Context-Aware Prediction With Uncertainty of Traffic Behaviors Using Mixture Density Networks


In autonomous driving, prediction is a measure of the future behavior of “agents” on the road. This typically includes behaviors (such as “lane following”, “turning left”, “turning right”), coupled with predicted trajectories consisting of (x, y) coordiantes. These predictions are then used by the Autonomous Vehicle’s (AV’s) planning and decision making modules, to ensure safe and optimal maneuvers that avoid collisions — for example, if a vehicle is predicted to come into the AV’s lane, the AV should respond accordingly by slowing down and allowing them to enter. The prediction problem can be complex, as scenes on the road can evolve very uniquely depending on the context such as number of agents on the road, road features, weather conditions, and so on. Additionally, the AV’s own behaviors affect the agents around it that predictions are sought for.

There is a plenty of literature proposing solutions to the traffic prediction problem using neural networks ([1], [2], [3] and many others) or probabilistic graphical models ([4], [5] and many others). However, I have not come across any work on solving it using Mixture Density Networks.

Mixture Density Networks (MDNs) [6], which I covered in a previous blog post, are neural networks that output the parameters of a mixture model, such as a Gaussian Mixture Model. MDNs are useful for tasks where a given input can potentially produce multiple outputs according to some probability. My post on MDNs shows the iconic toy problem where given an x coordinate, there could be multiple y coordinates. For example, given x=0.25, y could be {0, 0.5, 1} with perhaps equal probability of 1/3, 1/3, 1/3 per y coordinate. If such a problem is solved with neural networks trained with Mean Squared Error to produce a singular y output, it will likely only lead to the average y values — in the example aforementioned, y=0.5 for x=0.25.

Toy dataset of (x, y) coordiantes where certain x values map to multiple y values. The first image shows a 2D graph of the dataset’s ground truths. The second image shows the predictions of output y (in red) given input x, of a neural network trained on the dataset using Mean Squared Error. The third image shows the outputs (red) of a Mixture Density Network trained on the same dataset. The Mixture Density Network is adequately able to learn the distribution of the data.

The above example extends to traffic prediction very naturally. Let’s set up an example scene to explain why. An agent (a vehicle we are predicting for) approaches a Y-split in the road, and can go either right, or left. Depending on the agent’s past trajectory, perhaps one of the two options has a higher probability, though either is still probable up to a certain point. In this case, say we train a neural network regressor at this specific Y-split on past data of cars flowing through, using Mean Squared Error, to output a singular trajectory.

We can frame the inputs at timestep t as a series of (x, y) coordinates representing the past trajectory of the agent up to some P timesteps in the past, thus X_t = \{ (x_{t-P}, y_{t-P}), (x_{t-P+1}, y_{t-P+1}), ..., (x_t, y_t) \}. And the ouputs, the prediction, can be the agent’s trajectory up to some time horizon H in the future again consisting of coordinates at each timestep, thus Y_t = \{ (x_{t+1}, y_{t+1}), (x_{t+2}, y_{t+2}), ..., (x_{t+H}, y_{t+H}) \}.

Take the scenario of traffic at a Y-split where agents can either go right or left with a 50/50 probability (up to a certain point before the Y-split) given a similar past trajectory. And say we have a balanced dataset of examples of these. Again, as seen in the toy example above, the Neural Network regressor trained with Mean Squared Error or the like, will end up giving an average of the two possible future trajectories, as depicted below. The image on the left shows the typical trajectories agents can take at this scene, and the image on the right shows the likely output trajectory of a vanilla neural network regressor trained on this scene.

As seen above, traffic scene prediction is a good candidate for where Mixture Density Networks can be used. In traffic scene prediction, we need a distribution over behaviors an agent can exhibit — for example, an agent could turn left, turn right, or go straight. Thus, Mixture Density Networks can be used to represent “behaviors” in each of the mixture it learns, where the behavior consists of a probability and trajectory ((x, y) coordinates up to some time horizon in the future). In the autonomous driving domain, it is helpful to have not just the predictions (the trajectories of agents around an autonomous vehicle), but the uncertainty attached to predictions, for better planning and decision making. A Mixture Density Network is a good candidate here additionally, as it will output a mixture of behaviors that have a probability (certainty) attached to each along with a standard deviation over the trajectories.

