I Introduction
Reinforcement learning has demonstrated remarkable performance on a variety of sequential decision making tasks such as Go [1], Atari games [2], autonomous driving [3, 4], and continuous robotic control [5, 6]. Reinforcement learning (RL) methods fall under two broad categories: modelfree and modelbased. In modelfree RL, the environment’s physics are not modeled, and such methods require substantial environment interaction and can have prohibitive sample complexity [7]
. In contrast, modelbased methods allow for systematic analysis of environment physics, and in principle should lead to better sample complexity and more robust policies. These methods, however, have to date been challenging to integrate with deep neural networks and to generalize across multiple environment dimensions
[8, 9], or in truly novel scenarios, which are expected in unrestricted realworld applications such as driving.In this work, we focus on modelfree methods, but include explicit modeling of risk. We additionally focus on a framework that includes an adversary in addition to the main (i.e., protagonist) agent. By modeling risk, we can train stronger adversaries and through competition, more robust policies for the protagonist (see Figure 1 for an overview). We envision this as enabling training of more robust agents in simulation and then using simtoreal techniques [10] to generalize to real world applications, such as household robots or autonomous driving, with high reliability and safety requirements.
A recent algorithm combining robustness in reinforcement learning and the adversarial framework is robust adversarial reinforcement learning (RARL) [11], which trained a robust protagonist agent by having an adversary providing random and systematic attacks on input states and dynamics. The adversary is itself trained using reinforcement learning, and tries to minimize the long term expected reward while the protagonist tries to maximize it. As the adversary gets stronger, the protagonist experiences harder challenges.
RARL, along with similar methods [12], is able to achieve some robustness, but the level of variation seen during training may not be diverse enough to resemble the variety encountered in the realworld. Specifically, the adversary does not actively seek catastrophic outcomes as does the agent constructed in this paper. Without such experiences, the protagonist agent will not learn to guard against them. Consider autonomous driving: a car controlled by the protagonist may suddenly be hit by another car. We call this and other similar events catastrophic since they present extremely negative rewards to the protagonist, and should not occur under a reasonable policy. Such catastrophic events are highly unlikely to be encountered if an adversary only randomly perturbs the environment parameters or dynamics, or if the adversary only tries to minimize total reward.
In this paper, we propose risk averse robust adversarial reinforcement learning (RARARL) for training risk averse policies that are simultaneously robust to dynamics changes. Inspired by [13], we model risk as the variance of value functions. To emphasize that the protagonist be averse to catastrophes, we design an asymmetric reward function (see Section IVA): successful behavior receives a small positive reward, whereas catastrophes receive a very negative reward.
A robust policy should not only maximize long term expected reward, but should also select actions with low variance of that expected reward. Maximizing the expectation of the value function only maximizes the point estimate of that function without giving a guarantee on the variance. While
[13] proposed a method to estimate that variance, it assumes that the number of states is limited, while we don’t assume limited number of states and that assumption makes it impractical to apply it to real world settings where the number of possible states could be infinitely large. Here, we use an ensemble of Qvalue networks to estimate variance. A similar technique was proposed in Bootstrapped DQNs [14] to assist exploration, though in our case, the primary purpose of the ensemble is to estimate variance.We consider a twoagent reinforcement learning scenario (formalized in Section III). Unlike in [11], where the agents performed actions simultaneously, here they take turns executing actions, so that one agent may take multiple steps to bring the environment in a more challenging state for the other. We seek to enable the adversarial agent to actively explore the parameter variation space, so that the perturbations are generated more efficiently. We use a discrete control task, autonomous driving with the TORCS [15] simulator, to demonstrate the benefits of RARARL.
Ii Related Work
Reinforcement Learning with Adversaries. A recent technique in reinforcement learning involves introducing adversaries and other agents that can adjust the environment difficulty for a main agent. This has been used for robust grasping [16], simulated fighting [17], and RARL [11], the most relevant prior work to ours. RARL trains an adversary to appropriately perturb the environment for a main agent. The perturbations, though, were limited to a few parameters such as mass or friction, and the trained protagonist may be vulnerable to other variations.
The works of [12] and [18] proposed to add noise to state observations to provide adversarial perturbations, with the noise generated using fast gradient sign method [19]. However, they did not consider training an adversary or training risk averse policies. The work of [20] proposed to introduce Bayesian optimization to actively select environment variables that may induce catastrophes, so that models trained can be robust to these environment dynamics. However, they did not systematically explore dynamics variations and therefore the model may be vulnerable to changing dynamics even if it is robust to a handful of rare events.
