MASTERING ATARI WITH DISCRETE WORLD MODELS
Year: 2021

Abstract:
Intelligent agents need to generalize from past experience to achieve goals in complex environments. World models facilitate such generalization and allow learning behaviors from imagined outcomes to increase sample-efficiency. While learning world models from image inputs has recently become feasible for some tasks, modeling Atari games accurately enough to derive successful behaviors has remained an open challenge for many years. We introduce DreamerV2, a reinforcement learning agent that learns behaviors purely from predictions in the compact latent space of a powerful world model. The world model uses discrete representations and is trained separately from the policy. DreamerV2 constitutes the first agent that achieves human-level performance on the Atari benchmark of 55 tasks by learning behaviors inside a separately trained world model. With the same computational budget and wall-clock time, Dreamer V2 reaches 200M frames and surpasses the final performance of the top single-GPU agents IQN and Rainbow. DreamerV2 is also applicable to tasks with continuous actions, where it learns an accurate world model of a complex humanoid robot and solves stand-up and walking from only pixel inputs.

text: 1 INTRODUCTION
To successfully operate in unknown environments, reinforcement learning agents need to learn about their environments over time. World models are an explicit way to represent an agent’s knowledge about its environment. Compared to model-free reinforcement learning that learns through trial and error, world models facilitate generalization and can predict the outcomes of potential actions to enable planning (Sutton, 1991). Capturing general aspects of the environment, world models have been shown to be effective for transfer to novel tasks (Byravan et al., 2019), directed exploration (Sekar et al., 2020), and generalization from offline datasets (Yu et al., 2020). When the inputs are high-dimensional images, latent dynamics models predict ahead in an abstract latent space (Watter et al., 2015; Ha and Schmidhuber, 2018; Hafner et al., 2018; Zhang et al., 2019). Predicting compact representations instead of images has been hypothesized to reduce accumulating errors and their small memory footprint enables thousands of parallel predictions on a single GPU (Hafner et al., 2018; 2019). Leveraging this approach, the recent Dreamer agent (Hafner et al., 2019) has solved a wide range of continuous control tasks from image inputs.
Despite their intriguing properties, world models have so far not been accurate enough to compete with the stateof-the-art model-free algorithms on the most competitive benchmarks. The well-established Atari benchmark
∗Correspondence to: Danijar Hafner <mail@danijar.com>.
(Bellemare et al., 2013) historically required model-free algorithms to achieve human-level performance, such as DQN (Mnih et al., 2015), A3C (Mnih et al., 2016), or Rainbow (Hessel et al., 2018). Several attempts at learning accurate world models of Atari games have been made, without achieving competitive performance (Oh et al., 2015; Chiappa et al., 2017; Kaiser et al., 2019). On the other hand, the recently proposed MuZero agent (Schrittwieser et al., 2019) shows that planning can achieve impressive performance on board games and deterministic Atari games given extensive engineering effort and a vast computational budget. However, its implementation is not available to the public and it would require over 2 months of computation to train even one agent on a GPU, rendering it impractical for most research groups.
In this paper, we introduce DreamerV2, the first reinforcement learning agent that achieves humanlevel performance on the Atari benchmark by learning behaviors purely within a separately trained world model, as shown in Figure 1. Learning successful behaviors purely within the world model demonstrates that the world model learns to accurately represent the environment. To achieve this, we apply small modifications to the Dreamer agent (Hafner et al., 2019), such as using discrete latents and balancing terms within the KL loss. Using a single GPU and a single environment instance, DreamerV2 outperforms top single-GPU Atari agents Rainbow (Hessel et al., 2018) and IQN (Dabney et al., 2018), which rest upon years of model-free reinforcement learning research (Van Hasselt et al., 2015; Schaul et al., 2015; Wang et al., 2016; Bellemare et al., 2017; Fortunato et al., 2017). Moreover, aspects of these algorithms are complementary to our world model and could be integrated into the Dreamer framework in the future. To rigorously compare the algorithms, we report scores normalized by both a human gamer (Mnih et al., 2015) and the human world record (Toromanoff et al., 2019) and make a suggestion for reporting scores going forward.

heading: 2 DREAMERV2
text: We present DreamerV2, an evolution of the Dreamer agent (Hafner et al., 2019). We refer to the original Dreamer agent as DreamerV1 throughout this paper. This section describes the complete DreamerV2 algorithm, consisting of the three typical components of a model-based agent (Sutton, 1991). We learn the world model from a dataset of past experience, learn an actor and critic from imagined sequences of compact model states, and execute the actor in the environment to grow the experience dataset. In Appendix C, we include a list of changes that we applied to DreamerV1 and which of them we found to increase empirical performance.

heading: 2.1 WORLD MODEL LEARNING
text: World models summarize an agent’s experience into a predictive model that can be used in place of the environment to learn behaviors. When inputs are high-dimensional images, it is beneficial to learn compact state representations of the inputs to predict ahead in this learned latent space (Watter et al., 2015; Karl et al., 2016; Ha and Schmidhuber, 2018). These models are called latent dynamics models. Predicting ahead in latent space not only facilitates long-term predictions, it also allows to efficiently predict thousands of compact state sequences in parallel in a single batch, without having to generate images. DreamerV2 builds upon the world model that was introduced by PlaNet (Hafner et al., 2018) and used in DreamerV1, by replacing its Gaussian latents with categorical variables.
Experience dataset The world model is trained from the agent’s growing dataset of past experience that contains sequences of images x1:T , actions a1:T , rewards r1:T , and discount factors γ1:T . The discount factors equal a fixed hyper parameter γ = 0.999 for time steps within an episode and are set to zero for terminal time steps. For training, we use batches of B = 50 sequences of fixed length L = 50 that are sampled randomly within the stored episodes. To observe enough episode ends during training, we sample the start index of each training sequence uniformly within the episode and then clip it to not exceed the episode length minus the training sequence length.
Model components The world model consists of an image encoder, a Recurrent State-Space Model (RSSM; Hafner et al., 2018) to learn the dynamics, and predictors for the image, reward, and discount factor. The world model is summarized in Figure 2. The RSSM uses a sequence of deterministic recurrent states ht, from which it computes two distributions over stochastic states at each step. The posterior state zt incorporates information about the current image xt, while the prior state ẑt aims to predict the posterior without access to the current image. The concatenation of deterministic and
stochastic states forms the compact model state. From the posterior model state, we reconstruct the current image xt and predict the reward rt and discount factor γt. The model components are:
RSSM  Recurrent model: ht = fφ(ht−1, zt−1, at−1) Representation model: zt ∼ qφ(zt | ht, xt) Transition predictor: ẑt ∼ pφ(ẑt | ht) Image predictor: x̂t ∼ pφ(x̂t | ht, zt) Reward predictor: r̂t ∼ pφ(r̂t | ht, zt) Discount predictor: γ̂t ∼ pφ(γ̂t | ht, zt).
(1)
All components are implemented as neural networks and φ describes their combined parameter vector. The transition predictor guesses the next model state only from the current model state and the action but without using the next image, so that we can later learn behaviors by predicting sequences of model states without having to observe or generate images. The discount predictor lets us estimate the probability of an episode ending when learning behaviors from model predictions.
Neural networks The representation model is implemented as a Convolutional Neural Network (CNN; LeCun et al., 1989) followed by a Multi-Layer Perceptron (MLP) that receives the image embedding and the deterministic recurrent state. The RSSM uses a Gated Recurrent Unit (GRU; Cho et al., 2014) to compute the deterministic recurrent states. The model state is the concatenation of deterministic GRU state and a sample of the stochastic state. The image predictor is a transposed CNN and the transition, reward, and discount predictors are MLPs. We down-scale the 84 × 84 grayscale images to 64× 64 pixels so that we can apply the convolutional architecture of DreamerV1.
Algorithm 1: Straight-Through Gradients with Automatic Differentiation
sample = one_hot(draw(logits)) # sample has no gradient probs = softmax(logits) # want gradient of this sample = sample + probs - stop_grad(probs) # has gradient of probs
We use the ELU activation function for all components of the model (Clevert et al., 2015). The world model uses a total of 20M trainable parameters.
Distributions The image predictor outputs the mean of a diagonal Gaussian likelihood with unit variance, the reward predictor outputs a univariate Gaussian with unit variance, and the discount predictor outputs a Bernoulli likelihood. In prior work, the latent variable in the model state was a diagonal Gaussian that used reparameterization gradients during backpropagation (Kingma and Welling, 2013; Rezende et al., 2014). In DreamerV2, we instead use a vector of several categorical variables and optimize them using straight-through gradients (Bengio et al., 2013), which are easy to implement using automatic differentiation as shown in Algorithm 1. We discuss possible benefits of categorical over Gaussian latents in the experiments section.
Loss function All components of the world model are optimized jointly. The distributions produced by the image predictor, reward predictor, discount predictor, and transition predictor are trained to maximize the log-likelihood of their corresponding targets. The representation model is trained to produce model states that facilitates these prediction tasks, through the expectation below. Moreover, it is regularized to produce model states with high entropy, such that the model becomes robust to many different model states during training. The loss function for learning the world model is:
L(φ) .= Eqφ(z1:T | a1:T ,x1:T ) [∑T
t=1 − ln pφ(xt | ht, zt) image log loss − ln pφ(rt | ht, zt) reward log loss − ln pφ(γt | ht, zt) discount log loss
+βKL [ qφ(zt | ht, xt) ∥∥ pφ(zt | ht)] KL loss ] . (2)
We jointly minimize the loss function with respect to the vector φ that contains all parameters of the world model using the Adam optimizer (Kingma and Ba, 2014). We scale the KL loss by β = 0.1 for Atari and by β = 1.0 for continuous control (Higgins et al., 2016).
KL balancing The world model loss function in Equation 2 is the ELBO or variational free energy of a hidden Markov model that is conditioned on the action sequence. The world model can thus be interpreted as a sequential VAE, where the representation model is the approximate posterior and the transition predictor is the temporal prior. In the ELBO objective, the KL loss serves two purposes: it trains the prior toward the representations, and it regularizes the representations toward the prior. However, learning the transition function is difficult and we want to avoid regularizing the representations toward a poorly trained prior. To solve this problem, we minimize the KL loss faster with respect to the prior than the representations by using different learning rates, α = 0.8 for the prior and 1 − α for the approximate posterior. We implement this technique as shown in Algorithm 2 and refer to it as KL balancing. KL balancing encourages learning an accurate prior over increasing posterior entropy, so that the prior better approximates the aggregate posterior. KL balancing is different from and orthogonal to beta-VAEs (Higgins et al., 2016).

