Embed to Control: A Locally Linear Latent Dynamics Model for Control from Raw Images (1506.07365v3)

Abstract: We introduce Embed to Control (E2C), a method for model learning and control of non-linear dynamical systems from raw pixel images. E2C consists of a deep generative model, belonging to the family of variational autoencoders, that learns to generate image trajectories from a latent space in which the dynamics is constrained to be locally linear. Our model is derived directly from an optimal control formulation in latent space, supports long-term prediction of image sequences and exhibits strong performance on a variety of complex control problems.

• The paper introduces a method that encodes raw pixel images into a low-dimensional, locally linear latent space to enable efficient control of non-linear systems.
• It employs variational autoencoders and deep generative models to learn state transitions, optimizing control via locally linear dynamics.
• Experimental results demonstrate superior performance with lower trajectory costs and high success rates in tasks like planar navigation, inverted pendulum, and robotic arm control.

Embed to Control: A Locally Linear Latent Dynamics Model for Control from Raw Images

The paper "Embed to Control (E2C): A Locally Linear Latent Dynamics Model for Control from Raw Images" by Manuel Watter, Jost Tobias Springenberg, Joschka Boedecker, and Martin Riedmiller presents a novel approach for model learning and control of non-linear dynamical systems using raw pixel images as sensory input. The core idea behind the E2C method is to transform a high-dimensional, non-linear control problem into a lower-dimensional latent state space where the dynamics are constrained to be locally linear, facilitating efficient control.

Methodology

The E2C model integrates concepts from deep generative models and variational autoencoders (VAEs). The approach can be summarized in the following key steps:

1. Learning a Latent Representation: An encoder network transforms raw pixel images $x_t$ into a low-dimensional latent space $z_t$, where $z_t \sim N(m(x_t), \Sigma_{\omega})$.
2. Locally Linear State Transitions: In the latent space, the transitions are modeled to be locally linear, which permits the use of SOC methods like iterative Linear Quadratic Gaussian (iLQG) for control. The linear transition function is given by:

$$z_{t+1} = A_t \cdot z_t + B_t \cdot u_t + o_t + \omega$$

1. Generative Model: The model generates image sequences from the latent space via a decoder network, ensuring that the generated images correspond to valid observations.
2. Training via Variational Bayes: The entire network is trained using stochastic gradient variational Bayes, optimizing a joint loss that includes both reconstruction error and Kullback-Leibler (KL) divergence to enforce agreement between the predicted and true state distributions.

Experimental Evaluation

The paper validates the E2C approach on four challenging control tasks: a planar system with obstacles, an inverted pendulum swing-up, balancing a cart-pole, and controlling a three-link arm. The evaluation highlights several aspects of the model's efficacy:

• Planar System: E2C effectively discovered the latent structure of a 2D plane with obstacles, achieving trajectory costs close to those of a true system model, indicating successful long-term planning capabilities.
• Inverted Pendulum: The model managed to encode and control the swing-up task, despite the non-Markovian nature of standard image observations, requiring minimal action sequences for stabilization.
• Cart-Pole and Robotic Arm: In more complex settings with high-dimensional images, E2C, leveraging convolutional and up-convolutional networks, successfully performed control tasks by encoding high-dimensional observations into a lower-dimensional latent space.

Performance and Comparison

In terms of numerical results, the model demonstrated superior performance compared to deep autoencoders and standard VAEs. Specifically, for the planar system, E2C achieved a mean real trajectory cost of 25.1, significantly lower than the VAE+Slowness variant at 89.1, and showed a 100% success rate for achieving goal states. Similar trends were observed in the inverted pendulum task, with E2C showing lower costs and higher success rates than baseline methods.

Implications and Future Work

The implications of E2C are substantial in fields demanding autonomous control from high-dimensional sensory inputs, including robotics and reinforcement learning. By reducing the complexity of high-dimensional control problems to tractable latent spaces with locally linear dynamics, E2C facilitates efficient and effective control strategies suitable for real-world applications. Additionally, the generative modeling aspect enhances the robustness and adaptability of the model.

Future directions might involve extending E2C to handle partially observable environments and integrating it with advanced reinforcement learning frameworks. Moreover, leveraging more powerful generative models like GANs or exploring hierarchical latent spaces could further improve the cognitive capabilities and scalability of the approach.

In conclusion, Embed to Control presents a methodical stride in controlling non-linear dynamical systems from raw pixel data, underpinned by robust theoretical constructs and promising empirical results, offering significant advancements in autonomous agent control and predictive modeling.