Robustness and Safety in RL. More generally, robustness and safety have long been explored in reinforcement learning [21, 22, 23]. Chow et al. [23] proposed to model risk via constraint or chance constraint on the conditional value at risk (CVaR). This paper provided strong convergence guarantees but made strong assumptions: value and constrained value functions are assumed to be known exactly and to be differentiable and smooth. Risk is estimated by simply sampling trajectories which may never encounter adverse outcomes, whereas with sparse risks (as is the case here) adversarial sampling provides more accurate estimates of the probability of a catastrophe.
A popular ingredient is to enforce constraints on an agent during exploration [24] and policy updates [25, 26]. Alternative techniques include random noise injection during various stages of training [27, 28], injecting noise to the transition dynamics during training [29], learning when to reset [30] and even physically crashing as needed [31]. However, Rajeswaran et al. [29] requires training on a target domain and experienced performance degradation when the target domain has a different model parameter distribution from the source. We also note that in control theory, [32, 33] have provided theoretical analysis for robust control, though their focus lies in model based RL instead of model free RL. These prior techniques are orthogonal to our contribution, which relies on model ensembles to estimate variance.
UncertaintyDriven Exploration. Prior work on exploration includes [34], which measures novelty of states using state prediction error, and [35], which uses pseudo counts to explore novel states. In our work, we seek to measure the risk of a state by the variance of value functions. The adversarial agent explores states with high variance so that it can create appropriate challenges for the protagonist.
Simulation to Real Transfer. Running reinforcement learning on physical hardware can be dangerous due to exploration and slow due to high sample complexity. One approach to deploying RLtrained agents safely in the real world is to experience enough environment variation during training in simulation so that the realworld environment looks just like another variation. These simulationtoreal techniques have grown popular, including domain randomization [10, 36] and dynamics randomization [37]. However, their focus is on transferring policies to the real world rather than training robust and risk averse policies.
Iii Risk Averse Robust Adversarial RL
In this section, we formalize our risk averse robust adversarial reinforcement learning (RARARL) framework.
Iiia Two Player Reinforcement Learning
We consider the environment as a Markov Decision Process (MDP)
, where defines the state space, defines the action space, is the reward function, is the state transition model, and is the reward discount rate. There are two agents: the protagonist and the adversary .Definition. Protagonist Agent. A protagonist learns a policy to maximize discounted expected reward . The protagonist should be risk averse, so we define the value of action at state to be
(1) 
where is the modified Q function, is the original Q function, and is the variance of the Q function across different models, and is a constant; The term is called the riskaverse term thereafter, and encourages the protagonist to seek lower variance actions. The reward for is the environment reward at time .
Definition. Adversarial Agent. An adversary learns a policy to minimize long term expected reward, or to maximize the negative discounted reward . To encourage the adversary to systematically seek adverse outcomes, its modified value function for action selection is
(2) 
where is the modified Q function, is the original Q function, is the variance of the Q function across different models, and is a constant; the interaction between agents becomes a zerosum game by setting . The term is called the riskseeking term thereafter. The reward of is the negative of the environment reward , and its action space is the same as for the protagonist.
The necessity of having two agents working separately instead of jointly is to provide the adversary more power to create challenges for the protagonist. For example, in autonomous driving, a single risky action may not put the vehicle in a dangerous condition. In order to create a catastrophic event (e.g., a traffic accident) the adversary needs to be stronger. In our experiments (a vehicle controller with discrete control), the protagonist and adversary alternate full control of a vehicle, though our methods also apply to settings in [11], where the action applied to the environment is a sum of contributions from the protagonist and the adversary.
IiiB Reward Design and Risk Modeling
To train risk averse agents, we propose an asymmetric reward function design such that good behavior receives small positive rewards and risky behavior receives very negative rewards. See Section IVA and Equation 4 for details.
The risk of an action can be modeled by estimating the variance of the value function across different models trained on different sets of data. Inspired by [14], we estimate the variance of Q value functions by training multiple Q value networks in parallel. Hereafter, we use to denote the entire Q value network, and use to denote the th head of the multiheads Q value network.^{1}^{1}1We use and to represent functions that could apply to either the protagonist or adversary. If it is necessary to distinguish among the two agents, we add the appropriate subscript of or . As shown in Figure 2, the network takes in input , which consists of stacked frames of consecutive observations. It passes through three shared convolutional layers, followed by one (shared) dense layer. After this, the input is passed to different heads which perform one dense layer to obtain actionvalue outputs: . Defining the mean as , the variance of a single action is,
(3) 
where we use the subscripts to indicate variance over models, as in Equations 1 and 2. The variance in Equation 3 measures risk, and our goal is for the protagonist and adversarial agents to select actions with low and high variance, respectively.