heading: 2.2 BEHAVIOR LEARNING
text: DreamerV2 learns long-horizon behaviors purely within its world model using an actor and a critic. The actor chooses actions for predicting imagined sequences of compact model states. The critic accumulates the future predicted rewards to take into account rewards beyond the planning horizon. Both the actor and critic operate on top of the learned model states and thus benefit from the representations learned by the world model. The world model is fixed during behavior learning, so the actor and value gradients do not affect its representations. Not predicting images during behavior learning lets us efficiently simulate 2500 latent trajectories in parallel on a single GPU.
Imagination MDP To learn behaviors within the latent space of the world model, we define the imagination MPD as follows. The distribution of initial states ẑ0 in the imagination MDP is the distribution of compact model states encountered during world model training. From there, the transition predictor pφ(ẑt | ẑt−1, ât−1) outputs sequences ẑ1:H of compact model states up to the
Algorithm 2: KL Balancing with Automatic Differentiation
kl_loss = alpha * compute_kl(stop_grad(approx_posterior), prior) + (1 - alpha) * compute_kl(approx_posterior, stop_grad(prior))
imagination horizon H = 15. The mean of the reward predictor pφ(r̂t | ẑt) is used as reward sequence r̂1:H . The discount predictor pφ(γ̂t | ẑt) outputs the discount sequence γ̂1:H that is used to down-weight rewards. Moreover, we weigh the loss terms of the actor and critic by the cumulative predicted discount factors to softly account for the possibility of episode ends.
Model components To learn long-horizon behaviors in the imagination MDP, we leverage a stochastic actor that chooses actions and a deterministic critic. The actor and critic are trained cooperatively, where the actor aims to output actions that lead to states that maximize the critic output, while the critic aims to accurately estimate the sum of future rewards achieved by the actor from each imagined state. The actor and critic use the parameter vectors ψ and ξ, respectively:
Actor: ât ∼ pψ(ât | ẑt) Critic: vξ(ẑt) ≈ Epφ,pψ [∑ τ≥t γ̂ τ−tr̂τ ] .
(3)
In contrast to the actual environment, the latent state sequence is Markovian, so that there is no need for the actor and critic to condition on more than the current model state. The actor and critic are both MLPs with ELU activations (Clevert et al., 2015) and use 1M trainable parameters each. The actor outputs a categorical distribution over actions and the critic has a deterministic output. The two components are trained from the same imagined trajectories but optimize separate loss functions.
Critic loss function The critic aims to predict the discounted sum of future rewards that the actor achieves in a given model state, known as the state value. For this, we leverage temporal-difference learning, where the critic is trained toward a value target that is constructed from intermediate rewards and critic outputs for later states. A common choice is the 1-step target that sums the current reward and the critic output for the following state. However, the imagination MDP lets us generate on-policy trajectories of multiple steps, suggesting the use of n-step targets that incorporate reward information into the critic more quickly. We follow DreamerV1 in using the more general λ-target (Sutton and Barto, 2018; Schulman et al., 2015) that is defined recursively as follows:
V λt . = r̂t + γ̂t { (1− λ)vξ(ẑt+1) + λV λt+1 if t < H, vξ(ẑH) if t = H.
(4)
Intuitively, the λ-target is a weighted average of n-step returns for different horizons, where longer horizons are weighted exponentially less. We set λ = 0.95 in practice, to focus more on long horizon
targets than on short horizon targets. Given a trajectory of model states, rewards, and discount factors, we train the critic to regress the λ-return using a squared loss:
L(ξ) .= Epφ,pψ [∑H−1 t=1 1 2 ( vξ(ẑt)− sg(V λt ) )2] . (5)
We optimize the critic loss with respect to the critic parameters ξ using the Adam optimizer. There is no loss term for the last time step because the target equals the critic at that step. We stop the gradients around the targets, denoted by the sg(·) function, as typical in the literature. We stabilize value learning using a target network (Mnih et al., 2015), namely, we compute the targets using a copy of the critic that is updated every 100 gradient steps.
Actor loss function The actor aims to output actions that maximize the prediction of long-term future rewards made by the critic. To incorporate intermediate rewards more directly, we train the actor to maximize the same λ-return that was computed for training the critic. There are different gradient estimators for maximizing the targets with respect to the actor parameters. DreamerV2 combines unbiased but high-variance Reinforce gradients with biased but low-variance straightthrough gradients. Moreover, we regularize the entropy of the actor to encourage exploration where feasible while allowing the actor to choose precise actions when necessary.
Learning by Reinforce (Williams, 1992) maximizes the actor’s probability of its own sampled actions weighted by the values of those actions. The variance of this estimator can be reduced by subtracting the state value as baseline, which does not depend on the current action. Intuitively, subtracting the baseline centers the weights and leads to faster learning. The benefit of Reinforce is that it produced unbiased gradients and the downside is that it can have high variance, even with baseline.
DreamerV1 relied entirely on reparameterization gradients (Kingma and Welling, 2013; Rezende et al., 2014) to train the actor directly by backpropagating value gradients through the sequence of sampled model states and actions. DreamerV2 uses both discrete latents and discrete actions. To backpropagate through the sampled actions and state sequences, we leverage straight-through gradients (Bengio et al., 2013). This results in a biased gradient estimate with low variance. The combined actor loss function is:
L(ψ) .= Epφ,pψ [∑H−1 t=1 ( −ρ ln pψ(ât | ẑt) sg(V λt − vξ(ẑt))
reinforce
−(1− ρ)V λt dynamics backprop −ηH[at|ẑt] entropy regularizer
)] .
(6)
We optimize the actor loss with respect to the actor parameters ψ using the Adam optimizer. We consider both Reinforce gradients and straight-through gradients, which backpropagate directly through the learned dynamics. Intuitively, the low-variance but biased dynamics backpropagation could learn faster initially and the unbiased but high-variance could to converge to a better solution. For Atari, we find Reinforce gradients to work substantially better and use ρ = 1 and η = 10−3. For continuous control, we find dynamics backpropagation to work substantially better and use ρ = 0 and η = 10−4. Annealing these hyper parameters can improve performance slightly but to avoid the added complexity we report the scores without annealing.

heading: 3 EXPERIMENTS
text: We evaluate DreamerV2 on the well-established Atari benchmark with sticky actions, comparing to four strong model-free algorithms. DreamerV2 outperforms the four model-free algorithms in all scenarios. For an extensive comparison, we report four scores according to four aggregation protocols and give a recommendation for meaningfully aggregating scores across games going forward. We also ablate the importance of discrete representations in the world model. Our implementation of DreamerV2 reaches 200M environment steps in under 10 days, while using only a single NVIDIA V100 GPU and a single environment instance. During the 200M environment steps, DreamerV2 learns its policy from 468B compact states imagined under the model, which is 10,000× more than the 50M inputs received from the real environment after action repeat. Refer to the project website for videos, the source code, and training curves in JSON format.1
1https://danijar.com/dreamerv2
Experimental setup We select the 55 games that prior works in the literature from different research labs tend to agree on (Mnih et al., 2016; Brockman et al., 2016; Hessel et al., 2018; Castro et al., 2018; Badia et al., 2020) and recommend this set of games for evaluation going forward. We follow the evaluation protocol of Machado et al. (2018) with 200M environment steps, action repeat of 4, a time limit of 108,000 steps per episode that correspond to 30 minutes of game play, no access to life information, full action space, and sticky actions. Because the world model integrates information over time, DreamerV2 does not use frame stacking. The experiments use a single-task setup where a separate agent is trained for each game. Moreover, each agent uses only a single environment instance. We compare the algorithms based on both human gamer and human world record normalization (Toromanoff et al., 2019).
Model-free baselines We compare the learning curves and final scores of DreamerV2 to four model-free algorithms, IQN (Dabney et al., 2018), Rainbow (Hessel et al., 2018), C51 (Bellemare et al., 2017), and DQN (Mnih et al., 2015). We use the scores of these agents provided by the Dopamine framework (Castro et al., 2018) that use sticky actions. These may differ from the reported results in the papers that introduce these algorithms in the deterministic Atari setup. The training time of Rainbow was reported at 10 days on a single GPU and using one environment instance.