Problem Definition

In autonomous driving, data is typically gathered by the perception module, which consists of reading and logging sensors such as through video cameras, LiDaR, Radar, and so on. These sensors are meant to perceive the world and the state around the Autonomous Vehicle (AV), which include perceiving other agents on the road (vehicles, pedestrians, etc). In prediction (and other modules in the AV), raw sensor data such as images or point clouds can be used as input. Though I’ve read prediction papers where this raw data is used as input, a landmark paper by Waymo points to a number of papers that “show the brittleness of using raw sensor data” for learning to drive by means of prediction and planning [1] . Thus, some alternate pre-processing of raw sensor data is necessary. This data can be used to feed through neural network based object detectors, and a tracking module that uses perhaps the likes of Kalman filters to create traces of other agents on the road — individual agents tracked over time, using numeric representations of their dimensions (the size of the agent in meters, for example), position as a (x, y) coordinate, heading, velocity, and so on. Additionally, an HD map of the current location which includes features of the environment such as lanes represented as a series of (x, y) coordinates, can be used to place the perceived agents in the scene around the AV.

Example of an intermediate rasterized representation of a particular traffic scene at a particular timestep [2].

Where work on traffic prediction by Waymo [1], Uber [2], and others [3], create an intermediate rasterized (image) representation using this pre-processed data to feed to a convolutional network, I propose keeping it simple and feeding these numbers directly into a neural network without an intermediate representation expressed as an image. Thus, the input, X can consists of a series of vectors representing the past positions (past trajectory) up to the current timestep, heading at the current timestep, and velocity at the current timestep, of individual agents in the scene. These inputs can be mapped to outputs, Y — predictions of where the agents will be in some future time horizon such in the next 2 to 8 seconds from the present moment (their future trajectory).

Context is paramount in traffic prediction. For example, you observe a cyclist in a dedicated bike lane on a road going straight. And you also observe a vehicle stopped in the bike lane some distance ahead of the cyclist. Without taking the stopped vehicle into account, you would likely not be able to predict that the cyclist will have to pass the stopped vehicle by leaving the bike lane and coming into the vehicle traffic lanes. Thus, context must include other agents that may or may not affect a particular agent we are predicting for. Features of the current scene also provide important context, that must be taken into account. For example, the contours of the lane (if they curve right, an agent following the lane will need to adjust accordingly), locations and states of traffic lights (traffic light is red, so agents will likely stop as they approach), traffic rules (one way roads, etc), and so on.

For the traffic prediction problem, I propose a single feedforward neural network that at a given timestep t can take up to some N agents as inputs (their past trajectories up to some t-P timesteps) and produce N-1 outputs for predictions of their future trajectories up to some time horizon t+H. The N agents include the ego vehicle (the AV), as the position and movement of the ego vehicle itself may affect other agents around it and can provide context for predictions for the others, and then N-1 of the closest agents to the ego based on euclidean distance, as predictions for irrelevant agents such as those much further away than the AV are not necessary. This consolidated input will provide multi-agent context to the network, allowing it to learn interactions among agents and how that relates to predictions of their future trajectories. The outputs need only include the trajectories of the N-1 closest agents, as we need not predict for the AV.

More specifically, at a given timestep t, the input for each agent, X, can include their past trajectory of (x, y) coordinates up to some P previous timesteps, their heading, their velocity, and additionally I propose the lane contours of the lane they are currently in expressed as (x, y) coordinates. Thus, for each agent n, X^n_t = \{ (x_{t-P}, y_{t-P}), (x_{t-P+1}, y_{t-P+1}), ..., (x_t, y_t), heading_t, veolcity_t, (x^{lane}_1, y^{lane}_1),  (x^{lane}_2, y^{lane}_2), ..., (x^{lane}_?, y^{lane}_?) \}. The outputs for each agent n includes their trajectory up to some future time horizon H, thus Y^n_t = \{ (x_{t+1}, y_{t+1}), (x_{t+2}, y_{t+2}), ..., (x_{t+H}, y_{t+H}) \}. Following these, the full input to the neural network at a single timestep would be the ego-centric (from the perspective of the AV) “scene” I_{t} = \{ X^1_t, X^2_t, ..., X^N_t \} and full output would be O_{t} = \{ Y^1_t, Y^2_t, ..., Y^{N-1}_t \}. Note that in order to make this a feasible and somewhat consistent (“normalized”) problem for the neural network to learn, all euclidean (x, y) coordinates can be made relevant to the AV’s current position at the current timestep. That is, setting (x^0_{t=0}, y^0_{t=0}) = (0.0, 0.0), and having all other coordinates of all agents trajectory positions and lane contours offset relatively.

