucial role in other fields of research. For instance, modelfree RL has benefited greatly from the introduction of benchmarking code bases and environments such as rllab duan2016benchmarking , OpenAI Gym gym , and Deepmind Control Suite tassa2018deepmind
; where the latter two have been the de facto benchmarking platforms. Besides RL, benchmarking platforms have also accelerated areas such as computer vision
deng2009imagenet ; lin2014microsoft , machine translation koehn2007moses and speech recognition panayotov2015librispeech .In this paper, we benchmark 11 MBRL algorithms and 4 MFRL algorithms across 18 environments based on the standard OpenAI Gym gym . The environments, designed to hold the common assumptions in modelbased methods, range from simple 2D tasks, such as CartPole, to complex domains that are usually not evaluated on, such as Humanoid. The benchmark is further extended by characterizing the robustness of the different methods when stochasticity in the observations and actions is introduced. Based on the empirical evaluation, we propose three main causes that stagnate the performance of modelbased methods: 1) Dynamics bottleneck: algorithms with learned dynamics are stuck at performance local minima significantly worse than using groundtruth dynamics, i.e. the performance does not increase when more data is collected. 2) Planning horizon dilemma
: while increasing the planning horizon provides more accurate reward estimation, it can result in performance drops due to the curse of dimensionality and modelling errors. 3)
Early termination dilemma: early termination is commonly used in MFRL for more directed exploration, to achieve faster learning. However, similar performance gain are not yet observed in MBRL algorithms, which limits their effectiveness in complex environments.We formulate all of our tasks as a discretetime finitehorizon Markov decision process (MDP), which is defined by the tuple
. Here, denotes the state space, denotes the action space, is transition dynamics density function, defines the reward function, is the initial state distribution, is the discount factor, and is the horizon of the problem. Contrary to standard modelfree RL, we assume access to an analytic differentiable reward function. The aim of RL is to learn an optimal policy that maximizes the expected total reward .Dynamics Learning: MBRL algorithms are characterized by learning a model of the environment. After repeated interactions with the environment, the experienced transitions are stored in a dataset which is then used to learn a dynamics function . In the case where groundtruth dynamics are deterministic, the learned dynamics function
predicts the next state. In stochastic settings, it is common to represent the dynamics with a Gaussian distribution, i.e.,
and the learned dynamics model corresponds to .In this section, we introduce the benchmarked MBRL algorithms, which are divided into: 1) Dynastyle Algorithms, 2) Policy Search with Backpropagation through Time, and 3) Shooting Algorithms.
In the Dyna algorithm sutton1990integrated ; sutton1991dyna ; sutton1991planning , training iterates between two steps. First, using the current policy, data is gathered from interaction with the environment and then used to learn the dynamics model. Second, the policy is improved with imagined data generated by the learned model. This class of algorithms learn policies using modelfree algorithms with rich imaginary experience without interaction with the real environment.
ModelEnsemble TrustRegion Policy Optimization (METRPO) kurutach2018model
: Instead of using a single model, METRPO uses an ensemble of neural networks to model the dynamics, which effectively combats modelbias. The ensemble
is trained using standard squared L2 loss. In the policy improvement step, the policy is updated using TrustRegion Policy Optimization (TRPO) trpo , on experience generated by the learned dynamics models.Stochastic Lower Bound Optimization (SLBO) luo2018algorithmic : SLBO is a variant of METRPO with theoretical guarantees of monotonic improvement. In practice, instead of using singlestep squared L2 loss, SLBO uses a multistep L2norm loss to train the dynamics.
ModelBased MetaPolicyOptimzation (MBMPO) clavera2018mbmpo : MBMPO forgoes the reliance on accurate models by metalearning a policy that is able to adapt to different dynamics. Similar to METRPO, MBMPO learns an ensemble of neural networks. However, each model in the ensemble is considered as a different task to metatrain maml2017finn on. MBMPO metatrains a policy that quickly adapts to any of the different dynamics of the ensemble, which is more robust against modelbias.
Contrary to Dynastyle algorithms, where the learned dynamics models are used to provide imagined data, policy search with backpropagation through time exploits the model derivatives. Consequently, these algorithms are able to compute the analytic gradient of the RL objective with respect to the policy, and improve the policy accordingly.
Probabilistic Inference for Learning Control (PILCO) deisenroth2011pilco ; deisenroth2015gaussian ; kamthe2017data : In PILCO, Gaussian processes (GPs) are used to model the dynamics of the environment. The dynamics model is a probabilistic and nonparametric function of the collected data . The policy is trained to maximize the RL objective by computing the analytic derivatives of the objective with respect to the policy parameters . The training process iterates between collecting data using the current policy and improving the policy. Inference in GPs does not scale in high dimensional environments, limiting its application to simpler domains.
Iterative Linear QuadraticGaussian (iLQG) tassa2012synthesis : In iLQG, the groundtruth dynamics are assumed to be known by the agent. The algorithm uses a quadratic approximation on the RL reward function and a linear approximation on the dynamics, converting the problem solvable by linearquadratic regulator (LQR) bemporad2002explicit . By using dynamic programming, the optimal controller for the approximated problem is a linear timevarying controller. iLQG is an model predictive control (MPC) algorithm, where replanning is performed at each timestep.
Guided Policy Search (GPS) levine2014learning ; levine2015learning ; zhang2016learning ; finn2016guided ; montgomery2016guided ; chebotar2017path : Guided policy search essentially distills the iLQG controllers into a neural network policy by behavioural cloning, which minimizes . The dynamics are modelled to be Gaussianlinear timevarying. To prevent overconfident policy improvement that deviates from the last realworld trajectory, the reward function is augmented as , where is the passive dynamics distribution from last trajectories. In this paper, we use the MDGPS variant montgomery2016guided .
Stochastic Value Gradients (SVG) heess2015learning : SVG tackles the problem of compounding model errors by using observations from the real environment, instead of the imagined one. To accommodate mismatch between model predictions and real transitions, the dynamics models in SVG are probabilistic. The policy is improved by computing the analytic gradient of the real trajectories with respect to the policy. Reparametrization trick is used to permit backpropagation through the stochastic sampling.
This class of algorithms provide a way to approximately solve the receding horizon problem posed in model predictive control (MPC) when dealing with nonlinear dynamics and nonconvex reward functions. Their popularity has increased with the use of neural networks for modelling dynamics.