heading: 3.1 ATARI PERFORMANCE
text: The performance curves of DreamerV2 and four standard model-free algorithms are visualized in Figure 4. The final scores at 200M environment steps are shown in Table 1 and the scores on individual games are included in Table K1. There are different approaches for aggregating the scores across the 55 games and we show that this choice can have a substantial impact on the relative performance between algorithms. To extensively compare DreamerV2 to the model-free algorithms, we consider the following four aggregation approaches:
• Gamer Median Atari scores are commonly normalized based on a random policy and a professional gamer, averaged over seeds, and the median over tasks is reported (Mnih et al., 2015; 2016). However, if almost half of the scores would be zero, the median would not be affected. Thus, we argue that median scores are not reflective of the robustness of an algorithm and results in wasted computational resources for games that will not affect the score.
• Gamer Mean Compared to the task median, the task mean considers all tasks. However, the gamer performed poorly on a small number of games, such as Crazy Climber, James Bond, and Video Pinball. This makes it easy for algorithms to achieve a high normalized score on these few games, which then dominate the task mean so it is not informative of overall performance.
• Record Mean Instead of normalizing based on the professional gamer, Toromanoff et al. (2019) suggest to normalize based on the registered human world record of each game. This partially addresses the outlier problem but the mean is still dominated by games where the algorithms easily achieve superhuman performance.
• Clipped Record Mean To overcome these limitations, we recommend normalizing by the human world record and then clipping the scores to not exceed a value of 1, so that performance above the record does not further increase the score. The result is a robust measure of algorithm performance on the Atari suite that considers performance across all games.
From Figure 4 and Table 1, we see that the different aggregation approaches let us examine agent performance from different angles. Interestingly, Rainbow clearly outperforms IQN in the first aggregation method but IQN clearly outperforms Rainbow in the remaining setups. DreamerV2 outperforms the model-free agents in all four metrics, with the largest margin in record normalized mean performance. Despite this, we recommend clipped record normalized mean as the most meaningful aggregation method, as it considers all tasks to a similar degree without being dominated by a small number of outlier scores. In Table 1, we also include DreamerV2 with schedules that anneal the actor entropy loss scale and actor gradient mixing over the course of training, which further increases the gamer median score of DreamerV2.
Individual games The scores on individual Atari games at 200M environment steps are included in Table K1, alongside the model-free algorithms and the baselines of random play, human gamer, and human world record. We filled in reasonable values for the 2 out of 55 games that have no registered world record. Figure E1 compares the score differences between DreamerV2 and each model-free algorithm for the individual games. DreamerV2 achieves comparable or higher performance on most games except for Video Pinball. We hypothesize that the reconstruction loss of the world model does not encourage learning a meaningful latent representation because the most important object in the game, the ball, occupies only a single pixel. One the other hand, DreamerV2 achieves the strongest improvements over the model-free agents on the games James Bond, Up N Down, and Assault.
Agent Gamer Median Gamer Mean Record Mean Clipped Record Mean

heading: 3.2 ABLATION STUDY
text: To understand which ingredients of DreamerV2 are responsible for its success, we conduct an extensive ablation study. We compare equipping the world model with categorical latents, as in DreamerV2, to Gaussian latents, as in DreamerV1. Moreover, we study the importance of KL balancing. Finally, we investigate the importance of gradients from image reconstruction and reward prediction for learning the model representations, by stopping one of the two gradient signals before entering the model states. The results of the ablation study are summarized in Figure 5 and Table 2. Refer to the appendix for the score curves of the individual tasks.
Categorical latents Categorical latent variables outperform than Gaussian latent variables on 42 tasks, achieve lower performance on 8 tasks, and are tied on 5 tasks. We define a tie as being within 5% of another. While we do not know the reason why the categorical variables are beneficial, we state several hypotheses that can be investigated in future work:
• A categorical prior can perfectly fit the aggregate posterior, because a mixture of categoricals is again a categorical. In contrast, a Gaussian prior cannot match a mixture of Gaussian posteriors, which could make it difficult to predict multi-modal changes between one image and the next. • The level of sparsity enforced by a vector of categorical latent variables could be beneficial for
generalization. Flattening the sample from the 32 categorical with 32 classes each results in a sparse binary vector of length 1024 with 32 active bits.
• Despite common intuition, categorical variables may be easier to optimize than Gaussian variables, possibly because the straight-through gradient estimator ignores a term that would otherwise scale the gradient. This could reduce exploding and vanishing gradients.
• Categorical variables could be a better inductive bias than unimodal continuous latent variables for modeling the non-smooth aspects of Atari games, such as when entering a new room, or when collected items or defeated enemies disappear from the image.
KL balancing KL balancing outperforms the standard KL regularizer on 44 tasks, achieves lower performance on 6 tasks, and is tied on 5 tasks. Learning accurate prior dynamics of the world model is critical because it is used for imagining latent state trajectories using policy optimization. By scaling up the prior cross entropy relative to the posterior entropy, the world model is encouraged to minimize the KL by improving its prior dynamics toward the more informed posteriors, as opposed to reducing the KL by increasing the posterior entropy. KL balancing may also be beneficial for probabilistic models with learned priors beyond world models.
Model gradients Stopping the image gradients increases performance on 3 tasks, decreases performance on 51 tasks, and is tied on 1 task. The world model of DreamerV2 thus heavily relies on the learning signal provided by the high-dimensional images. Stopping the reward gradients increases performance on 15 tasks, decreases performance on 22 tasks, and is tied on 18 tasks. Figure H1 further shows that the difference in scores is small. In contrast to MuZero, DreamerV2 thus learns general representations of the environment state from image information alone. Stopping reward gradients improved performance on a number of tasks, suggesting that the representations that are not specific to previously experienced rewards may generalize better to unseen situations.
Policy gradients Using only Reinforce gradients to optimize the policy increases performance on 18 tasks, decreases performance on 24 tasks, and is tied on 13 tasks. This shows that DreamerV2 relies mostly on Reinforce gradients to learn the policy. However, mixing Reinforce and straight-through gradients yields a substantial improvement on James Bond and Seaquest, leading to a higher gamer normalized task mean score. Using only straight-through gradients to optimize the policy increases performance on 5 tasks, decreases performance on 44 tasks, and is tied on 6 tasks. We conjecture that straight-through gradients alone are not well suited for policy optimization because of their bias.

heading: 4 RELATED WORK
text: Model-free Atari The majority of agents applied to the Atari benchmark have been trained using model-free algorithms. DQN (Mnih et al., 2015) showed that deep neural network policies can be trained using Q-learning by incorporating experience replay and target networks. Several works have extended DQN to incorporate bias correction as in DDQN (Van Hasselt et al., 2015), prioritized experience replay (Schaul et al., 2015), architectural improvements (Wang et al., 2016), and distributional value learning (Bellemare et al., 2017; Dabney et al., 2017; 2018). Besides value learning, agents based on policy gradients have targeted the Atari benchmark, such as ACER (Schulman et al., 2017a), PPO (Schulman et al., 2017a), ACKTR (Wu et al., 2017), and Reactor (Gruslys et al., 2017). Another line of work has focused on improving performance by distributing data collection, often while increasing the budget of environment steps beyond 200M (Mnih et al., 2016; Schulman et al., 2017b; Horgan et al., 2018; Kapturowski et al., 2018; Badia et al., 2020).
World models Several model-based agents focus on proprioceptive inputs (Watter et al., 2015; Gal et al., 2016; Higuera et al., 2018; Henaff et al., 2018; Chua et al., 2018; Wang et al., 2019; Wang and Ba, 2019), model images without using them for planning (Oh et al., 2015; Krishnan et al., 2015; Karl et al., 2016; Chiappa et al., 2017; Babaeizadeh et al., 2017; Gemici et al., 2017; Denton and Fergus, 2018; Buesing et al., 2018; Doerr et al., 2018; Gregor and Besse, 2018), or combine the benefits of model-based and model-free approaches (Kalweit and Boedecker, 2017; Nagabandi et al., 2017; Weber et al., 2017; Kurutach et al., 2018; Buckman et al., 2018; Ha and Schmidhuber, 2018; Wayne et al., 2018; Igl et al., 2018; Srinivas et al., 2018; Lee et al., 2019). Risi and Stanley (2019) optimize discrete latents using evolutionary search. Parmas et al. (2019) combine reinforce and reparameterization gradients. Most world model agents with image inputs have thus far been limited to relatively simple control tasks (Watter et al., 2015; Ebert et al., 2017; Ha and Schmidhuber, 2018; Hafner et al., 2018; Zhang et al., 2019; Hafner et al., 2019). We explain the two model-based approaches that were applied to Atari in detail below.
SimPLe The SimPLe agent (Kaiser et al., 2019) learns a video prediction model in pixel-space and uses its predictions to train a PPO agent (Schulman et al., 2017a), as shown in Table 3. The model directly predicts each frame from the previous four frames and receives an additional discrete latent variable as input. The authors evaluate SimPLe on a subset of Atari games for 400k and 2M environment steps, after which they report diminishing returns. Some recent model-free methods have followed the comparison at 400k steps (Srinivas et al., 2020; Kostrikov et al., 2020). However, the highest performance achieved in this data-efficient regime is a gamer normalized median score of 0.28 (Kostrikov et al., 2020) that is far from human-level performance. Instead, we focus on the well-established and competitive evaluation after 200M frames, where many successful model-free algorithms are available for comparison.
MuZero The MuZero agent (Schrittwieser et al., 2019) learns a sequence model of rewards and values (Oh et al., 2017) to solve reinforcement learning tasks via Monte-Carlo Tree Search (MCTS; Coulom, 2006; Silver et al., 2017). The sequence model is trained purely by predicting task-specific information and does not incorporate explicit representation learning using the images, as shown in Table 3. MuZero shows that with significant engineering effort and a vast computational budget, planning can achieve impressive performance on several board games and deterministic Atari games. However, MuZero is not publicly available, and it would require over 2 months to train an Atari agent on one GPU. By comparison, DreamerV2 is a simple algorithm that achieves human-level performance on Atari on a single GPU in 10 days, making it reproducible for many researchers. Moreover, the advanced planning components of MuZero are complementary and could be applied to the accurate world models learned by DreamerV2. DreamerV2 leverages the additional learning signal provided by the input images, analogous to recent successes by semi-supervised image classification (Chen et al., 2020; He et al., 2020; Grill et al., 2020).