As a simple feedforward neural network (as opposed to a more complex LSTM setup) will be used, unfortunately the input and output sizes will need to be fixed. This is somewhat problematic because we have to know up front how many agents are in a given scene, and additionally, how many behaviors we want the MDN to predict (as in, number of mixtures — more on that below). For example, if the network is trained using input that includes the ego vehicle plus the 9 closest agents, but only 3 other agents are observed. However, this can be overcome as we can zero-pad all inputs and outputs if none are available at any given timestep. In the aforementioned example, all inputs and outputs for the additional agents can just consist of all 0’s.

Mixture Density Network Setup

As a MDN is being used for this problem, the neural network does not output O_{t} directly. Instead, it outputs a set of mixture coefficients (probabilities attached to each mixture) \alpha, means \mu, and standard deviations \sigma — the parameters of a Gaussian Mixture Model (GMM). These parameters can then be sampled from to get O_{t} for predicted future trajectories for the individual agents, which are used along with the uncertainty of each trajectory expressed as a probability and standard deviation of the trajectory, to feed into the AV’s planning module.

While the MDN can be framed to output a Gaussian mixture, \{\alpha^{1:K}_t, \mu^{1:K}_t, \sigma^{1:K}_t \} where K is the number of desired mixtures, for each individual (x, y) euclidean coordinate in the trajectory, it can instead be biased to learn behaviors in the mixtures. Instead of making every point a mixture, we can move the mixture to a higher level where the mixture represents a set of (x, y) coordinates — the trajectory — as a whole, and thus, map to a certain behavior. In this setup, each “behavior” is intended to represent a high-level behavior an agent on the road could exhibit, such as “go left”, “go right”, “go straight”, and so on.

We can represent behaviors as B^{1:K}_t, where K is the number of mixtures output for the GMM being represented by the MDN. Each behavior B^k_t can have a coefficient \alpha^{k}_{t} representing the probability of the likelihood of that mixture, so \sum_{k=1}^{K} \alpha^k_t = 1. Additionally, each behavior contains a set of (x, y) coordinates up to a time horizon H represented as means M^k_t = (\mu^{x, k}_{t+1:t+H}, \mu^{y, k}_{t+1:t+H}), to produce the predicted trajectory defined earlier: Y_t = \{ (x_{t+1}, y_{t+1}), (x_{t+1}, y_{t+1}), ..., (x_{t+H}, y_{t+H}) \}. For the standard deviation \sigma, it can be taken per (x, y) coordinate as a whole, rather than individually for each x and each y. The standard deviation then represents the uncertainty of the coordinate as a whole — so per behavior k, in K total behaviors, we have S^k_t = \sigma^{k}_{t+1:t+H}.

The predicted behaviors per agent per timestep t can be represented as B^k_t = \{\alpha^k_t, M^k_t, S^k_t\}. If for our MDN we have N agents as input — the ego vehicle (the AV) and N-1 closest agents in euclidean distance — the desired outputs include predicted behavior of the N-1 closest agents. The full output O_t of the neural network in the MDN is then \{ B^{1, 1:K}_t, B^{2, 1:K}_t, ..., B^{N-1, 1:K}_t \}.

Inputs and Outputs of Mixture Density Network for traffic behavior prediction. Zero-filled where data is not available.

The number of mixtures (the number of behaviors) is an open question, and will likely require some thought and experimentation. However, a good starting point would be 3, simply based on the intuition that in traffic, a vehicle can go left, right, or straight. Left or right could mean switch lanes or a full left or right turn, which the network should learn based on the usual behavior and lane contours in the scene.