Random Shooting (RS) richards2005robust ; rao2009survey : RS optimizes the action sequence to maximize the expected planning reward under the learned dynamics model, i.e., . In particular, the agent generates
candidate random sequences of actions from a uniform distribution, and evaluates each candidate using the learned dynamics. The optimal action sequence is approximated as the one with the highest return. A RS agent only applies the first action from the optimal sequence and replans at every timestep.
ModeFree ModelBased (MBMF) nagabandi2017neural : Generally, random shooting has worse asymptotic performance when compared with modelfree algorithms. In MBMF, the authors first train a RS controller , and then distill the controller into a neural network policy using DAgger ross2011reduction , which minimizes . After the policy distillation step, the policy is finetuned using standard modelfree algorithms. In particular the authors use TRPO trpo .
Probabilistic Ensembles with Trajectory Sampling (PETSRS and PETSCEM) chua2018deep : In the PETS algorithm, the dynamics are modelled by an ensemble of probabilistic neural networks models, which captures both epistemic uncertainty from limited data and network capacity, and aleatoric uncertainty from the stochasticity of the groundtruth dynamics. Except for the difference in modeling the dynamics, PETSRS is the same as RS. Instead, in PETSCEM, the online optimization problem is solved using crossentropy method (CEM) de2005tutorial ; botev2013cross to obtain a better solution.
In our benchmark, we include MFRL baselines to quantify the sample complexity and asymptotic performance gap between MFRL and MBRL. Specifically, we compare against representative MFRL algorithms including TrustRegion Policy Optimization (TRPO) trpo , ProximalPolicy Optimization (PPO) ppo ; deepmindppo , Twin Delayed Deep Deterministic Policy Gradient (TD3) fujimoto2018addressing , and Soft ActorCritic (SAC) haarnoja2018soft . The former two are stateoftheart onpolicy MFRL algorithms, and the latter two are considered the stateoftheart offpolicy MFRL algorithms.
In this section, we present the results of our benchmarking and examine the causes that stagnate the performance of MBRL methods. Specifically, we designed the benchmark to answer the following questions: 1) How do existing MBRL approaches compare against each other and against MFRL methods across environments with different complexity (Section 4.3)? 2) Are MBRL algorithms robust against observation and action noise (Section 4.4)? and 3) What are the main bottlenecks in the MBRL methods?
Aiming to answer the last question, we present three phenomena inherent of MBRL methods, which we refer to as dynamics bottleneck (Section 4.5), planning horizon dilemma (Section 4.6), and early termination dilemma (Section 4.7).
Our benchmark consists of 18 environments with continuous state and action space based on OpenAI Gym gym . We include a full spectrum of environments with different difficulty and episode length, from CartPole to Humanoid. More specifically, we have the following modifications:
[leftmargin=*]
To accommodate traditional MBRL algorithms such as iLQG and GPS, we modify the reward function so that the gradient with respect to observation always exists or can be approximated.
We note that early termination has not been applied in MBRL, and we specifically have both the raw environments and the variants with early termination, indicated by the suffix ET.
The original Swimmerv0 in OpenAI Gym was unsolvable for all algorithms. Therefore, we modified the position of the velocity sensor so that it’s easier to solve. We name this easier version as Swimmer while still keep the original one as a reference, named as Swimmerv0.
For a detailed description of the environments and the reward functions used, we refer readers to Appendix A.
Performance Metric and Hyperparameter Search: Each algorithm is run with 4 random seeds. In the learning curves, the performance is averaged with a sliding window of 5 algorithm iterations. The error bars were plotted by the default Seaborn seaborn
smoothing scheme from the mean and standard deviation of the results. Similarly, in the tables, we show the performance averaged across different random seeds with a window size of 5000 timesteps. We perform a grid search for each algorithm separately, which is summarized in appendix
B. For each algorithm, We show the results using the hyperparameters producing the best average performance.Training Time: In MFRL, 1 million timestep training is common, but for many environments, MBRL algorithms converge much earlier than 200k timesteps and it takes an impractically long time to train for 1 million timesteps for some of the MBRL algorithms. We therefore show both the performance of 200k timestep training for all algorithms and show the performance of 1M timestep training for algorithms where computation is not a major bottleneck.