heading: 5 DISCUSSION
text: We present DreamerV2, a model-based agent that achieves human-level performance on the Atari 200M benchmark by learning behaviors purely from the latent-space predictions of a separately trained world model. Using a single GPU and a single environment instance, DreamerV2 outperforms top model-free single-GPU agents Rainbow and IQN using the same computational budget and training time. To develop DreamerV2, we apply several small modifications to the Dreamer agent (Hafner et al., 2019). We confirm experimentally that learning a categorical latent space and using KL balancing improves the performance of the agent. Moreover, we find the DreamerV2 relies on image information for learning generally useful representations — its performance is not impacted by whether the representations are especially learned for predicting rewards.
DreamerV2 serves as proof of concept, showing that model-based RL can outperform top model-free algorithms on the most competitive RL benchmarks, despite the years of research and engineering effort that modern model-free agents rest upon. Beyond achieving strong performance on individual tasks, world models open avenues for efficient transfer and multi-task learning, sample-efficient learning on physical robots, and global exploration based on uncertainty estimates.
Acknowledgements We thank our anonymous reviewers for their feedback and Nick Rhinehart for an insightful discussion about the potential benefits of categorical latent variables.

heading: A HUMANOID FROM PIXELS
text: Figure A1: Behavior learned by DreamerV2 on the Humanoid Walk task from pixel inputs only. The task is provided by the DeepMind Control Suite and uses a continuous action space with 21 dimensions. The frames show the agent inputs.
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800 Humanoid Walk
DreamerV2
Figure A2: Performance on the humanoid walking task from only pixel inputs. While the main experiments of this paper focus on the Atari benchmark with discrete actions, DreamerV2 is also applicable to control tasks with continuous actions. For this, we the actor outputs a truncated normal distribution instead of a categorical distribution. To demonstrate the abilities of DreamerV2 for continuous control, we choose the challenging humanoid environment with only image inputs, shown in Figure A1. We find that for continuous control tasks, dynamics backpropagation substantially outperforms reinforce gradients and thus set ρ = 0. We also set η = 10−5 and β = 2 and leave all other hyper parameters at their defaults. We find that DreamerV2 reliably solves both the stand-up motion required at the beginning of the episode and the subsequent walking. The score is shown in Figure A2. To the best of our knowledge, this constitutes the first published result of solving the humanoid environment from only pixel inputs.

heading: B MONTEZUMA’S REVENGE
text: Figure B1: Behavior learned by DreamerV2 on the Atari game Montezuma’s Revenge, that poses a hard exploration challenge. Without any explicit exploration mechanism, DreamerV2 reaches about the same performance as the exploration method ICM.
0.0 0.5 1.0 1.5 2.0 1e8
0
1000
2000
3000 Montezuma Revenge
DreamerV2 ( =0.99) Rainbow + Curiosity Rainbow
IQN C51 DQN
Figure B2: Performance on the Atari game Montezuma’s Revenge. While our main experiments use the same hyper parameters across all tasks, we find that DreamerV2 achieves higher performance on Montezuma’s Revenge by using a lower discount factor of γ = 0.99, possibly to stabilize value learning under sparse rewards. Figure B2 shows the resulting performance, with all other hyper parameters left at their defaults. DreamerV2 outperforms existing modelfree approaches on the hard-exploration game Montezuma’s Revenge and matches the performance of the explicit exploration algorithm ICM (Pathak et al., 2017) that was applied on top of Rainbow by Taiga et al. (2019). This suggests that the world model may help with solving sparse reward tasks, for example due to improved generalization, efficient policy optimization in the compact latent space enabling more actor critic updates, or because the reward predictor generalizes and thus smooths out the sparse rewards.

heading: C SUMMARY OF MODIFICATIONS
text: To develop DreamerV2, we used the Dreamer agent (Hafner et al., 2019) as a starting point. This subsection describes the changes that we applied to the agent to achieve high performance on the Atari benchmark, as well as the changes that were tried but not found to increase performance and thus were not not included in DreamerV2.
Summary of changes that were tried and were found to help:
• Categorical latents Using categorical latent states using straight-through gradients in the world model instead of Gaussian latents with reparameterized gradients. • KL balancing Separately scaling the prior cross entropy and the posterior entropy in the KL
loss to encourage learning an accurate temporal prior, instead of using free nats. • Reinforce only Reinforce gradients worked substantially better for Atari than dynamics back-
propagation. For continuous control, dynamics backpropagation worked substantially better. • Model size Increasing the number of units or feature maps per layer of all model components,
resulting in a change from 13M parameters to 22M parameters. • Policy entropy Regularizing the policy entropy for exploration both in imagination and during
data collection, instead of using external action noise during data collection.
Summary of changes that were tried but were found to not help substantially:
• Binary latents Using a larger number of binary latents for the world model instead of categorical latents, which could have encouraged a more disentangled representation, was worse.
• Long-term entropy Including the policy entropy into temporal-difference loss of the value function, so that the actor seeks out states with high action entropy beyond the planning horizon.
• Mixed actor gradients Combining Reinforce and dynamics backpropagation gradients for learning the actor instead of Reinforce provided marginal or no benefits.
• Scheduling Scheduling the learning rates, KL scale, actor entropy loss scale, and actor gradient mixing (from 0.1 to 0) provided marginal or no benefits.
• Layer norm Using layer normalization in the GRU that is used as part of the RSSM latent transition model, instead of no normalization, provided no or marginal benefits.
Due to the large computational requirements, a comprehensive ablation study on this list of all changes is unfortunately infeasible for us. This would require 55 tasks times 5 seeds for 10 days per change to run, resulting in over 60,000 GPU hours per change. However, we include ablations for the most important design choices in the main text of the paper.

heading: D HYPER PARAMETERS
text: Name Symbol Value
World Model
Dataset size (FIFO) — 2 · 106 Batch size B 50 Sequence length L 50 Discrete latent dimensions — 32 Discrete latent classes — 32 RSSM number of units — 600 KL loss scale β 0.1 KL balancing α 0.8 World model learning rate — 2 · 10−4 Reward transformation — tanh
Behavior
Imagination horizon H 15 Discount γ 0.995 λ-target parameter λ 0.95 Actor gradient mixing ρ 1 Actor entropy loss scale η 1 · 10−3 Actor learning rate — 4 · 10−5 Critic learning rate — 1 · 10−4 Slow critic update interval — 100
Common
Environment steps per update — 4 MPL number of layers — 4 MPL number of units — 400 Gradient clipping — 100 Adam epsilon 10−5 Weight decay (decoupled) — 10−6
Table D1: Atari hyper parameters of DreamerV2. When tuning the agent for a new task, we recommend searching over the KL loss scale β ∈ {0.1, 0.3, 1, 3}, actor entropy loss scale η ∈ {3 · 10−5, 10−4, 3 · 10−4, 10−3}, and the discount factor γ ∈ {0.99, 0.999}. The training frequency update should be increased when aiming for higher data-efficiency.