[2]’s approach to traffic prediction involves using a neural network outputting a single behavior for an agent as a trajectory of (x, y) coordinates up to some future time horizon. Additionally, they output a standard deviation per each (x, y) coordinate in the trajectory. [2] find empirically that training such a network requires curriculum, where the network is first trained with the Mean Displacement Error, and then the standard deviation outputs are trained. Note that the Mean Displacement Error simply involves taking the mean of the euclidean distance of each point in the predicted trajectory with the ground truth trajectory. The equations for calculating the Mean Displacement Error, and the half-Gaussian loss including the standard deviation can be seen in the figure below.

[2]’s loss functions. Equation 1 represents the displacement error between the predicted and ground truth (x, y) coordinates. Equation 2 represents the Mean Displacement Error which the network is first trained with. Equation 5 represents the full half-Gaussian loss function which includes the displacement error.

The MDN can be trained similarly via curriculum where first the network is trained with Mean Displacement Error, where the means (mu‘s) of the (x, y) coordinates output for each mixture for each agent are trained, and then the full outputs including the standard deviation’s (\sigma‘s) are trained using the half-Gaussian distribution loss. The main difference is that we need to train additionally for multiple outputs (behaviors/mixtures) per individual agent, and thus also train the MDN coefficients \alpha‘s. This can simply be done by taking a softmax of the coefficients and multiplying them with the mean loss for each mixture, and then taking the sum of the loss across all mixtures. That is, L_{total} = \sum_{k=1}^{K} \alpha^k L_{ij} where L_{ij} are the losses either from equation (2) or (5) displayed from [1]’s approach, depending on which part of the curriculum is being trained.

Additionally, as this is a complex problem for the network to learn, I propose applying another curriculum to the training by progressively training for the number of agents in the output starting from number of trainable agents = 1. Meaning, first train the mixtures output for the first agent (agent 1) only, by a) using Mean Displacement Error first and b) full half-Gaussian second, and then training the first two agents together (agents 1 and 2) similarly, and so on up to all agents in the full output.

Code for this loss function in Python, using Chainer, is displayed below. The input variable holds the full input at some timestep as mentioned earlier (past trajectories and lane contours of an ego vehicle and its closest N-1 agents). The num_agents represents the total number of agents in the inputs and outputs. trainable_agents represents the number of agents in the curriculum being currently trained for as described earlier (1, 1:2, 1:3, 1:4, …, 1:N-1). And finally output holds the ground truth trajectory of (x, y) coordinates for each agent being predicted for, up to future some time horizon.

coef, mu, ln_var = self.fprop(input)

coef = coef.reshape(output.shape[0], num_agents, num_mixtures)
coef = F.softmax(coef, axis=2)
mu = F.reshape(mu, (output.shape[0], num_agents, num_mixtures, -1, 2))
ln_var = F.reshape(ln_var, (output.shape[0], num_agents, num_mixtures, -1))

output = output.reshape(output.shape[0], num_agents, 1, -1, 2)
output = F.repeat(output, num_mixtures, 2)

coef = coef[:, :trainable_agents, :]
ln_var = ln_var[:, :trainable_agents, :, :]

x_true = output[:, :trainable_agents, :, :, 0]
y_true = output[:, :trainable_agents, :, :, 1]
x_pred = mu[:, :trainable_agents, :, :, 0]
y_pred = mu[:, :trainable_agents, :, :, 1]

if curriculum_step == 1:
    loss = F.sum(coef * F.sum(((x_true-x_pred)**2) + ((y_true-y_pred)**2), axis=3))
    displacement_sq = ((x_true-x_pred)**2) + ((y_true-y_pred)**2)
    ln_var = F.clip(ln_var, -300., 300.) #Numerical stability
    var = F.exp(ln_var)
    loss = F.sum(coef * F.sum(((displacement_sq/(2*var)) + F.log(F.sqrt(2 * math.pi * var))), axis=3))


The data can originate from either sensors on an AV driving around, or fixed CCTV style cameras pointed at a road. In either case, it will be necessary to track agents in the scene at a given moment with an identifier, and preprocess the sensor or video footage into “traces” — a vector of (x, y) coordinates observed for the agent’s position over time, expressed in a particular reference system, as the agents enter and exit the vicinity of the AV or fixed camera.