Pendulum  InvertedPendulum  Acrobot  CartPole  Mountain Car  Reacher  
Random  202.6 249.3  205.1 13.6  374.5 17.1  38.4 32.5  105.1 1.8  45.7 4.8 
ILQG  160.8 29.8  0.0 0.0  195.5 28.7  199.3 0.6  55.9 8.3  6.0 2.6 
GTCEM  170.5 35.2  0.2 0.1  13.9 40.5  199.9 0.1  58.0 2.9  3.6 1.2 
GTRS  171.5 31.8  0.0 0.0  2.5 39.4  200.0 0.0  68.5 2.2  25.7 3.5 
RS  164.4 9.1  0.0 0.0  4.9 5.4  200.0 0.0  71.3 0.5  27.1 0.6 
MBMF  157.5 13.2  182.3 24.4  92.5 15.8  199.7 1.2  4.2 18.5  15.1 1.7 
PETSCEM  167.4 53.0  20.5 28.9  12.5 29.0  199.5 3.0  57.9 3.6  12.3 5.2 
PETSRS  167.9 35.8  12.1 25.1  71.5 44.6  195.0 28.0  78.5 2.1  40.1 6.9 
METRPO  177.3 1.9  126.2 86.6  68.1 6.7  160.1 69.1  42.5 26.6  13.4 0.2 
GPS  162.7 7.6  74.6 97.8  193.3 11.7  14.4 18.6  10.6 32.1  19.8 0.9 
PILCO  132.6 410.1  194.5 0.8  394.4 1.4  1.9 155.9  59.0 4.6  13.2 5.9 
SVG  141.4 62.4  183.1 9.0  79.7 6.6  82.1 31.9  27.6 32.6  11.0 1.0 
MBMPO  171.2 26.9  0.0 0.0  87.8 12.9  199.3 2.3  30.6 34.8  5.6 0.8 
SLBO  173.5 2.5  240.4 7.2  75.6 8.8  78.0 166.6  44.1 6.8  4.1 0.1 
PPO  163.4 8.0  40.8 21.0  95.3 8.9  86.5 7.8  21.7 13.1  17.2 0.9 
TRPO  166.7 7.3  27.6 15.8  147.5 12.3  47.3 15.7  37.2 16.4  10.1 0.6 
TD3  161.4 14.4  224.5 0.4  64.3 6.9  196.0 3.1  60.0 1.2  14.0 0.9 
SAC  168.2 9.5  0.2 0.1  52.9 2.0  199.4 0.4  52.6 0.6  6.4 0.5 
HalfCheetah  Swimmerv0  Swimmer  Ant  AntET  Walker2D  
Random  288.3 65.8  1.2 11.2  9.5 11.6  473.8 40.8  124.6 145.0  2456.9 345.3 
iLQG  2142.6 137.7  47.8 2.4  306.7 0.8  9739.8 745.0  1506.2 459.4  1186.2 126.3 
GTCEM  14777.2 13964.2  111.0 4.6  335.9 1.1  12115.3 209.7  226.0 178.6  7719.7 486.7 
GTRS  815.7 38.5  35.8 3.0  42.2 5.3  2709.1 631.1  2519.0 469.8  1641.4 137.6 
RS  421.0 55.2  31.1 2.0  92.8 8.1  535.5 37.0  239.9 81.7  2060.3 228.0 
MBMF  126.9 72.7  51.8 30.9  284.9 25.1  134.2 50.4  85.7 27.7  2218.1 437.7 
PETSCEM  2795.3 879.9  22.1 25.2  306.3 37.3  1165.5 226.9  81.6 145.8  260.2 536.9 
PETSRS  966.9 471.6  42.1 20.2  170.1 8.1  1852.1 141.0  130.0 148.1  312.5 493.4 
METRPO  2283.7 900.4  30.1 9.7  336.3 15.8  282.2 18.0  42.6 21.1  1609.3 657.5 
GPS  52.3 41.7  14.5 5.6  35.3 8.4  445.5 212.9  275.4 309.1  1730.8 441.7 
PILCO  41.9 267.0  13.8 16.1  18.7 10.3  770.7 153.0  N. A.  2693.8 484.4 
SVG  336.6 387.6  77.2 99.0  75.2 85.3  377.9 33.6  185.0 141.6  1430.9 230.1 
MBMPO  3639.0 1185.8  85.0 98.9  268.5 125.4  705.8 147.2  30.3 22.3  1545.9 216.5 
SLBO  1097.7 166.4  41.6 18.4  125.2 93.2  718.1 123.3  200.0 40.1  1277.7 427.5 
PPO  17.2 84.4  38.0 1.5  306.8 4.2  321.0 51.2  80.1 17.3  1893.6 234.1 
TRPO  12.0 85.5  37.9 2.0  215.7 10.4  323.3 24.9  116.8 47.3  2286.3 373.3 
TD3  3614.3 82.1  40.4 8.3  331.1 0.9  956.1 66.9  259.7 1.0  73.8 769.0 
SAC  4000.7 202.1  41.2 4.6  309.8 4.2  506.7 165.2  2012.7 571.3  415.9 588.1 
Walker2DET  Hopper  HopperET  SlimHumanoid  SlimHumanoidET  HumanoidET  
Random  2.8 4.3  2572.7 631.3  12.7 7.8  1172.9 757.0  41.8 47.3  50.5 57.1 
iLQG  229.0 74.7  1157.6 224.7  83.4 21.7  13225.2 1344.9  520.0 240.9  255.0 94.6 
GTCEM  254.8 233.4  3232.3 192.3  256.8 16.3  45979.8 1654.9  1242.7 676.0  1236.2 668.0 
GTRS  207.9 27.2  2467.2 55.4  209.5 46.8  8074.4 441.1  361.5 103.8  312.9 167.8 
RS  201.1 10.5  2491.5 35.1  247.1 6.1  99.2 388.5  332.8 13.4  295.5 10.9 
MBMF  350.0 107.6  1047.4 1098.7  926.9 154.1  1320.2 735.3  809.7 57.5  776.8 62.9 
PETSCEM  2.5 6.8  1125.0 679.6  129.3 36.0  1472.4 738.3  355.1 157.1  110.8 91.0 
PETSRS  0.8 3.2  1469.8 224.1  205.8 36.5  2055.1 771.5  320.7 182.2  106.9 102.6 
METRPO  9.5 4.6  1272.5 500.9  4.9 4.0  154.9 534.3  76.1 8.8  72.9 8.9 
GPS  2400.6 610.8  768.5 200.9  2303.9 338.1  592.6 214.1  N. A.  N. A. 
PILCO  N. A.  1729.9 1611.1  N. A.  N. A.  N. A.  N. A. 
SVG  252.4 48.4  877.9 427.9  435.2 163.8  1096.8 791.0  1084.3 77.0  811.8 241.5 
MBMPO  10.3 1.4  333.2 1189.7  8.3 3.6  674.4 982.2  115.5 31.9  73.1 23.1 
SLBO  207.8 108.7  741.7 734.1  805.7 142.4  588.9 332.1  776.1 252.5  1377.0 150.4 
PPO  306.1 17.2  103.8 1028.0  758.0 62.0  1466.7 278.5  454.3 36.7  451.4 39.1 
TRPO  229.5 27.1  2100.1 640.6  237.4 33.5  1140.9 241.8  281.3 10.9  289.8 5.2 
TD3  3299.7 1951.5  2245.3 232.4  1057.1 29.5  1319.1 1246.1  1070.0 168.3  147.7 0.7 
SAC  2216.4 678.7  726.4 675.5  1815.5 655.1  1328.4 468.2  843.6 313.1  1794.4 458.3 
In Table 1, we summarize the performance of each algorithm trained with 200,000 timesteps. We also include some representative performance curves in Figure 1. The learning curves for all the environments can be seen in appendix C. The engineering statistics shown in Table 2 include the computational resources, the estimated wallclock time, and whether the algorithm is fast enough to run at realtime at test time, namely, if the action selection can be done faster than the default timestep of the environment. In Table 5, we summarize the performance ranking.