heading: E AGENT COMPARISON
text: Ja m
es B
on d
Up N
D ow
n Kr ul l
Go ph er De m on A tta ck As sa ul t Ro ad R un ne r Ti m e Pi lo t Br ea ko ut As te rix Ph oe ni x Qb er t At la nt is Za xx on Ice H oc ke y W iza rd O f W or Ya rs R ev en ge Na m e Th is Ga m e Ku ng F u M as te r Ro bo ta nk Cr az y Cl im be r As te ro id s Fr os tb ite Ce nt ip ed e Gr av ita r Be am R id er Ka ng ar oo Fi sh in g De rb y Sk iin g Am id ar Ba nk H ei st Tu ta nk ha m Ri ve rra id Bo wl in g M s P ac m an Be rz er k Pi tfa ll Fr ee wa y Po ng Ba ttl e Zo ne Pr iv at e Ey e So la ris M on te zu m a Re v. Al ie n Se aq ue st Bo xi ng He ro Te nn is En du ro Ch op pe r C om . Ve nt ur e Sp ac e In va de rs Do ub le D un k St ar G un ne r Vi de o Pi nb al l
100 10 0
10 100
DreamerV2 vs IQN
Ja m
es B
on d
Up N
D ow
n Kr ul l
As sa
ul t
Go ph er De m on A tta ck Ro ad R un ne r Ti m e Pi lo t At la nt is Br ea ko ut As te rix Ph oe ni x Qb er t Za xx on Ice H oc ke y Ya rs R ev en ge Ku ng F u M as te r Sk iin g Ro bo ta nk W iza rd O f W or
Na m
e Th
is Ga
m e
Te nn is As te ro id s Gr av ita r
Be am
R id er Fr os tb ite Cr az y Cl im be r Ce nt ip ed e Ka ng ar oo Fi sh in g De rb y M s P ac m an Tu ta nk ha m Al ie n Ba nk H ei st Bo wl in g Am id ar Ba ttl e Zo ne Pi tfa ll Fr ee wa y Be rz er k Po ng Se aq ue st So la ris
M on
te zu
m a
Re v.
Pr iv
at e
Ey e
Ri ve
rra id
Bo xi ng En du ro He ro
Sp ac
e In
va de rs Ve nt ur e Ch op pe r C om . Do ub le D un k St ar G un ne r Vi de o Pi nb al l
100 10 0
10 100
DreamerV2 vs Rainbow
Ja m
es B
on d
Up N
D ow n As sa ul t De m on A tta ck Kr ul l Go ph er Ro ad R un ne r Ti m e Pi lo t At la nt is Do ub le D un k As te rix Ph oe ni x Qb er t Za xx on Br ea ko ut Ya rs R ev en ge Ice H oc ke y W iza rd O f W or Ka ng ar oo Ro bo ta nk Ku ng F u M as te r Fr os tb ite Cr az y Cl im be r Fi sh in g De rb y Sk iin g Gr av ita r Bo xi ng As te ro id s Be am R id er Am id ar Ba nk H ei st En du ro Ce nt ip ed e Ba ttl e Zo ne
Na m
e Th
is Ga
m e
M s P
ac m
an Al ie n
Ri ve
rra id
Tu ta
nk ha m Bo wl in g Be rz er k Po ng Pi tfa ll Fr ee wa y
Pr iv
at e
Ey e
So la ris M on te zu m a Re v. He ro Ch op pe r C om . Te nn is Se aq ue st Sp ac e In va de rs Ve nt ur e St ar G un ne r Vi de o Pi nb al l
100 10 0
10 100
DreamerV2 vs C51
Ja m
es B
on d
Up N
D ow n As sa ul t De m on A tta ck Kr ul l Go ph er Ro ad R un ne r Ti m e Pi lo t Do ub le D un k As te rix At la nt is Br ea ko ut Ph oe ni x Qb er t Za xx on Ice H oc ke y Fr os tb ite Ya rs R ev en ge W iza rd O f W or Cr az y Cl im be r Ro bo ta nk
Ku ng
F u
M as
te r
Na m
e Th
is Ga
m e
Fi sh
in g
De rb y En du ro Bo xi ng Gr av ita r Tu ta nk ha m Te nn is Ce nt ip ed e As te ro id s Ka ng ar oo Am id ar Be am R id er Ba nk H ei st Ba ttl e Zo ne Sk iin g
Sp ac
e In
va de rs M s P ac m an Ri ve rra id Fr ee wa y Al ie n He ro Se aq ue st Bo wl in g Be rz er k Po ng Pr iv at e Ey e Ch op pe r C om . M on te zu m a Re v. Pi tfa ll Ve nt ur e So la ris St ar G un ne r Vi de o Pi nb al l
100 10 0
10 100
DreamerV2 vs DQN
Figure E1: Atari agent comparison. The bars show the difference in gamer normalized scores at 200M steps. DreamerV2 outperforms the four model-free algorithms IQN, Rainbow, C51, and DQN while learning behaviors purely by planning within a separately learned world model. DreamerV2 achieves higher or similar performance on all tasks besides Video Pinball, where we hypothesize that the reconstruction loss does not focus on the ball that makes up only one pixel on the screen.

heading: F MODEL-FREE COMPARISON
text: 0.0 0.5 1.0 1.5 2.00 2000 4000 6000 8000
Alien
0.0 0.5 1.0 1.5 2.0 0
1000 2000 3000 4000
Amidar
0.0 0.5 1.0 1.5 2.0 0
10000 20000 30000 40000
Assault
0.0 0.5 1.0 1.5 2.0 0
30000 60000 90000
Asterix
0.0 0.5 1.0 1.5 2.0 40000
0 40000 80000
120000 Asteroids
0.0 0.5 1.0 1.5 2.0 0.00 0.25 0.50 0.75 1.00
1e6Atlantis
0.0 0.5 1.0 1.5 2.0 0
300 600 900
1200 Bank Heist
0.0 0.5 1.0 1.5 2.0
15000
30000
45000
Battle Zone
0.0 0.5 1.0 1.5 2.0 0
5000 10000 15000 20000
Beam Rider
0.0 0.5 1.0 1.5 2.0 200 400 600 800
Berzerk
0.0 0.5 1.0 1.5 2.0 15 30 45 60
Bowling
0.0 0.5 1.0 1.5 2.0
0 40 80
Boxing
0.0 0.5 1.0 1.5 2.0 0
150 300 450
Breakout
0.0 0.5 1.0 1.5 2.0 3000 6000 9000
12000 15000
Centipede
0.0 0.5 1.0 1.5 2.0 0
3000 6000 9000
12000
Chopper Com.
0.0 0.5 1.0 1.5 2.0
40000 80000
120000 160000
Crazy Climber
0.0 0.5 1.0 1.5 2.0 0
40000 80000
120000 160000
Demon Attack
0.0 0.5 1.0 1.5 2.0
15
0
15
Double Dunk
0.0 0.5 1.0 1.5 2.0 0
800 1600 2400
Enduro
0.0 0.5 1.0 1.5 2.0 80 40 0
40 80 Fishing Derby
0.0 0.5 1.0 1.5 2.0 0
10 20 30 40
Freeway
0.0 0.5 1.0 1.5 2.0 0
6000 12000 18000 24000
Frostbite
0.0 0.5 1.0 1.5 2.0 0
30000 60000 90000
120000 Gopher
0.0 0.5 1.0 1.5 2.0 0
1500 3000 4500
Gravitar
0.0 0.5 1.0 1.5 2.0 0
15000
30000
45000 Hero
0.0 0.5 1.0 1.5 2.0 15
0
15
30 Ice Hockey
0.0 0.5 1.0 1.5 2.0 0
15000 30000 45000 60000
James Bond
0.0 0.5 1.0 1.5 2.0 0
4000 8000
12000 16000
Kangaroo
0.0 0.5 1.0 1.5 2.0 0
25000 50000 75000
100000 Krull
0.0 0.5 1.0 1.5 2.0 0
20000 40000 60000 80000
Kung Fu Master
0.0 0.5 1.0 1.5 2.0 500
0 500
1000 1500 Montezuma Rev.
0.0 0.5 1.0 1.5 2.0
2000 4000 6000 8000
Ms Pacman
0.0 0.5 1.0 1.5 2.0
4000 8000
12000 16000 Name This Game
0.0 0.5 1.0 1.5 2.0 0
20000 40000 60000 80000
Phoenix
0.0 0.5 1.0 1.5 2.0 240 160 80 0
Pitfall
0.0 0.5 1.0 1.5 2.0 20 10 0
10 20
Pong
0.0 0.5 1.0 1.5 2.0 0
10000 20000 30000
Private Eye
0.0 0.5 1.0 1.5 2.0 100000
0 100000 200000 300000
Qbert
0.0 0.5 1.0 1.5 2.00 5000
10000 15000 20000
Riverraid
0.0 0.5 1.0 1.5 2.0 0
150000 300000 450000
Road Runner
0.0 0.5 1.0 1.5 2.00 20 40 60 80
Robotank
0.0 0.5 1.0 1.5 2.0 0
6000 12000 18000 24000
Seaquest
0.0 0.5 1.0 1.5 2.0 30000 24000 18000 12000 6000
Skiing
0.0 0.5 1.0 1.5 2.0 0
800 1600 2400
Solaris
0.0 0.5 1.0 1.5 2.0 0
2000 4000 6000
Space Invaders
0.0 0.5 1.0 1.5 2.0 0
25000 50000 75000
Star Gunner
0.0 0.5 1.0 1.5 2.0
15 0
15 30
Tennis
0.0 0.5 1.0 1.5 2.0 0
15000 30000 45000
Time Pilot
0.0 0.5 1.0 1.5 2.0
80 160 240
Tutankham
0.0 0.5 1.0 1.5 2.0 0
150000 300000 450000 600000
Up N Down
0.0 0.5 1.0 1.5 2.0 0
500 1000 1500
Venture
0.0 0.5 1.0 1.5 2.0 0
150000 300000 450000 600000
Video Pinball
0.0 0.5 1.0 1.5 2.0 0
6000 12000 18000 24000
Wizard Of Wor
0.0 0.5 1.0 1.5 2.0 0
50000 100000 150000 200000
Yars Revenge
0.0 0.5 1.0 1.5 2.0 0
15000 30000 45000 60000
Zaxxon
0.0 0.5 1.0 1.5 2.0 0.0 0.6 1.2 1.8 2.4 Gamer Median 0.0 0.5 1.0 1.5 2.0 0 10 20 30 40 Gamer Mean 0.0 0.5 1.0 1.5 2.0 0.00 0.15 0.30 0.45 Record Mean 0.0 0.5 1.0 1.5 2.0 0.00 0.08 0.16 0.24 Clip Record Mean
DreamerV2 IQN Rainbow Figure F1: Comparison of DreamerV2 to the top model-free RL methods IQN and Rainbow. The curves show mean and standard deviation over 5 seeds. IQN and Rainbow additionally average each point over 10 evaluation episodes, explaining the smoother curves. DreamerV2 outperforms IQN and Rainbow in all four aggregated scores. While IQN and Rainbow tend to succeed on the same tasks, DreamerV2 shows a different performance profile.