These traces can then be divided into inputs and outputs to form a dataset in the following manner. As an example setup, if an agent is seen for a total of 15 seconds at a frame rate of 25 Hz, that gives us 375 total observed positions of the agent over time. If we set the prediction time horizon (referenced as H previously) to 2 seconds at 25 Hz (thus 50 future positions desired for prediction), based on an input of the past 1 second of the agent’s past positions (25 past positions), our inputs and outputs can be divided from the total 375 frames available for the agent, for every timestep t \in \{0:325\}. At each timestep t, the dataset for this agent can contain input = \{(x_{t-25}, y_{t-25}), ..., (x_t, y_t)\} and output = \{(x_{t+1}, y_{t+1}), ..., (x_{t+50}, y_{t+50})\}. At timestep t = 0, as positions are not available from the previous 1 second for the agent, the input can be zero-filled.

Additionally, for adding in the environmental context, an HD map of the locations the agent traces are collected from will be required. This map can provide the lane contours for each agent for the inputs.

All data for inputs, outputs, and lane contours, should be ego-centered. Meaning, for the ego vehicle in the input, the current (x, y) position at t=0 would be (0.0, 0.0), and all other (x, y) positions should be relative to this, to normalize the data for the neural network.


Experiments were performed on a proprietary dataset consisting of 24 hours of detection at a fixed scene, using the following setup and hyperparameters:

Number of mixtures3
Past positions of agents, P1 second = 25
Time horizon for prediction, H2 seconds = 50
Ego vehicle + number of closest agents1+9 = 10
Distance to find closest agents40 meters
Number of layers10
Hidden units1024

The dataset was split into train, validation and test sets. After curriculum training the MDN as described above to a point where satisfactory results were achieved on the validation set, predictions on the test set were compared against the ground truths using Root Mean Squared Error (RMSE), using the Manhattan distance between the coordiantes. This RMSE was compared to an implementation of DESIRE [3], which was trained and tested on the same dataset.

While DESIRE achieved an RMSE of 0.38979 on the test set, the MDN achieved an RMSE of 0.401587, when testing only the first agent’s output. When testing all 9 agent’s outputs against the ground truths, the MDN achieved an RMSE of 0.478878 (when using the trajectory from the most likely behavior). Note that with DESIRE, only a single agent’s prediction is tested at a time. Furthermore, DESIRE constitutes a series of complex LSTMs with several expensive intermediate steps, whereas the MDN can achieve similar results for multiple agents at a time, in a single forward pass with a simpler feedforward network.

Upon inspecting the results in detail, I also noticed that the mixtures were appropriately mapping to plausible high-level behaviors with adequate probabilities given the agent’s history, particularly in situations where there was ambiguity in where the agent could go next. I also observed in many cases the probability being almost 1 (\alpha=1) for a behavior for agents where the network strongly expressed they could only go in a certain direction given their past trajectory. Otherwise, the performance and learning of the behaviors will require more quantitative experiments to measure, as the definition and ground truths of behavior are harder to measure as opposed to raw predicted trajectories being compared using RMSE.


For the traffic prediction problem, Mixture Density Networks (MDN) can be used to learn a distribution over agents’ trajectories in a traffic scene. If the MDN is framed to produce a full trajectory within each Gaussian mixture as a whole, as opposed to framing each point in the trajectory to be a mixture in its own, the MDN can be biased to learn higher-level behaviors of agents, leading to more accurate and plausible predictions. These behaviors come with a probabilities at the behavior level, and standard deviation over the trajectory predictions, allowing for uncertainty-aware decision making in the Autonomous Vehicle context.

This method can achieve similar prediction accuracy to state-of-the-art methods such as DESIRE [3], when measuring the Root Mean Squared Error of the Manhattan distance of the predicted coordinates in the trajectories to the ground truths. While state-of-the art methods may require series of complex neural networks such as LSTMs, with expensive intermediate processing steps, the MDN can achieve similar results for multiple agents at a time, in a single forward pass with a simple feedforward network, using no intermediate rasterized representation.