Reacher 2D  HalfCheetah  Ant  HumanoidET  SlimHumanoidET  SlimHumanoid  Realtime testing  Resources  
RS  9.23  8.83  8.2  13.9  9.5  11.73  ✗  20 CPUs 
MBMF  4.03  4.05  5.25  5.05  3.3  4.8  ✓  20 CPUs 
PETS  4.64  15.3  6.5  7.03  4.76  6.6  ✗  4CPUs + 1GPU 
PETSRS  2.68  6.76  5.01  5.1  3.35  5.06  ✗  4CPUs + 1GPU 
METRPO  4.76  5.23  3.46  5.68  2.58  2.36  ✓  4CPUs + 1GPU 
GPS  1.1  3.3  5.1  N. A.  N. A.  17.24  ✓  5 CPUs 
PILCO  120  N. A.  N. A.  N. A.  N. A.  N. A.  ✓  4CPUs + 1GPU 
SVG  1.61  1.41  1.49  1.92  1.06  1.05  ✓  2CPUs 
MBMPO  30.9  17.38  55.2  41.4  41.5  41.6  ✓  8CPUs 
SLBO  2.38  4.96  5.46  5.5  6.86  6.8  ✓  10CPUs 
PPO  0.02  0.04  0.07  0.05  0.034  0.04  ✓  5 CPUs 
TRPO  0.02  0.02  0.05  0.043  0.031  0.034  ✓  5 CPUs 
TD3  2.9  4.3  3.6  5.37  3.13  3.95  ✓  12 CPUs 
SAC  2.38  2.21  3.15  3.35  4.05  3.15  ✓  12 CPUs 
Shooting Algorithms: RS is very effective on simple tasks such as InvertedPendulum, CartPole and Acrobot, but as task difficulty increases RS gradually gets surpassed by PETSRS and PETSCEM, which indicates that modelling uncertainty aware dynamics is crucial for the performance of shooting algorithms. At the same time, PETSCEM is better than PETSRS in most of the environments, showing the importance of an effective planning module. However, PETSCEM search is not as effective as PETSRS in Ant, Walker2D and SlimHumanoid, indicating that we need more expressive and general planning module for more complex environments. MBMF does not have obvious gains compared to other shooting algorithms, but like other modelfree controllers, MBMF can jump out of performance localminima in MountainCar. Shooting algorithms are effective and robust across different environments.
DynaStyle Algorithms: MBMPO surpasses the performance of METRPO in most of the environments and achieves the best performance in domains like HalfCheetah. Both algorithms seems to perform the best when the horizon is short. SLBO can solve MountainCar and Reacher very efficiently, but more interestingly in complex environment it achieves better performance than METRPO and MBMPO, except for in SlimHumanoid. This category of algorithms is not efficient to solve long horizon complex domains due to the compounding error effect.
SVG: For the majority of the tasks, SVG does not have the best sample efficiency. But for Humanoid environments, SVG is very effective compared with other MBRL algorithms. Complex environments exacerbate compounding errors; SVG which uses real observations and a value function to look into future returns, is able to surpass other MBRL algorithms in these highdimensional domains.
PILCO: In our benchmarking, PILCO can be applied to the simple environments, but fails to solve most of the other environments with bigger episode length and observation size. PILCO is unstable across different random seeds and timeconsuming to train.
GPS: GPS has the best performance in AntET, but cannot match the best algorithms in other environments. In the original GPS, the environment is usually 100 timestep long, while most of our environments are 200 or 1000 timestep. Also GPS assumes several separate constant initial states, while our environments sample the initial state from a distribution. The deviation of trajectories between iterations can be the reason of GPS’s performance drop.
MF baselines: SAC and TD3 are two very powerful baselines with very stable performance across different environments. In general modelfree and modelbased methods are two almost evenly matched rivals when trained for 200,000 timesteps.
MB with Groundtruth Dynamics: Algorithms with groundtruth dynamics can solve the majority of the tasks, except for some of the tasks such as MountainCar. With the increasing complexity of the environments, shooting methods gradually have much better performance than the policy search methods such as iLQG, whose linear quadratic assumption is not a good approximation anymore. Early termination cause a lot of troubles for modelbased algorithms, both with and without groundtruth dynamics, which is further studied in section 4.7.
HalfCheetah  Original Performance  Change /  Change /  Change /  Change / 
iLQG  2142.6  2167.9  1955.4  1881.4  1832.5 
GTPETS  14777.2  13138.7  5550.7  3292.7  1616.6 
RS  421  274.8  +2.1  +24.8  +21.3 
PETS  2795.3  915.8  385  367.8  368.1 
METRPO  2283.7  1874.3  886.8  963.9  160.8 
SVG  336.6  336.5  95.8  173.1  314.7 
MBMPO  3639.0  1282.6  3.5  266.1  +79.7 
SLBO  1097.7  885.2  +147.1  +495.5  366.6 
In this section, we study the robustness of each algorithm with respect to the noise added to the observation and actions. Specifically, we added Gaussian white noise to the observations and actions with standard deviation
and , respectively. In Table 3 we show the results for the HalfCheetah environment, for the full results we refer the reader to appendix D.As expected, adding noise is in general detrimental to the performance of the MBRL algorithms. METRPO and SLBO are more likely to suffer from a catastrophic performance drop when compared to shooting methods such as PETS and RS, suggesting that replanning successfully compensates for the uncertainty. On the other hand, the Dynastyle method MBMPO presents to be very robust against noise. Due to the limited exploration in baseline, the performance is sometimes increased after adding noise that encourages exploration.
We further run MBRL algorithms for 1M timesteps on HalfCheetah, Walker2D, Hopper, and Ant environments to capture the asymptotic performance, as are shown in Table 4 and Figure 2.
The results show that MBRL algorithms plateau at a performance level well below their modelfree counterparts and themselves with groundtruth dynamics. This points out that when learning models, more data does not result in better performance. For instance, PETS’s performance plateaus after 400k timesteps at a value much lower than the performance when using the groundtruth dynamics.
The following assumptions can potentially explain the dynamics bottleneck. 1) The prediction error accumulates with time, and MBRL inevitably involves prediction on unseen states. While techniques such as probabilistic ensemble were proposed to capture uncertainty, it can be seen empirically in our paper as well as in chua2018deep , that prediction becomes unstable and inaccurate with time. 2) The policy and the learning of dynamics is coupled, which makes the agents more prone to performance localminima. While exploration and offpolicy learning have been studied in NIPS2016_6383 ; Dearden:1999:MBB:2073796.2073814 ; Wiering98efficientmodelbased ; NIPS2016_6591 ; schaul2019ray ; fujimoto2018addressing , it has been barely addressed on current modelbased approaches.
