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Title: D3PG: Deep Differentiable Deterministic Policy Gradients

Paper content:
Over the last decade, two competing control strategies have emerged for solving complex control tasks with high efficacy.
Model-based control algorithms, such as model-predictive control (MPC) and trajectory optimization, peer into the gradients of underlying system dynamics in order to solve control tasks with high sample efficiency.
However, like all gradient-based numerical optimization methods,model-based control methods are sensitive to intializations and are prone to becoming trapped in local minima.
Deep reinforcement learning (DRL), on the other hand, can somewhat alleviate these issues by exploring the solution space through sampling — at the expense of computational cost.
In this paper, we present a hybrid method that combines the best aspects of gradient-based methods and DRL.
We base our algorithm on the deep deterministic policy gradients (DDPG) algorithm and propose a simple modification that uses true gradients from a differentiable physical simulator to increase the convergence rate of both the actor and the critic.
We demonstrate our algorithm on seven 2D robot control tasks, with the most complex one being a differentiable half cheetah with hard contact constraints.
Empirical results show that our method boosts the performance of DDPGwithout sacrificing its robustness to local minima.
In recent years, deep reinforcement learning (DRL) has emerged as a flexible and robust means of teaching simulated robots to complete complex tasks, from manipulation (Kumar et al., 2016) and locomotion (Haarnoja et al., 2018b) , to navigating complex terrain (Peng et al., 2016) .
Compared with more direct optimization methods such as gradient descent or second-order optimization, DRL naturally incorporates exploration into its planning, allowing it to learn generalizable policies and robust state value estimations across simulated environments.
Perhaps the most salient reason for DRL's surge in popularity is its ability to operate on black-box simulators where the underlying dynamics model is not available.
DRL's model-free, Monte-Carlo-style methods have made it applicable to a wide range of physical (and non-physical) simulation environments, including those where a smooth, well-behaved dynamical model does not exist.
This comes at two striking costs.
First, such sampling procedures may be inefficient, requiring a large number of samples for adequate learning.
Second, in order to be generally applicable to any model-free environment, underlying dynamical gradients are not used, even if they are available.
In other words, valuable information that could greatly aid control tasks is not taken advantage of in these schemes.
When an accurate model of robot dynamics is given, model-based methods such as model-predictive control (MPC) or trajectory optimization have historically been employed.
These methods can solve tasks with higher sample efficiency than model-free DRL algorithms.
Models provide access to ground-truth, analytical gradients of robot physics without the need for sample-based estimation.
However, such methods don't incorporate exploration or learning into their procedures, and are especially prone to becoming trapped in poor local minima.
While there has been a recent surge in fast and accurate differentiable simulators not previously available, most applications for control have relied on established local methods such as MPC (de Avila Belbute-Peres et al., 2018) , gradient descent (Degrave et al., 2019) , or trajectory optimization (Hu et al., 2019) to solve control tasks.
An ideal algorithm would exploit the efficiency of model-based methods while maintaining DRL's relative robustness to poor local minima.
In this paper, we propose an actor-critic algorithm that leverages differentiable simulation and combines the benefits of model-based methods and DRL.
We build our method upon standard actor-critic DRL algorithms and use true model gradients in order to improve the efficacy of learned critic models.
Our main insights are twofold: First, gradients of critics play an important role in certain DRL algorithms, but optimization of these critics' gradients has not been explored by previous work.
Second, the emergence of differentiable simulators enables computation of advantage estimation (AE) gradients with little additional computational overhead.
Based on these observations, we present an algorithm that uses AE gradients in order to co-learn critic value and gradient estimation, demonstrably improving convergence of both actor and critic.
In this paper, we contribute the following: 1) An efficient hybrid actor-critic method which builds upon deep deterministic policy gradients (DDPG, (Lillicrap et al., 2015) ), using gradient information in order to improve convergence in a simple way.