The limitations with this approach are that the number of agents being input and output, and the number of mixtures and thus behaviors, are fixed at train time. To overcome this limitation, perhaps in future work RNNs can be explored for feeding inputs and receiving outputs as sequences that are terminated with separator markers, similar to how the dataset is composed to train and test the Differentiable Neural Computer: Furthermore, in future work, this method should be trained and tested on a variety of scenes, with an ablation study to determine how much the multi-agent and environmental context helps with prediction. The number of Gaussian mixtures also requires further consideration and experimentation.


[1] Mayank Bansal et al., ChauffeurNet: Learning to Drive
by Imitating the Best and Synthesizing the Worst
, 2018,
[2] Nemanja Djuric et al., Short-term Motion Prediction of Traffic Actors for Autonomous Driving using Deep Convolutional Networks, 2018,
[3] Namhoon Lee et al., DESIRE: Distant Future Prediction in Dynamic Scenes with Interacting Agents, 2017,
[4] Jens Schulz et al., Interaction-Aware Probabilistic Behavior Prediction in Urban Environments, 2018,
[5] Jingbo Zhou et al., R2-D2: a System to Support Probabilistic Path Prediction in Dynamic Environments via “Semi-Lazy” Learning, 2013,
[6] Christopher M. Bishop, Mixture Density Networks, 1994,

Probabilistic Model-Based Reinforcement Learning Using The Differentiable Neural Computer

My experiments found that a model learned in a Differentiable Neural Computer outperformed a vanilla LSTM based model, on two gaming environments.

➡ Thesis PDF:


For my MSc Artificial Intelligence at the University of Edinburgh, my dissertation included 4 months of research. I investigated the use of the Differentiable Neural Computer (DNC) for model-based Reinforcement Learning / Evolution Strategies. A predictive, probabilistic model of the environment was learned in a DNC, and used to train a controller in video gaming environments to maximize rewards (score).

The difference between Reinforcement Learning (RL) and Evolution Strategies (ES) are detailed here. However, in this post and my dissertation, the two are used interchangeably as either is used to accomplish the same goal — given an environment (MDP or partially observable MDP), learn to maximize the cumulative rewards in the environment. The focus is rather on learning a model of the environment, which can be queried while training an ES or RL agent.

The authors of the DNC conducted some simple RL experiments using the DNC, given coded states. However, to the best of my knowledge, this is the first time the DNC was used in learning a model of the environment entirely from pixels, in order to train a complex RL or ES agent. The experiments I conducted showed the DNC outperforming Long Short Term Memory (LSTM) used similarly to learn a model of the environment.

Learning a Model

The model architecture is borrowed from the World Models framework (see my World Models implementation onGitHub). Given a state in an environment at timestep t, s_t, and an action a_t performed at that state, the task of the model is to predict the next state s_{t+1}. Thus the input to the model is [s_t + a_t], which produces output (prediction) s_{t+1}. Note that the states in this case consist of frames from the game at each timestep consisting of pixels. These states are compressed down to latent variables z using a Convolutional Variational Autoencoder (CVAE), therefore more specifically the model maps [z_t + a_t] => z_{t+1}.

The model consists of a Recurrent Neural Network, such as LSTM, that outputs the parameters of a Mixture Density Model — in this cased, a mixture of Gaussians. This type of architecture is known as a Mixture Density Network (MDN), where a neural network is used to output the parameters of a Mixture Density Model. My blog post on MDNs goes into more details. When coupled with a Recurrent Neural Network (RNN), the architecture is known as MDN-RNN.

Thus, in learning a model of an environment, the output of the MDN-RNN “model” is not simply z_{t+1}, but the parameters of a Gaussian Mixture Model (\alpha, \mu, \sigma) which are then used to sample the prediction of the next state z_{t+1}. This allows the model to be more powerful by becoming probabilistic, and encode stochastic environments where the next state after a given state and action can be one of multiple.

For the experiments I conducted, the architecture of the model used a DNC where the RNN is used in World Models, thus, the model is composed of a MDN-DNC. Simply, the recurrent layers used in the MDN are replaced with a DNC (which itself contains recurrent layers that are additionally coupled with external memory).

The hypothesis was that using the DNC instead of vanilla RNNs such as LSTM, will allow for a more robust and algorithmic model of the environment to be learned, thus allowing the agent to perform better. This would particularly be true in complex environments with long term dependencies (meaning, a state perhaps hundreds or thousands of timesteps ago needs to be kept in context for another state down the line).