GTCEM  PETSCEM  METRPO  MBMPO  SLBO  TD3  SAC  
HalfCheetah  14777.2 13964.2  2875.9 1132.2  2672.7 1481.6  4513.1 1045.4  2041.4 932.7  5072.9 815.8  6095.5 936.1 
Walker2D  7719.7 486.7  1931.7 667.3  2947.1 640.0  1793.7 80.6  1371.7 2761.7  3293.6 644.4  3941.0 985.3 
Hopper  3232.3 192.3  288.4 988.2  948.0 854.3  495.2 265.0  2963.1 323.4  2745.7 546.7  3020.3 134.6 
Ant  12115.3 209.7  1675.0 108.6  262.7 36.5  810.8 240.6  513.6 182.0  3073.8 773.8  2989.9 1182.8 
One of the critical choices in shooting methods is the planning horizon. In Figure 3, we show the performance of iLQG, CEM and RS, using the same number of candidate planning sequences, but with different planning horizon. We notice that increasing the planning horizon does not necessarily increase the performance, and more often instead decreases the performance. A planning horizon between 20 to 40 works the best both for the models using groundtruth dynamics and the ones using learned dynamics. We argue that this is result of insufficient planning in a search space which increases exponentially with planning depth, i. e., the curse of dimensionality. However, in more complex environments such as the ones with early terminations, short planning horizon can lead to catastrophic performance drop, which we discuss in appendix G. We further experiment with the imaginary environment length in Dyna algorithms. We have similar results that increasing horizon does not necessarily help the performance, which is summarized in appendix F.
Early termination, when the episode is finalized before the horizon has been reached, is a standard technique used in MFRL algorithms to prevent the agent from visiting unpromising states or damaging states for real robots. When early termination is applied to the real environments, MBRL can correspondingly also apply early termination in the planned trajectories, or generate early terminated imaginary data. However, we find this technique hard to integrate into the existing MB algorithms. The results, shown in Table 1, indicates that early termination does in fact decrease the performance for MBRL algorithms of different types. We further experiment with addition schemes to incorporate early termination, summarized in appendix G. However none of them were successful. We argue that to perform efficient learning in complex environments, such as Humanoid, early termination is almost necessary. We leave it as an important request for research.
RS  MBMF  PETSCEM  PETSRS  METRPO  GPS  PILCO  SVG  MBMPO  SLBO  
Mean rank  5.2 / 10  5.5 / 10  4.0 / 10  4.8 / 10  5.7 / 10  7.7 / 10  9.5 / 10  4.8 / 10  4.7 / 10  4.0 / 10 
Median rank  5.5 / 10  7 / 10  4 / 10  5 / 10  6 / 10  8.5 / 10  10 / 10  4 / 10  4.5 / 10  3.5 / 10 
In this paper, we benchmark the performance of a wide collection of existing MBRL algorithms, evaluating their sample efficiency, asymptotic performance and robustness. Through systematic evaluation and comparison, we characterize three key research challenges for future MBRL research. Across this very substantial benchmarking, there is no clear consistent best MBRL algorithm, suggesting lots of opportunities for future work bringing together the strengths of different approaches.
Proceedings of the Fifteenth Conference on Uncertainty in Artificial Intelligence
, UAI’99, pages 150–159, San Francisco, CA, USA, 1999. Morgan Kaufmann Publishers Inc.Proceedings of the 28th International Conference on machine learning (ICML11)
, pages 465–472, 2011.2009 IEEE conference on computer vision and pattern recognition
, pages 248–255. Ieee, 2009.A reduction of imitation learning and structured prediction to noregret online learning.
In Proceedings of the fourteenth international conference on artificial intelligence and statistics, pages 627–635, 2011.Seaborn: statistical data visualization.
http://seaborn.pydata.org/. Accessed: 20100930.We provide an overview of the environments in this section. Table 6 shows the dimensionality and horizon lengths of those environments, and Table 7 specifies their reward functions.
Environment Name 


Horizon  
Acrobot  6  1  200  
Pendulum  3  1  200  
InvertedPendulum  4  1  100  
CartPole  4  1  200  
MountainCar  2  1  200  
Reacher2D (Reacher)  11  2  50  
HalfCheetah  17  6  1000  
Swimmerv0  8  2  1000  
Swimmer  8  2  1000  
Hopper  11  3  1000  
Ant  28  8  1000  
Walker 2D  17  6  1000  
Humanoid  376  17  1000  
SlimHumanoid  45  17  1000 
Environment Name  Reward Function 
Acrobot  
Pendulum  
InvertedPendulum  
CartPole  
MountainCar  
Reacher2D (Reacher)  
HalfCheetah  
Swimmerv0  
Swimmer  
Hopper  
Ant  
Walker2D  
Humanoid  
SlimHumanoid 
In this section, we provide details about environmentspecific dynamics and goals.
The dynamical system consists of a pendulum with two links. The joint between the two links is actuated. Initially, both links point downwards. The goal is to swing up the pendulum, such that the tip of the pendulum reaches a given height. Let , be the joint angles of the first (with one end fixed to a hinge) and second link at time . The 6dimensional observation at time is the tuple: . The reward is the height of the tip of the pendulum: .
A singlelinked pendulum is fixed on the one end, with an actuator located on the joint. The goal is to keep the pendulum at the upright position. Let be the joint angle at time . The 3dimensional observation at time is The reward penalizes the position and velocity deviation from the upright equilibrium, as well as the magnitude of the control input.
The dynamical system consists of a cart that slides on a rail, and a pole connected through an unactuated joint to the cart. The only actuator applies force on the cart along the rail. The actuator force is a real number. Let be the angle of the pole away from the upright vertical position, and be the position of the cart away from the centre of the rail at time . The 4dimensional observation at time is . The reward penalizes the angular deviation from the upright position.
The dynamical system of CartPole is very similar to that of the Inverted Pendulum environment. The differences are: 1) the realvalued actuator input is discretized to , with a threshold at zero; 2) the reward indicates that the goal is to make the pole stay upright, and the cart stay at the centre of the rail.
A car is initially positioned between two “mountains", and can drive on a onedimensional track. The goal is to reach the top of the “mountain" on the right. However, the engine of the car is not strong enough for it to drive up the valley in one go, so the solution is to drive back and forth to accumulate momentum. The observation at time is the tuple , where both the position and velocity are onedimensional, with respect to the track. The reward at time is simply . Note that we use a fixed horizon, so that the agent is encouraged to reach the goal as soon as possible.
