Implicit Bias of Policy Gradient in Linear Quadratic Control: Extrapolation to Unseen Initial States

In modern machine learning, models can often fit training data in numerous ways, some of which perform well on unseen (test) data, while others do not. Remarkably, in such cases gradient descent frequently exhibits an implicit bias that leads to excellent performance on unseen data. This implicit bias was extensively studied in supervised learning, but is far less understood in optimal control (reinforcement learning). There, learning a controller applied to a system via gradient descent is known as policy gradient, and a question of prime importance is the extent to which a learned controller extrapolates to unseen initial states. This paper theoretically studies the implicit bias of policy gradient in terms of extrapolation to unseen initial states. Focusing on the fundamental Linear Quadratic Regulator (LQR) problem, we establish that the extent of extrapolation depends on the degree of exploration induced by the system when commencing from initial states included in training. Experiments corroborate our theory, and demonstrate its conclusions on problems beyond LQR, where systems are non-linear and controllers are neural networks. We hypothesize that real-world optimal control may be greatly improved by developing methods for informed selection of initial states to train on.

What Makes Data Suitable for a Locally Connected Neural Network? A Necessary and Sufficient Condition Based on Quantum Entanglement

The question of what makes a data distribution suitable for deep learning is a fundamental open problem. Focusing on locally connected neural networks (a prevalent family of architectures that includes convolutional and recurrent neural networks as well as local self-attention models), we address this problem by adopting theoretical tools from quantum physics. Our main theoretical result states that a certain locally connected neural network is capable of accurate prediction over a data distribution if and only if the data distribution admits low quantum entanglement under certain canonical partitions of features. As a practical application of this result, we derive a preprocessing method for enhancing the suitability of a data distribution to locally connected neural networks. Experiments with widespread models over various datasets demonstrate our findings. We hope that our use of quantum entanglement will encourage further adoption of tools from physics for formally reasoning about the relation between deep learning and real-world data.