At training time, when we sample one action using the Q values, we randomly choose one of heads from to , and use this head throughout one episode to choose the action that will be applied by the agent. When updating Q functions, our algorithm (like DQN [2]) samples a batch of data of size from the replay buffer which, for each data point, includes the state, action, next state, reward, and task completion signal. Then we sample a
sized mask. Each mask value is sampled using a Poisson distribution (modeling a true Bootstrap sample with replacement) instead of the Bernoulli distribution in
[14] (sample without replacement). At test time, the mean value is used for selecting actions.IiiC Risk Averse RARL
In our twoplayer framework, the agents take actions sequentially, not simultaneously: the protagonist takes steps, the adversary takes steps, and the cycle repeats. The experience of each agent is only visible to itself, which means each agent changes the environment transition dynamics for another agent. The Q learning Bellman equation is modified to be compatible with this case. Let the current and target value functions be and for the protagonist, and (respectively) and for the adversary. Given the current state and action pair , we denote actions executed by the protagonist as and actions taken by the adversary as . The target value functions are and, similarly, To increase training stability for the protagonist, we designed a training schedule of the adversarial agent. For the first steps, only the protagonist agent takes actions. After that, for every steps taken by the protagonist, the adversary takes steps. The reason for this training schedule design is that we observed if the adversarial agent is added too early (e.g., right at the start), the protagonist is unable to attain any rewards. Thus, we let the protagonist undergo a sufficient amount of training steps to learn basic skills. The use of masks in updating Q value functions is similar to [14], where the mask is a integer vector of size equal to batch size times number of ensemble Q networks, and is used to determine which model is to be updated with the sample batch. Algorithm 1 describes our training algorithm.
Iv Experiments
We evaluated models trained by RARARL on an autonomous driving environment, TORCS [15]. Autonomous driving has been explored in recent contexts for policy learning and safety [38, 39, 40] and is a good testbed for riskaverse reinforcement learning since it involves events (particularly crashes) that qualify as catastrophes.
Iva Simulation Environment
For experiments, we use the Michigan Speedway environment in TORCS [15], which is a round way racing track; see Figure 6 for sample observations. The states are dimensional RGB images. The vehicle can execute nine actions: (1) move left and accelerate, (2) move ahead and accelerate, (3) move right and accelerate, (4) move left, (5) do nothing, (6) move right, (7) move right and decelerate, (8) move ahead and decelerate, (9) move right and decelerate.
We next define our asymmetric reward function. Let be the magnitude of the speed, be the angle between the speed and road direction, be the distance of the vehicle to the center of the road, and be the road width. We additionally define two binary flags: and , with if the vehicle is stuck (and 0 otherwise) and if the vehicle is damaged (and 0 otherwise). Letting , the reward function is defined as:
(4) 
with the intuition being that encourages speed direction along the road direction, penalizes moving across the road, and penalizes driving on the side of the road. We set the catastrophe reward as and set as a tunable constant which ensures that the magnitude of the noncatastrophe reward is significantly less than that of the catastrophe reward. The catastrophe reward measures collisions, which are highly undesirable events to be avoided. We note that constants used to blend reward and variance terms in the riskaugmented Qfunctions in Equations 1 and 2 were set to .
We consider two additional reward functions to investigate in our experiments. The total progress reward excludes the catastrophe reward:
(5) 
and the pure progress reward is defined as
(6) 
The total progress reward considers both moving along the road and across the road, and penalizes large distances to the center of the road, while the pure progress only measures the distance traveled by the vehicle, regardless of the vehicle’s location. The latter can be a more realistic measure since vehicles do not always need to be at the center of the road.
IvB Baselines and Our Method
All baselines are optimized using Adam [41] with learning rate 0.0001 and batch size 32. In all our ensemble DQN models, we trained with 10 heads since empirically that provided a reasonable balance between having enough models for variance estimation but not so much that training time would be overbearing. For each update, we sampled 5 models using Poisson sampling with to generate the mask for updating Q value functions. We set the training frequency as 4, the target update frequency as 1000, and the replay buffer size as 100,000. For training DQN with an epsilongreedy strategy, the decreased linearly from 1 to 0.02 from step 10,000 to step 500,000. The time point to add in perturbations is steps, and for every steps taken by protagonist agent, the random agent or adversary agent will take step.