An arm with two links is fixed at one end, and is free to move on the horizontal 2D plane. There are two actuators, located at the two joints respectively. At each episode, a target is randomly placed on the 2D plane within reach of the arm. The goal is to make the tip of the arm reach the target as fast as possible, and with the smallest possible control input. Let be the two joint positions, be the position of the target, and be the position of the tip of the arm at time , respectively.The observation is . The reward at time is , where the first term is the Euclidean distance between the tip and the target.
Half Cheetah is a 2D robot with 7 rigid links, including 2 legs and a torso. There are 6 actuators located at 6 joints respectively. The goal is to run forward as fast as possible, while keeping control inputs small. The observation include the (angular) position and velocity of all the joints (including the root joint, whose position specifies the robot’s position in the world coordinate), except for the position of the root joint. The reward is the direction velocity plus penalty for control inputs.
Swimmerv0 is a 2D robot with 3 rigid links, sliding on a 2D plane. There are 2 actuators, located on the 2 joints between the links. The root joint is located at the centre of the middle link. The observation include the (angular) position and velocity of all the joints, except for the position of the two slider joints (indicating the and positions). The reward is the direction velocity plus penalty for control inputs.
The dynamical system of Swimmer is similar to that of Swimmerv0, except that the root joint is located at the tip of the first link (i.e. the “head" of the swimmer).
Hopper is a 2D “robot leg" with 4 rigid links, including the torso, thigh, leg and foot. There are 3 actuators, located at the three joints connecting the links. The observation include the (angular) position and velocity of all the joints, except for the position of the root joint. The reward is the direction velocity plus penalty for the distance to a target height and control input. The intended goal is to hop forward as fast as possible, while approximately maintaining the standing height, and with the smallest control input possible. We also add an alive bonus of 1 to the agents at every timestep, which is also applied to Ant, Walker2D.
Ant is a 3D robot with 13 rigid links, including a torso 4 legs. There are 8 actuators, 2 for each leg, located at the joints. The observation include the (angular) position and velocity of all the joints, except for the and positions of the root joint. The reward is the direction velocity plus penalty for the distance to a target height and control input. The intended goal is to go forward, while approximately maintaining the normal standing height, and with the smallest control input possible.
Walker 2D is a planar robot, consisting of 7 rigid links, including a torso and 2 legs. There are 6 actuators, 3 for each leg. The observation include the (angular) position and velocity of all the joints, except for the position of the root joint. The reward is the direction velocity plus penalty for the distance to a target height and control input. The intended goal is to walk forward as fast as possible, while approximately maintaining the standing height, and with the smallest control input possible.
Humanoid is a 3D human shaped robot consisting of 13 rigid links. There are 17 actuators, located at the humanoid’s abdomen, hips, knees, shoulders and elbows. The observation space include the joint (angular) positions and velocities, centre of mass based inertia, velocity, external force, and actuator force. The reward is the scaled direction velocity, plus penalty for control input, impact (external force) and undesired height.
The dynamical system of Slim Humanoid is similar to that of Humanoid, except that the observation is simply the joint positions and velocities, without the center of mass based quantities, external force and actuator force. Also, the reward no longer penalizes the impact (external force).
In this section, we provide a more detailed description of the hyperparameters we search for each algorithm. Note that we select the best hyperparameter combination for each algorithm, but we still provide a reference hyperparameter combination that is generally good for all environments.
For iLQG algorithm, the hyperparameters searched are summarized in 8. While the recommended hyperparameters usually have the best performance, they can result in more computation resources needed. In the following sections, number of planning trajectory is also refereed as search population size.
Hyperparameter  Value Tried  Recommended Value 
planning horizon  10, 20, 30, 50, 100  20 
max linesearch backtrack  10, 15, 20  10 
number iLQG update per timestep  10, 20  10 
number of planning trajectory  1, 2, …, 10, 20  10 
For the CEM and RS with groundtruth dynamics, we search only with different planning horizon, search population size. which include 10, 20, 30, 40, 50, 60, 70, 80, 90, 100. As also mentioned in planning horizon dilemma in section 4.6, the best planning horizon is usually 20 to 30.
Hyperparameter  Value Tried  Recommended Value 
planning horizon  10, 20, 30, …, 90, 100  30 
search population size  500, 1000, 2000  1000 
We mention that in [36], RS has very different hyperparameter sets from the RS studied in PETSRS [7]. The search of hyperparameters for RS is the same for RS using groundtruth dynamics as illustrated in the Table 9. The PETS and PETS is searched with the hyperparameters in Table 10. For simpler environments, it is usually better to use a planning horizon of 30. For environments such as Walker2D and Hopper, 100 is the best planning horizon.
Hyperparameter  Value Tried  Recommended Value 
planning horizon  30, 50, 60, 70, 80, 90, 100  30 / 100 
search population size  500, 1000, 2000  500 
elite size  50, 100, 150  50 
PETS combination  DE, DEE, PEE, PETSinf, PETS1, PEDS  PEE 
We note that for the dynamicspropagation combination, we choose PEE not only because of performance. PEE is among the best models, with comparable performance to other combinations such as PEDS, PETS1, PETSinf. However PEE is very computation efficient compared to other variants. For example, PEDS, which costs 68 hours for one random seed for HalfCheetah with planning horizon of 30 to train for 200,000 timesteps. While PEE usually only takes about 5 hours, and is suitable for research. The best models for HalfCheetah uses planning horizon of 100, and takes about 15 hours.
PETSRS uses the search scheme in Table 10, except for it does not have hyperparameters of elite size.
Originally MBMF was designed to run for 1 million timesteps [36]. Therefore, to accommodate the algorithm with 200,000 timesteps, we perform the search in Table 11.
Hyperparameter  Value Tried  Recommended Value 
trust region method  TRPO, PPO  PPO 
search population size  1000, 5000, 2000  5000 
planning horizon  10, 20, 30  20 
timesteps per iteration  1000, 2000, 5000  1000 
model based timesteps  5000, 7000, 10000  7000 
dagger epoch 
300, 500  300 
For METRPO and SLBO, we search for the following hyperparameters. We note that for environments with episode length of 100 or 200, we always use the same length for imaginary episodes.