heading: G LATENTS AND KL BALANCING ABLATIONS
text: 0.0 0.5 1.0 1.5 2.00
1500
3000
4500 Alien
0.0 0.5 1.0 1.5 2.0 0
800 1600 2400 3200
Amidar
0.0 0.5 1.0 1.5 2.0 0
3000 6000 9000
12000 Assault
0.0 0.5 1.0 1.5 2.0 0
8000 16000 24000 32000
Asterix
0.0 0.5 1.0 1.5 2.0 0
4000 8000
12000 16000
Asteroids
0.0 0.5 1.0 1.5 2.0 0
250000 500000 750000
1000000 Atlantis
0.0 0.5 1.0 1.5 2.0 0
400 800
1200 1600
Bank Heist
0.0 0.5 1.0 1.5 2.00 10000 20000 30000 40000
Battle Zone
0.0 0.5 1.0 1.5 2.0 0
4000 8000
12000
Beam Rider
0.0 0.5 1.0 1.5 2.0 200 400 600 800
1000 Berzerk
0.0 0.5 1.0 1.5 2.0 20 40 60 80
100 Bowling
0.0 0.5 1.0 1.5 2.030 0
30 60 90
Boxing
0.0 0.5 1.0 1.5 2.0 0
150
300
450 Breakout
0.0 0.5 1.0 1.5 2.00 2500 5000 7500
10000 Centipede
0.0 0.5 1.0 1.5 2.0
1500 3000 4500
Chopper Com.
0.0 0.5 1.0 1.5 2.0
40000 80000
120000 160000
Crazy Climber
0.0 0.5 1.0 1.5 2.0 6000
0 6000
12000
Demon Attack
0.0 0.5 1.0 1.5 2.0 20 10 0
10 20 Double Dunk
0.0 0.5 1.0 1.5 2.0 0
600 1200 1800 2400
Enduro
0.0 0.5 1.0 1.5 2.0 80 40 0
40
Fishing Derby
0.0 0.5 1.0 1.5 2.0
8 16 24 32
Freeway
0.0 0.5 1.0 1.5 2.0 0
20000 40000 60000 80000
Frostbite
0.0 0.5 1.0 1.5 2.0 0
25000 50000 75000
100000 Gopher
0.0 0.5 1.0 1.5 2.0 0
1500
3000
4500 Gravitar
0.0 0.5 1.0 1.5 2.0 0
8000 16000 24000 32000
Hero
0.0 0.5 1.0 1.5 2.0 15
0
15
30 Ice Hockey
0.0 0.5 1.0 1.5 2.0 0
6000 12000 18000
James Bond
0.0 0.5 1.0 1.5 2.0 0
4000 8000
12000
Kangaroo
0.0 0.5 1.0 1.5 2.02000 4000 6000 8000
10000 Krull
0.0 0.5 1.0 1.5 2.00 25000 50000 75000
100000 Kung Fu Master
0.0 0.5 1.0 1.5 2.0 0
800 1600 2400 3200 Montezuma Rev.
0.0 0.5 1.0 1.5 2.00 1500 3000 4500 6000
Ms Pacman
0.0 0.5 1.0 1.5 2.0 4000 8000
12000 16000 Name This Game
0.0 0.5 1.0 1.5 2.0 0
10000 20000 30000
Phoenix
0.0 0.5 1.0 1.5 2.0 240 160 80 0
Pitfall
0.0 0.5 1.0 1.5 2.0 20 10 0
10 20
Pong
0.0 0.5 1.0 1.5 2.0 0
1500 3000 4500 6000
Private Eye
0.0 0.5 1.0 1.5 2.0 0
150000 300000 450000
Qbert
0.0 0.5 1.0 1.5 2.0 4000 8000
12000 16000
Riverraid
0.0 0.5 1.0 1.5 2.0 0
40000 80000
120000 160000
Road Runner
0.0 0.5 1.0 1.5 2.0 0
20 40 60
Robotank
0.0 0.5 1.0 1.5 2.0 80000
0 80000
160000
Seaquest
0.0 0.5 1.0 1.5 2.0 30000 24000 18000 12000 6000
Skiing
0.0 0.5 1.0 1.5 2.0 0
1500 3000 4500 6000
Solaris
0.0 0.5 1.0 1.5 2.00 1000 2000 3000 4000 Space Invaders 0.0 0.5 1.0 1.5 2.0 0 10000 20000 30000 Star Gunner
0.0 0.5 1.0 1.5 2.0
15 0
15 30
Tennis
0.0 0.5 1.0 1.5 2.0 0
15000
30000
45000 Time Pilot
0.0 0.5 1.0 1.5 2.060 120 180 240 300
Tutankham
0.0 0.5 1.0 1.5 2.0 0
150000 300000 450000 600000
Up N Down
0.0 0.5 1.0 1.5 2.0 20 0
20 40 60
Venture
0.0 0.5 1.0 1.5 2.0 0
15000 30000 45000
Video Pinball
0.0 0.5 1.0 1.5 2.0 0
6000 12000 18000 24000
Wizard Of Wor 0.0 0.5 1.0 1.5 2.00 20000 40000 60000 80000Yars Revenge
0.0 0.5 1.0 1.5 2.0 0
5000 10000 15000 20000
Zaxxon
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Gamer Median 0.0 0.5 1.0 1.5 2.0 0 6 12 18 24 Gamer Mean 0.0 0.5 1.0 1.5 2.0 0.0 0.1 0.2 0.3 0.4 Record Mean 0.0 0.5 1.0 1.5 2.0 0.00 0.06 0.12 0.18 0.24 Clip Record Mean
DreamerV2 Gaussian Latents No KL Balance Figure G1: Comparison of DreamerV2, Gaussian instead of categorical latent variables, and no KL balancing. The ablation experiments use a slightly earlier version of the agent. The curves show mean and standard deviation across two seeds. Categorical latent variables and KL balancing both substantially improve performance across many of the tasks. The importance of the two techniques is reflected in all four aggregated scores.

heading: H REPRESENTATION LEARNING ABLATIONS
text: 0.0 0.5 1.0 1.5 2.0 0
2000 4000 6000
Alien
0.0 0.5 1.0 1.5 2.0 0
800 1600 2400 3200
Amidar
0.0 0.5 1.0 1.5 2.0 0
4000 8000
12000
Assault
0.0 0.5 1.0 1.5 2.0 0
10000 20000 30000
Asterix
0.0 0.5 1.0 1.5 2.0 0
1500 3000 4500 6000
Asteroids
0.0 0.5 1.0 1.5 2.0 0
250000 500000 750000
1000000 Atlantis
0.0 0.5 1.0 1.5 2.0 0
300 600 900
1200 Bank Heist
0.0 0.5 1.0 1.5 2.0 0
10000 20000 30000 40000
Battle Zone
0.0 0.5 1.0 1.5 2.0 0
4000 8000
12000
Beam Rider
0.0 0.5 1.0 1.5 2.0 200 400 600 800
1000 Berzerk
0.0 0.5 1.0 1.5 2.0 0
20 40 60
Bowling
0.0 0.5 1.0 1.5 2.0 100 50 0
50 100
Boxing
0.0 0.5 1.0 1.5 2.0 0
150 300 450
Breakout
0.0 0.5 1.0 1.5 2.0 2000 4000 6000 8000
10000 Centipede
0.0 0.5 1.0 1.5 2.00 2000 4000 6000 8000
Chopper Com.
0.0 0.5 1.0 1.5 2.00 40000 80000
120000 160000
Crazy Climber
0.0 0.5 1.0 1.5 2.0 6000
0 6000
12000 18000
Demon Attack
0.0 0.5 1.0 1.5 2.0
15
0
15
Double Dunk
0.0 0.5 1.0 1.5 2.0 0
600 1200 1800 2400
Enduro
0.0 0.5 1.0 1.5 2.0 80 40 0
40
Fishing Derby
0.0 0.5 1.0 1.5 2.0 0 8
16 24 32
Freeway
0.0 0.5 1.0 1.5 2.0 0
8000 16000 24000 32000
Frostbite
0.0 0.5 1.0 1.5 2.0 0
25000 50000 75000
100000 Gopher
0.0 0.5 1.0 1.5 2.0 0
1500
3000
4500 Gravitar
0.0 0.5 1.0 1.5 2.0 0
8000 16000 24000 32000
Hero
0.0 0.5 1.0 1.5 2.0
15 0
15 30
Ice Hockey
0.0 0.5 1.0 1.5 2.0 0
6000 12000 18000
James Bond
0.0 0.5 1.0 1.5 2.0 0
4000 8000
12000 16000
Kangaroo
0.0 0.5 1.0 1.5 2.0 2500 5000 7500
10000 Krull
0.0 0.5 1.0 1.5 2.0 0
25000 50000 75000
100000 Kung Fu Master
0.0 0.5 1.0 1.5 2.0 0
800 1600 2400 3200 Montezuma Rev.
0.0 0.5 1.0 1.5 2.00 2000 4000 6000 8000
Ms Pacman
0.0 0.5 1.0 1.5 2.0 0
4000 8000
12000 16000 Name This Game
0.0 0.5 1.0 1.5 2.0 0
10000 20000 30000
Phoenix
0.0 0.5 1.0 1.5 2.0 600 450 300 150
0 Pitfall
0.0 0.5 1.0 1.5 2.0 20 10 0
10 20
Pong
0.0 0.5 1.0 1.5 2.0 1000
0 1000 2000
Private Eye
0.0 0.5 1.0 1.5 2.0 0
150000 300000 450000
Qbert
0.0 0.5 1.0 1.5 2.0 0
5000 10000 15000 20000
Riverraid
0.0 0.5 1.0 1.5 2.0 0
80000 160000 240000 320000
Road Runner
0.0 0.5 1.0 1.5 2.0 0
20 40 60 80
Robotank
0.0 0.5 1.0 1.5 2.0 80000
0 80000
160000
Seaquest
0.0 0.5 1.0 1.5 2.0 30000 24000 18000 12000 6000
Skiing
0.0 0.5 1.0 1.5 2.0 0
2000 4000 6000
Solaris
0.0 0.5 1.0 1.5 2.0 0
1000 2000 3000 4000 Space Invaders
0.0 0.5 1.0 1.5 2.0 0
10000 20000 30000
Star Gunner
0.0 0.5 1.0 1.5 2.0
15 0
15 30
Tennis
0.0 0.5 1.0 1.5 2.0 0
15000
30000
45000 Time Pilot
0.0 0.5 1.0 1.5 2.0 0
80 160 240 320
Tutankham
0.0 0.5 1.0 1.5 2.0 0
150000 300000 450000 600000
Up N Down
0.0 0.5 1.0 1.5 2.0 20 0
20 40 60
Venture
0.0 0.5 1.0 1.5 2.0 0
20000 40000 60000 80000
Video Pinball
0.0 0.5 1.0 1.5 2.0 0
6000 12000 18000 24000
Wizard Of Wor
0.0 0.5 1.0 1.5 2.0 0
25000 50000 75000
100000 Yars Revenge
0.0 0.5 1.0 1.5 2.0 0
5000 10000 15000 20000
Zaxxon
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Gamer Median 0.0 0.5 1.0 1.5 2.0 0 6 12 18 24 Gamer Mean 0.0 0.5 1.0 1.5 2.0 0.0 0.1 0.2 0.3 0.4 Record Mean 0.0 0.5 1.0 1.5 2.0 0.00 0.08 0.16 0.24 Clip Record Mean
DreamerV2 No Reward Gradients No Image Gradients Figure H1: Comparison of leveraging image prediction, reward prediction, or both for learning the model representations. While image gradients are crucial, reward gradients are not necessary for our world model to succeed and their gradients can be stopped. Representations learned purely from images are not biased toward previously encountered rewards and outperform reward-specific representations on a number of tasks, suggesting that they may generalize better to unseen situations.