Experiments And Results

Experimentation schematic, based on the World Models framework.

The model of the environment learned is then used to train an RL agent. More specifically, features from the model are used, and in the case of the World Models framework, this consists of the hidden and cell states h_t, c_t of the LSTM layers of the model at every timestep. These “features” of the model, coupled with the compressed latent representation of the environment state, z, at a given timestep t is used as input to a controller. Thus, the controller takes as input [z_t + h_t + c_t] to output action a_t to be taken to achieve a high reward. CMA-ES (more details in my blog post on CMA-ES in Python) was used to train the controller in my experiments.

The games the MDN-DNC was tested on were ViZDoom: Take Cover and Pommerman. For either game, a series of experiments were conducted to compare the results with a model of the environment learned in a MDN-DNC versus a MDN-LSTM.

ViZDoom: Take Cover

In the case of ViZDoom: Take Cover, a predictive model was trained in the environment using MDN-LSTM and MDN-DNC. Each was trained for 1 epoch, on random rollouts of 10,000 games which recorded the frames from the game (pixels) and actions taken at each timestep. The model was then used to train a CMA-ES controller. Note that the controllers were trained in the “dream” environment simulated by the model, as done in World Models.

A simulation of the environment — “dream” — where the controller is used to train using the learned model only, rather than the actual environment.

The controllers were tested in the environment throughout the generations for a 100 rollouts at each test. The results are plotted below. The MDN-DNC based controller clearly outperformed the MDN-LSTM based controller, and solved the game (achieving a mean score of 750 over 100 rollouts).

Comparison of a DNC based model versus a LSTM based model, used for training a controller in ViZDoom: Take Cover. The DNC based controller outperforms the LSTM controller.


In the case of Pommerman, only the model’s predictions were used to test the capacity of the predictive model learned in a MDN-DNC and a MDN-LSTM. A controller was not trained. This was possible given that the states in Pommerman are coded as integers, rather than pixels. Thus, given [s_t + a_t], the predicted state [s_{t+1}] could be compared with the ground truth state from the actual game for equality, and to measure how many components of the state (position, ammo available, etc) were correctly predicted.

Here again, the MDN-DNC model outperformed the MDN-LSTM model, where both were trained exactly the same way for the same number of epochs. The MDN-DNC was more accurately able to predict the individual components of the next state given a current state and an action.

The predictive power of a DNC based model versus a LSTM based model in the Pommerman environment. The DNC based model was able to predict future states more accurately.


Model-based Reinforcement Learning or Evolution Strategies involve using a model of the environment when training a Reinforcement Learning or Evolution Strategies agent. In my case, the World Models approach to learn a predictive, probabilistic model of the environment in an Mixture Density Network was used. The Mixture Density Network consisted of a Differentiable Neural Computer, which output the parameters of a Gaussian mixture model that were used to sample the next state in a game. My experiments found that a model learned in a Differentiable Neural Computer outperformed a vanilla LSTM based model, on two gaming environments.

Future work should include games with long term memory dependencies, whereas with the experiments performed for this work it is hard to justify there being such dependencies in the ViZDoom: Take Cover and Pommerman environments. Other such environments would perhaps magnify the capabilities of the Differentiable Neural Computer. Also, what exactly is going on in the memory of the Differentiable Neural Computer at each timestep? It would be useful to know what it has learned, and perhaps features from the external memory of the Differentiable Neural Computer itself could be used when training a controller. For example, the Differentiable Neural Computer emits read heads, r_t, at each timestep, which are selected from the full memory, and used to produce the output (a prediction of the next state). Perhaps the contents of the read heads, or other portions of the external memory, could provide useful information of the environment if exposed directly to the controller along with the hidden state and cell state of the underlying LSTM.

Full details on this work can be found in my MSc thesis at:


[1] Alex Graves et al., Hybrid computing using a neural network with dynamic external memory, 2016.
[2] David Ha, Jürgen Schmidhuber, World Models, 2018.
[3] Nikolaus Hansen, The CMA Evolution Strategy: A Tutorial, 2016.
[4] Christopher M. Bishop, Mixture Density Networks, 1994.