Hyperparameter  Value Tried  Recommended Value 
imaginary episode length  1000, 500, 200, 100  1000 
TRPO iterations  1, 10, 20, 30, 40  20 / 40 
network ensembles  1, 5, 10, 20  5 
Terminate imaginary episode  True, False  False 
The GPS is based on the codebase [15]. We note that in the original codebase, the agent samples the initial state from several separate conditions. For each condition, there is not randomness of the initial state. However, in our benchmarking environments, the initial state is sample from a Gaussian distribution, which is essentially making the environments harder to solve.
Hyperparameter  Value Tried  Recommended Value 
timestep per iteration  1000, 5000, 10000  5000 
kl step  1.0, 2.0, 0.5  1.0 
dynamics Gaussian mixture model clusters 
5, 10, 20, 30  20 
policy Gaussian mixture model clusters  10, 20  20 
For PILCO, we search for the following hyperparameter in Table 14. We note that PILCO is very unstable across random seeds. Also, it is quite common for PILCO algorithms to add additional penalty in existing codebases using human priors. We argue that it is unfair to other algorithms and we remove any additional reward functions. Also, for PILCO to train for 200,000 timesteps, we have to use a dataset to increase training efficiency.
Hyperparameter  Value Tried  Recommended Value 
Optimizing Horizon  30, 100, 200, adaptive  100 
episode per iteration  1, 2, 4  1 
dataset size  20000, 40000, 10000  20000 
For SVG, we reproduce the variant of SVG1 with experience replay, which is claimed in [21].
Hyperparameter  Value Tried  Recommended Value 
SVG learning rate  0.0001, 0.0003, 0.001  0.0001 
data buffer size  25000  25000 
KL penalty  0.001, 0.003  0.001 
In this algorithm, we use most the hyperparameters in the original paper [8], except in the ones the algorithm is more sensitive to.
Hyperparameter  Value Tried  Recommended Value 
inner learning rate  0.0005, 0.001, 0.01  0.0005 
rollouts per task  10, 20, 30  20 
MAML iterations  30, 50, 75  50 
For PPO and TRPO, we search for different timesteps samples in one iteration. For SAC and TD3, we use the default values from [19] and [18] respectively.
Hyperparameter  Value Tried  Recommended Value 
timesteps per iteration  1000, 2000, 5000, 20000  2000 
In this appendix section, we include all the curves of every algorithms in Figure 4 and Figure 5. Some of the GPS curves and PILCO curves are not shown in the figures. We note that this is because their reward scale is sometimes very different from other algorithms.
In this appendix section, we provide more details of the performance with noise for each algorithm. In Figure 6 and Figure 7, we show the curves of different algorithms, and in Table 18 and Table 19 we show the performance numbers at the end the of training. The pink color indicates a decrease of performance, while the green color indicates a increase of performance, and black color indicates a almost the same performance.
Cheetah  Cheetah,  Cheetah,  Cheetah,  Cheetah,  
iLQR  2142.6 137.7  25.3 127.5  187.2 102.9  261.2 106.8  310.1 112.6 
GTPETS  14777.2 13964.2  1638.5 188.5  9226.5 8893.4  11484.5 12264.7  13160.6 13642.6 
GTRS  815.7 38.5  6.6 52.5  493.3 38.3  604.8 42.7  645.7 39.6 
RS  421.0 55.2  146.2 19.9  423.1 28.7  445.8 19.2  442.3 26.0 
MBMF  126.9 72.7  146.1 87.8  232.1 122.0  184.0 148.9  257.0 96.6 
PETS  2795.3 879.9  1879.5 801.5  2410.3 844.0  2427.5 674.1  2427.2 1118.6 
PETSRS  966.9 471.6  217.0 193.4  814.9 678.6  1128.6 674.2  1017.5 734.9 
METRPO  2283.7 900.4  409.4 834.2  1396.9 834.8  1319.8 698.0  2122.9 889.1 
GPS  52.3 41.7  6.8 13.6  175.2 169.4  41.6 45.7  94.0 57.0 
PILCO  41.9 267.0  282.0 258.4  275.4 164.6  175.6 284.1  260.8 290.3 
SVG  336.6 387.6  0.1 271.3  240.8 236.6  163.5 338.6  21.9 81.0 
MBMPO  3639.0 1185.8  2356.4 734.4  3635.5 1486.8  3372.9 1373  3718.7 922.3 
SLBO  1097.7 166.4  212.5 279.6  1244.8 604.0  1593.2 265.0  731.1 215.8 
PPO  17.2 84.4  113.3 92.8  83.1 117.7  28.0 54.1  35.5 87.8 
TRPO  12.0 85.5  146.0 67.4  9.4 57.6  32.7 110.9  70.9 71.9 
TD3  3614.3 82.1  895.7 61.6  817.3 11.0  4256.5 117.4  3941.8 61.3 
SAC  4000.7 202.1  1146.7 67.9  3869.2 88.2  3530.5 67.8  3708.1 96.2 
Pendulum  Pendulum,  Pendulum,  CartPole  CartPole,  CartPole,  
iLQR  160.8 29.8  357.9 251.9  2.2 166.5  200.3 0.6  197.8 2.9  200.4 0.7 
GTPETS  170.5 35.2  171.4 26.2  157.3 66.3  200.9 0.1  199.5 1.2  200.9 0.1 
GTRS  171.5 31.8  125.2 40.3  157.8 39.1  201.0 0.0  200.2 0.3  201.0 0.0 
RS  164.4 9.1  154.5 12.9  160.1 6.7  201.0 0.0  197.7 4.5  200.9 0.0 
MBMF  157.5 13.2  162.8 14.7  165.9 8.5  199.7 1.2  152.3 48.3  137.9 48.5 
PETS  167.4 53.0  174.7 27.8  166.7 52.0  199.5 3.0  156.6 50.3  196.1 5.1 
PETSRS  167.9 35.8  148.0 58.6  113.6 124.1  195.0 28.0  192.3 20.6  200.8 0.2 
METRPO  177.3 1.9  173.3 3.2  173.7 4.8  160.1 69.1  174.9 21.9  165.9 58.5 
GPS  162.7 7.6  162.2 4.5  168.9 6.8  14.4 18.6  479.8 859.7  22.7 53.8 
PILCO  132.6 410.1  211.6 272.1  168.9 30.5  1.9 155.9  139.9 54.8  2060.1 14.9 
SVG  141.4 62.4  86.7 34.6  78.8 73.2  82.1 31.9  119.2 46.3  106.6 42.0 
MBMPO  171.2 26.9  178.4 22.2  183.8 19.9  199.3 2.3  65.1 542.6  198.2 1.8 
SLBO  173.5 2.5  171.1 1.5  173.6 2.4  78.0 166.6  691.7 801.0  141.8 167.5 
PPO  163.4 8.0  165.9 15.4  157.3 12.6  86.5 7.8  120.5 42.9  120.3 46.7 
TRPO  166.7 7.3  167.5 6.7  161.1 13.0  47.3 15.7  572.3 368.0  818.0 288.1 
TD3  161.4 14.4  169.2 13.1  170.2 7.2  196.0 3.1  190.4 4.7  180.9 8.2 
SAC  168.2 9.5  169.3 5.6  169.1 12.6  199.4 0.4  60.9 23.4  70.7 11.4 
We note that we also perform the dilemma search with different population size with learnt PETSCEM. We experiment both with the HalfCheetah in our benchmarking environments, as well as the environments from [7], whose observation is further preprocessed. It can be seen from the figure that, planning horizon dilemma exists with different population size. We also show that observation preprocessing can affect the performance by a large margin.