heading: I POLICY LEARNING ABLATIONS
text: 0.0 0.5 1.0 1.5 2.0 0
2500 5000 7500
10000 Alien
0.0 0.5 1.0 1.5 2.0 0
800 1600 2400 3200
Amidar
0.0 0.5 1.0 1.5 2.0 0
3000 6000 9000
12000 Assault
0.0 0.5 1.0 1.5 2.0 0
8000 16000 24000 32000
Asterix
0.0 0.5 1.0 1.5 2.0 0
1500 3000 4500 6000
Asteroids
0.0 0.5 1.0 1.5 2.0 0
250000 500000 750000
1000000 Atlantis
0.0 0.5 1.0 1.5 2.0 0
300 600 900
1200 Bank Heist
0.0 0.5 1.0 1.5 2.0
10000 20000 30000 40000
Battle Zone
0.0 0.5 1.0 1.5 2.0 0
4000 8000
12000 16000
Beam Rider
0.0 0.5 1.0 1.5 2.0 250 500 750
1000 1250
Berzerk
0.0 0.5 1.0 1.5 2.0 15 30 45 60 75
Bowling
0.0 0.5 1.0 1.5 2.0 0
30 60 90
Boxing
0.0 0.5 1.0 1.5 2.0 0
150
300
450 Breakout
0.0 0.5 1.0 1.5 2.00 2500 5000 7500
10000 Centipede
0.0 0.5 1.0 1.5 2.0
1500 3000 4500 6000
Chopper Com.
0.0 0.5 1.0 1.5 2.0 0
40000 80000
120000 160000
Crazy Climber
0.0 0.5 1.0 1.5 2.08000 0
8000 16000
Demon Attack
0.0 0.5 1.0 1.5 2.0 20 10 0
10 20 Double Dunk
0.0 0.5 1.0 1.5 2.0 0
600 1200 1800 2400
Enduro
0.0 0.5 1.0 1.5 2.0 80 40 0
40
Fishing Derby
0.0 0.5 1.0 1.5 2.0 0 8
16 24 32
Freeway
0.0 0.5 1.0 1.5 2.0 0
8000 16000 24000 32000
Frostbite
0.0 0.5 1.0 1.5 2.0 0
25000 50000 75000
100000 Gopher
0.0 0.5 1.0 1.5 2.0 0
1500
3000
4500 Gravitar
0.0 0.5 1.0 1.5 2.00 8000
16000 24000 32000
Hero
0.0 0.5 1.0 1.5 2.0 15
0
15
30 Ice Hockey
0.0 0.5 1.0 1.5 2.0 0
6000 12000 18000
James Bond
0.0 0.5 1.0 1.5 2.0 0
4000 8000
12000
Kangaroo
0.0 0.5 1.0 1.5 2.0 2500 5000 7500
10000
Krull
0.0 0.5 1.0 1.5 2.00 25000 50000 75000
100000 Kung Fu Master
0.0 0.5 1.0 1.5 2.0 0
800 1600 2400 3200 Montezuma Rev.
0.0 0.5 1.0 1.5 2.0 1500 3000 4500 6000
Ms Pacman
0.0 0.5 1.0 1.5 2.0
4000 8000
12000 16000 Name This Game
0.0 0.5 1.0 1.5 2.0 0
10000 20000 30000
Phoenix
0.0 0.5 1.0 1.5 2.0 240 160 80 0
Pitfall
0.0 0.5 1.0 1.5 2.0 20 10 0
10 20
Pong
0.0 0.5 1.0 1.5 2.0 250
0 250 500 750
Private Eye
0.0 0.5 1.0 1.5 2.0 0
150000 300000 450000
Qbert
0.0 0.5 1.0 1.5 2.0
5000 10000 15000 20000
Riverraid
0.0 0.5 1.0 1.5 2.0 0
150000
300000
450000 Road Runner
0.0 0.5 1.0 1.5 2.0 0
15 30 45 60
Robotank
0.0 0.5 1.0 1.5 2.0 80000
0 80000
160000
Seaquest
0.0 0.5 1.0 1.5 2.0 30000 24000 18000 12000 6000
Skiing
0.0 0.5 1.0 1.5 2.0 0
1500
3000
4500 Solaris
0.0 0.5 1.0 1.5 2.0 0
1500 3000 4500 6000 Space Invaders
0.0 0.5 1.0 1.5 2.0 0
10000 20000 30000 40000
Star Gunner
0.0 0.5 1.0 1.5 2.0 15 0
15 30
Tennis
0.0 0.5 1.0 1.5 2.0 0
15000 30000 45000
Time Pilot
0.0 0.5 1.0 1.5 2.00 80
160 240 320
Tutankham
0.0 0.5 1.0 1.5 2.0 0
150000 300000 450000 600000
Up N Down
0.0 0.5 1.0 1.5 2.0 20 0
20 40 60
Venture
0.0 0.5 1.0 1.5 2.00 15000 30000 45000 60000
Video Pinball
0.0 0.5 1.0 1.5 2.0 0
6000 12000 18000 24000
Wizard Of Wor
0.0 0.5 1.0 1.5 2.00 20000 40000 60000 80000
Yars Revenge
0.0 0.5 1.0 1.5 2.0 0
8000 16000 24000 32000
Zaxxon
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Gamer Median 0.0 0.5 1.0 1.5 2.0 0 6 12 18 24 Gamer Mean 0.0 0.5 1.0 1.5 2.0 0.0 0.1 0.2 0.3 0.4 Record Mean 0.0 0.5 1.0 1.5 2.0 0.00 0.06 0.12 0.18 0.24 Clip Record Mean
DreamerV2 No Straight-Through No ReinforceFigure I1: Comparison of leveraging Reinforce gradients, straight-through gradients, or both for training the actor. While Reinforce gradients are crucial, straight-through gradients are not important for most of the tasks. Nonetheless, combining both gradients yields substantial improvements on a small number of games, most notably on Seaquest. We conjecture that straight-through gradients have low variance and thus help the agent start learning, whereas Reinforce gradients are unbiased and help converging to a better solution.