Vanilla DQN. The purpose of comparing with vanilla DQN is to show that models trained in one environment may overfit to specific dynamics and fail to transfer to other environments, particularly those that involve random perturbations. We denote this as dqn.
Ensemble DQN. Ensemble DQN tends to be more robust than vanilla DQN. However, without being trained on different dynamics, even Ensemble DQN may not work well when there are adversarial attacks or simple random changes in the dynamics. We denote this as bsdqn.
Ensemble DQN with Random Perturbations Without Risk Averse Term. We train the protagonist and provide random perturbations according to the schedule . We do not include the variance guided exploration term here, so only the Q value function is used for choosing actions. The schedule is the same as in our method. We denote this as bsdqnrand.
Ensemble DQN with Random Perturbations With the Risk Averse Term. We only train the protagonist agent and provide random perturbations according to the adversarial training schedule . The protagonist selects action based on its Q value function and the risk averse term. We denote this as bsdqnrandriskaverse.
Ensemble DQN with Adversarial Perturbation. This is to compare our model with [11]. For a fair comparison, we also use Ensemble DQN to train the policy while the variance term is not used as either riskaverse or riskseeking term in either agents. We denote this as bsdqnadv.
Our method. In our method, we train both the protagonist and the adversary with Ensemble DQN. We include here the variance guided exploration term, so the Q function and its variance across different models will be used for action selection. The adversarial perturbation is provided according to the adversarial training schedule . We denote this as bsdqnadvriskaverse.
IvC Evaluation
To evaluate robustness of our trained models, we use the same trained models under different testing conditions, and evaluate using the previouslydefined reward classes of total progress (Equation 5), pure progress (Equation 6), and additionally consider the reward of catastrophes. We present three broad sets of results: (1) No perturbations. (Figure 3) We tested all trained models from Section IVB without perturbations. (2) Random perturbations. (Figure 4) To evaluate the robustness of trained models in the presence of random environment perturbations, we benchmarked all trained models using random perturbations. For every 10 actions taken by the main agent, 1 was taken at random. (3) Adversarial Perturbations. (Figure 5) To test the ability of our models to avoid catastrophes, which normally require deliberate, nonrandom perturbations, we test with a trained adversarial agent which took 1 action for every 10 taken by the protagonist.
All subplots in Figures 3, 4, and 5 include a vertical blue line at 0.55 million steps indicating when perturbations were first applied during training (if any). Before 0.55 million steps, we allow enough time for protagonist agents to be able to drive normally. We choose 0.55 million steps because the exploration rate decreases to 0.02 at 0.50 million steps, and we allow additional 50000 steps for learning to stabilize.
Does adding adversarial agent’s perturbation affect the robustness? In Table I, we compare the robustness of all models by their catastrophe rewards. The results indicate that adding perturbations improves a model’s robustness, especially to adversarial attacks. DQN trained with random perturbations is not as robust as models trained with adversarial perturbations, since random perturbations are weaker than adversarial perturbations.


Exp  Normal  Random Perturb  Adv. Perturb 


dqn  0.80  3.0  4.0 
bsdqn  0.90  1.1  2.5 
bsdqnrand  0.10  1.0  2.1 
bsdqnadv  0.30  0.5  1.0 
bsdqnrandriskaverse  0.09  0.4  2.0 
bsdqnadvriskaverse  0.08  0.1  0.1 

How does the risk term affect the robustness of the trained models? As shown in Figures 4 and 5, models trained with the risk term achieved better robustness under both random and adversarial perturbations. We attribute this to the risk term encouraging the adversary to aggressively explore regions with high risk while encouraging the opposite for the protagonist.
How do adversarial perturbations compare to random perturbations? A trained adversarial agent can enforce stronger perturbations than random perturbations. By comparing Figure 4 and Figure 5, we see that the adversarial perturbation provides stronger attacks, which causes the reward to be lower than with random perturbations.
We also visualize an example of the differences between a trained adversary and random perturbations in Figure 6, which shows that a trained adversary can force the protagonist (a vanilla DQN model) to drive into a wall and crash.
V Conclusion
We show that by introducing a notion of risk averse behavior, a protagonist agent trained with a learned adversary experiences substantially fewer catastrophic events during testtime rollouts as compared to agents trained without an adversary. Furthermore, a trained adversarial agent is able to provide stronger perturbations than random perturbations and can provide a better training signal for the protagonist as compared to providing random perturbations. In future work, we will apply RARARL in other safetycritical domains, such as in surgical robotics.
Acknowledgments
Xinlei Pan is supported by Berkeley Deep Drive. Daniel Seita is supported by a National Physical Science Consortium Fellowship.
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