2) A principled mathematical framework for fitting critic gradients, providing a roadmap for applying our method to any deterministic policy gradient method, and
3) Demonstrations of our algorithm on seven control tasks, ranging from contact-free classic control problems to complex tasks with accurate, hard contact, such as the HalfCheetah, along with comparisons to both model-based control and DRL baselines.
Immediately obvious from our results is the fact that DDPG and our algorithm are both competitive on all problems presented, regardless of problem difficulty.
While MPC dominates on the simplest control tasks, it struggles on the more complicated tasks with hard contacts, and DRL approaches dominate.
This underscores our thesis -that DRL's exploration properties make it better suited than model-based approaches for problems with a myriad of poor local minima.
More naïve model-based approaches, such as GD, can succeed when they begin very close to a local minimum -as is the case with CartPole -but show slow or no improvement in dynamical environments with nontrivial control schemes.
This is especially apparent in problems where the optimal strategy requires robots to make locally suboptimal motions in order to build up momentum to be used to escape local minima later.
Examples include Pendulum, CartPoleSwingUp, and MountainCar, where the robot must learn to build up momentum through local oscillations before attempting to reach a goal.
GD further fails on complex physical control tasks like HalfCheetah, where certain configurations, such as toppling, can be unrecoverable.
Finally, we note that although MPC is able to tractably find a good solution for the RollingDie problem, the complex nonlinearities in the contact-heavy dynamics require long planning horizons (100 steps, chosen by running hyperparameter search) in order to find a good trajectory.
Thus, although MPC eventually converges to a control sequence with very high reward, it requires abundant computation to converge.
DRL-based control approaches are able to find success on all problems, and are especially competitive on those with contact.
Compared with DDPG, our hybrid algorithm universally converges faster or to higher returns.
The rolling die example presents a particularly interesting contrast.
As the die is randomly initialized, it is more valuable to aim for higher return history rather than return mean due to the large variance in the initial state distribution.
It can be seen from Figure 4 that our method managed to reach a higher average return history over 16 runs.
Manually visualizing the controller from the best run in our method revealed that it discovered a novel two-bounce strategy for challenging initial poses (Figure 3) , while most of the strategies in DDPG typically leveraged one bounce only.
There are a few other reasons why our algorithm may be considered superior to MPC.
First, our algorithm is applicable to a wider range of reward structures.
While we had planned to demonstrate MPC on another classic control problem, namely the Acrobot, MPC is inapplicable to this robot's reward structure.
The Acrobot's rewards penalize it with −1 point for every second it has not reached its target pose.
MPC requires a differentiable reward, and this reward structure is not.
Thus, our Hybrid DDPG algorithm applies to a wider range of problems than MPC.
Second, closedloop network controllers are naturally more robust than MPC.
Even as noise is added or initial conditions and tasks change, learned controllers can generalize.
While MPC can recover from these scenarios, it requires expensive replanning.
In these scenarios, MPC becomes especially unattractive to deploy on physical hardware, where power and computational resource constraints can render MPC inapplicable to realtime applications.
Figure 3 : Visualization of the twobounce strategy discovered by our algorithm.
Solid red box: initial die.
Dash cyan curve: trajectory of the die.
Blue box: the target zone.
Light red boxes: states of the die at collisions and about to enter the target.
In this paper, we have presented an actor-critic algorithm that uses AE gradients to co-learn critic value and gradient estimation and improve convergence of both actor and critic.
Our algorithm leverages differentiable simulation and combines the benefits of model-based methods and DRL.
We designed seven 2D control tasks with three different contact scenarios and compared our method with several state-of-the-art baseline algorithms.
We demonstrated our method boosts the performance of DDPG and is much less sensitive to local minima than model-based approaches.
In the future, it would be interesting to see if our mathematical framework can be applied to improve the effectiveness of value functions used in other DRL algorithms.
A APPENDIX

assistant: text: We propose a novel method that leverages the gradients from differentiable simulators to improve the performance of RL for robotics control