heading: J ADDITIONAL ABLATIONS
text: 0.0 0.5 1.0 1.5 2.00
1500
3000
4500 Alien
0.0 0.5 1.0 1.5 2.0 0
800 1600 2400 3200
Amidar
0.0 0.5 1.0 1.5 2.0 0
3000 6000 9000
12000 Assault
0.0 0.5 1.0 1.5 2.0 0
8000 16000 24000 32000
Asterix
0.0 0.5 1.0 1.5 2.0 0
1500 3000 4500 6000
Asteroids
0.0 0.5 1.0 1.5 2.0 0
250000 500000 750000
1000000 Atlantis
0.0 0.5 1.0 1.5 2.0 0
300 600 900
1200 Bank Heist
0.0 0.5 1.0 1.5 2.00 15000 30000 45000
Battle Zone
0.0 0.5 1.0 1.5 2.0 0
4000 8000
12000
Beam Rider
0.0 0.5 1.0 1.5 2.0 250 500 750
1000 1250
Berzerk
0.0 0.5 1.0 1.5 2.0 15 30 45 60 75
Bowling
0.0 0.5 1.0 1.5 2.0 0
30 60 90
Boxing
0.0 0.5 1.0 1.5 2.0 0
150
300
450 Breakout
0.0 0.5 1.0 1.5 2.0 0
2500 5000 7500
10000 Centipede
0.0 0.5 1.0 1.5 2.0
1000 2000 3000 4000 5000
Chopper Com.
0.0 0.5 1.0 1.5 2.0
40000 80000
120000 160000
Crazy Climber
0.0 0.5 1.0 1.5 2.08000 0
8000 16000 24000
Demon Attack
0.0 0.5 1.0 1.5 2.0
15
0
15
Double Dunk
0.0 0.5 1.0 1.5 2.0 0
600 1200 1800 2400
Enduro
0.0 0.5 1.0 1.5 2.0 80 40 0
40 80 Fishing Derby
0.0 0.5 1.0 1.5 2.0 0 8
16 24 32
Freeway
0.0 0.5 1.0 1.5 2.0 0
8000 16000 24000 32000
Frostbite
0.0 0.5 1.0 1.5 2.0 0
25000 50000 75000
100000 Gopher
0.0 0.5 1.0 1.5 2.0 0
1500 3000 4500
Gravitar
0.0 0.5 1.0 1.5 2.00 10000 20000 30000 40000
Hero
0.0 0.5 1.0 1.5 2.0 15 0
15 30 45
Ice Hockey
0.0 0.5 1.0 1.5 2.0 0
8000 16000 24000 32000
James Bond
0.0 0.5 1.0 1.5 2.0 0
4000 8000
12000
Kangaroo
0.0 0.5 1.0 1.5 2.0 4000 6000 8000
10000 Krull
0.0 0.5 1.0 1.5 2.0 0
30000 60000 90000
120000 Kung Fu Master
0.0 0.5 1.0 1.5 2.0 0
800 1600 2400 3200 Montezuma Rev.
0.0 0.5 1.0 1.5 2.00 1500 3000 4500 6000
Ms Pacman
0.0 0.5 1.0 1.5 2.0 4000 8000
12000 16000 Name This Game
0.0 0.5 1.0 1.5 2.0 0
10000 20000 30000
Phoenix
0.0 0.5 1.0 1.5 2.0 240 160 80 0
Pitfall
0.0 0.5 1.0 1.5 2.0 20 10 0
10 20
Pong
0.0 0.5 1.0 1.5 2.0 0
5000 10000 15000 20000
Private Eye
0.0 0.5 1.0 1.5 2.0 0
150000 300000 450000
Qbert
0.0 0.5 1.0 1.5 2.0 4000 8000
12000 16000
Riverraid
0.0 0.5 1.0 1.5 2.0 0
60000 120000 180000 240000
Road Runner
0.0 0.5 1.0 1.5 2.0 0
15 30 45 60
Robotank
0.0 0.5 1.0 1.5 2.0 80000
0 80000
160000
Seaquest
0.0 0.5 1.0 1.5 2.0 30000 24000 18000 12000 6000
Skiing
0.0 0.5 1.0 1.5 2.0 0
1500 3000 4500
Solaris
0.0 0.5 1.0 1.5 2.00 1000 2000 3000 4000 Space Invaders 0.0 0.5 1.0 1.5 2.0 0 10000 20000 30000 40000 Star Gunner
0.0 0.5 1.0 1.5 2.0
15 0
15 30
Tennis
0.0 0.5 1.0 1.5 2.0 0
15000
30000
45000 Time Pilot
0.0 0.5 1.0 1.5 2.0
80 160 240 320
Tutankham
0.0 0.5 1.0 1.5 2.0 0
150000 300000 450000 600000
Up N Down
0.0 0.5 1.0 1.5 2.0 20 0
20 40 60
Venture
0.0 0.5 1.0 1.5 2.0 0
15000
30000
45000 Video Pinball
0.0 0.5 1.0 1.5 2.0 0
6000 12000 18000 24000
Wizard Of Wor
0.0 0.5 1.0 1.5 2.0 0
8000 16000 24000 32000
Yars Revenge
0.0 0.5 1.0 1.5 2.0 0
5000 10000 15000 20000
Zaxxon
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Gamer Median 0.0 0.5 1.0 1.5 2.0 0 8 16 24 32 Gamer Mean 0.0 0.5 1.0 1.5 2.0 0.0 0.1 0.2 0.3 0.4 Record Mean 0.0 0.5 1.0 1.5 2.0 0.00 0.08 0.16 0.24 Clip Record Mean
DreamerV2 No Layer Norm Random Data Figure J1: Comparison of DreamerV2 to a version without layer norm in the GRU and to training from experience collected over time by a uniform random policy. We find that the benefit of layer norm depends on the task at hand, increasing and decreasing performance on a roughly equal number of tasks. The comparison to random data collection highlights which of the tasks require non-trivial exploration, which can help guide future work on directed exploration using world models.

heading: K ATARI TASK SCORES
text: Baselines Algorithms Task Random Gamer Record Rainbow IQN DreamerV2
Alien 229 7128 251916 3457 4961 3967 Amidar 6 1720 104159 2529 2393 2577 Assault 222 742 8647 3229 4885 23625 Asterix 210 8503 1000000 18367 10374 72311 Asteroids 719 47389 10506650 1484 1585 41526 Atlantis 12850 29028 10604840 802548 890214 978778 Bank Heist 14 753 82058 1075 1052 1126 Battle Zone 2360 37188 801000 40061 40953 40325 Beam Rider 364 16926 999999 6290 7130 18646 Berzerk 124 2630 1057940 833 648 810 Bowling 23 161 300 43 39 49 Boxing 0 12 100 99 98 92 Breakout 2 30 864 120 79 312 Centipede 2091 12017 1301709 6510 3728 11883 Chopper Command 811 7388 999999 12338 9282 2861 Crazy Climber 10780 35829 219900 145389 132738 161839 Demon Attack 152 1971 1556345 17071 15350 82263 Double Dunk -19 -16 22 22 21 17 Enduro 0 860 9500 2200 2203 1656 Fishing Derby -92 -39 71 42 45 65 Freeway 0 30 38 34 34 33 Frostbite 65 4335 454830 8208 7812 11384 Gopher 258 2412 355040 10641 12108 92282 Gravitar 173 3351 162850 1272 1347 3789 Hero 1027 30826 1000000 46675 36058 21868 Ice Hockey -11 1 36 0 -5 26 James Bond 7 29 45550 1097 3166 40445 Kangaroo 52 3035 1424600 12748 12602 14064 Krull 1598 2666 104100 4066 8844 50061 Kung Fu Master 258 22736 1000000 26475 31653 62741 Montezuma Revenge 0 4753 1219200 500 500 81 Ms Pacman 307 6952 290090 3861 5218 5652 Name This Game 2292 8049 25220 9026 6639 14649 Phoenix 761 7243 4014440 8545 5102 49375 Pitfall -229 6464 114000 -20 -13 0 Pong -21 15 21 20 20 20 Private Eye 25 69571 101800 21334 4181 2198 Qbert 164 13455 2400000 17383 16730 94688 Riverraid 1338 17118 1000000 20756 15183 16351 Road Runner 12 7845 2038100 54662 58966 203576 Robotank 2 12 76 66 66 78 Seaquest 68 42055 999999 9903 17039 7480 Skiing -17098 -4337 -3272 -28708 -11162 -9299 Solaris 1236 12327 111420 1583 1684 922 Space Invaders 148 1669 621535 4131 4530 2474 Star Gunner 664 10250 77400 57909 80003 7800 Tennis -24 -8 21 0 23 14 Time Pilot 3568 5229 65300 12051 11666 37945 Tutankham 11 168 5384 239 251 264 Up N Down 533 11693 82840 34888 59944 653662 Venture 0 1188 38900 1529 1313 2 Video Pinball 16257 17668 89218328 466895 415833 41860 Wizard Of Wor 564 4756 395300 7879 5671 12851 Yars Revenge 3093 54577 15000105 45542 84144 156748 Zaxxon 32 9173 83700 14603 11023 50699
Table K1: Atari individual scores. We select the 55 games that are common among most papers in the literature. We compare the algorithms DreamerV2, IQN, and Rainbow to the baselines of random actions, DeepMind’s human gamer, and the human world record. Algorithm scores are highlighted in bold when they fall within 5% of the best algorithm. Note that these scores are already averaged across seeds, whereas any aggregated scores must be computed before averaging across seeds.

Review summaries:
The authors introduce DreamerV2, a modification of the influential Dreamer RL agent (hereafter refered to as DreamerV1). The primary changes from DreamerV1 are a discrete latent space and a modified loss function (and with it, a modified optimization scheme). As in DreamerV1, the agent trains a world model with environment experience, and the policy is learned by "imagining" within the learned latent space using the world model to simulate transitions and rewards. They demonstrate superior performance over a variety of successful benchmarks that use similar compute (1 GPU, 1 environment -- e.g. MuZero, which requires vastly more, is not considered) on Atari. Further, they analyze several ways of aggregating Atari scores, and (while their algorithm performs best of those tried in each aggregation), they recommend one aggregation method (along with several other choices made for benchmarking) going forward.

The authors build on the Dreamer architecture, that is able to learn models of an environment, to build DreamerV2, which learns a model of an environment in latent space. The authors then train their agent in this latent space. DreamerV2 was evaluated on the Atari learning environment and results showed that it was comparable to Rainbow and better, under certain metrics.