In this section, we study how the environment length and imaginary environment length (or planning horizon) affect the performance. More specifically, we test with HalfCheetah and Ant, using different environment length form [100, 200, 500, 1000]. For the planning horizon, besides the matching length, we also test all the length from [100, 200, 500, 800, 1000]. The figures are shown in Figure 9, and the tables are shown in Table 20.
Environment  Original Length  Horizon=100  Horizon=200  Horizon=500  Horizon=800  Horizon=1000 
HalfCheetah  Env100  250.7 32.1  290.3 44.5  222.0 34.1  253.0 22.3  243.7 41.7 
HalfCheetah  Env200  422.7 143.7  675.4 139.6  529.0 50.0  451.4 124.5  528.1 74.7 
HalfCheetah  Env500  816.6 466.0  583.4 392.7  399.2 250.5  986.9 501.9  1062.7 182.0 
HalfCheetah  Env1000  1312.1 656.1  1514.2 1001.5  1522.6 456.3  1544.2 1349.0  2027.5 1125.5 
Ant  Env100  1207.8 41.6  1142.2 25.7  1111.9 35.3  1103.7 70.9  1085.5 22.9 
Ant  Env200  1249.9 127.7  1172.7 36.4  1136.9 32.6  1079.7 37.3  1096.8 18.6 
Ant  Env500  1397.6 49.9  1319.1 50.1  1423.6 46.2  1287.3 118.7  1331.5 92.9 
Ant  Env1000  1666.2 201.9  1646.0 151.8  1680.7 255.3  1530.7 48.0  1647.2 118.5 
Note that we also include planning horizon longer than the actual environment length for reference. For example, for the Ant with 100 environment length, we also include results using 200, 500, 800, 1000 planning horizon. As we can see, for the HalfCheetah environment, increasing planning horizon does not have obvious affects on the performance. In the Ant environments with different environment lengths, a planning horizon of 100 usually produces the best performance, instead of the longer ones.
GTCEM  GTCEMET  GTCEMET,  learnedCEM  learnedCEMET  
Ant  12115.3 209.7  8074.2 210.2  4339.8 87.8  1165.5 226.9  162.6 142.1 
Hopper  3232.3 192.3  260.5 12.2  817.8 217.6  1125.0 679.6  801.9 194.9 
Walker2D  7719.7 486.7  105.3 36.6  6310.3 55.0  493.0 583.7  290.6 113.4 
In this appendix section, we include the results of several schemes we experiment with early termination. The early termination dilemma is universal in all MBRL algorithms we tested, including Dynaalgorithms, shooting algorithms, and algorithm that performs policy search with backpropagation through time. To study the problem, we majorly start with exploring shooting algorithms including RS, PETSRS and PETSCEM, which only relates to early termination during planning. In Table 22 and Table 23, we also include the results that the agent does not consider being terminated in planning, even if it will be terminated, which we represent as "Unaware".
GTCEM  GTCEM+ETUnaware  GTCEMET  GTCEMET,  
Ant  12115.3 209.7  226.0 178.6  8074.2 210.2  4339.8 87.8 
Hopper  3232.3 192.3  256.8 16.3  260.5 12.2  817.8 217.6 
Walker2D  7719.7 486.7  254.8 233.4  105.3 36.6  6310.3 55.0 
GTRS  GTRSETUnaware  GTRSET  GTRSET,  
Ant  2709.1 631.1  2519.0 469.8  2083.8 537.2  2083.8 537.2 
Hopper  2467.2 55.4  209.5 46.8  220.4 54.9  289.8 30.5 
Walker2D  1641.4 137.6  207.9 27.2  231.0 32.4  258.3 51.5 
Scheme A  Scheme B  Scheme D  Scheme C  Scheme E  
Ant  1165.5 226.9  81.6 145.8  171.0 177.3  110.8 171.8  162.6 142.1 
Hopper  1125.0 679.6  129.3 36.0  701.7 173.6  801.9 194.9  684.1 157.2 
Walker2D  493.0 583.7  2.5 6.8  79.1 172.4  290.6 113.4  142.8 150.6 
AntETUnaware  AntET  AntET2xPenalty  AntET5xPenalty  AntET10xPenalty  AntET20xPenalty  AntET30Penalty  
GTCEM  226.0 178.6  8074.2 210.2  1940.9 2051.9  8092.3 183.1  7968.8 179.6  7969.9 181.5  7601.5 1140.8 
GTRS  2519.0 469.8  2083.8 537.2  2474.3 636.4  2591.1 447.5  2541.1 827.9  2715.6 763.2  2728.8 855.5 
LearntPETS  1165.5 226.9  81.6 145.8  196.4 176.7  181.0 142.8  205.5 186.0  204.6 202.6  188.3 130.7 
For the algorithms with unknown dynamics, we specifically study PETS. We design the following schemes.
Scheme A: The episode will not be terminated and the agent does not consider being terminated during planning.
Scheme B: The episode will be terminated early and the agent adds penalty in planning to avoid being terminated.
Scheme C
: The episode will be terminated, and the agent pads zero rewards after the episode is terminated during planning.
Scheme D: The same as Scheme A except for that the episode will be terminated.
Scheme E: The same as Scheme C except for that agent is allow to interact with the environment for extra timesteps (100 timesteps for example) to learn dynamics around termination boundary.