1. A.C.B de Olivera, M. Siami, and E.D. Sontag. Remarks on the gradient flow for linear neural network based feedback for the LQR Problem. In Proc. 2024 63rd IEEE Conference on Decision and Control (CDC), 2024. Note: Submitted.Keyword(s): neural networks, overparametrization, gradient descent, input to state stability, gradient systems, feedback control, LQR. Abstract:

   Motivated by the current interest in using Artificial intelligence (AI) tools in control design, this paper takes the first steps towards bridging results from gradient methods for solving the LQR control problem, and neural networks. More specifically, it looks into the case where one wants to find a Linear Feed-Forward Neural Network (LFFNN) that minimizes the Linear Quadratic Regulator (LQR) cost. This work develops gradient formulas that can be used to implement the training of LFFNNs to solve the LQR problem, and derives an important conservation law of the system. This conservation law is then leveraged to prove global convergence of solutions and invariance of the set of stabilizing networks under the training dynamics. These theoretical results are then followed by and extensive analysis of the simplest version of the problem (the ``scalar case'') and by numerical evidence of faster convergence of the training of general LFFNNs when compared to traditional direct gradient methods. These results not only serve as indication of the theoretical value of studying such a problem, but also of the practical value of LFFNNs as design tools for data-driven control applications.




2. A.C.B de Olivera, M. Siami, and E.D. Sontag. Dynamics and perturbations of overparameterized linear neural networks. In Proc. 2023 62st IEEE Conference on Decision and Control (CDC), pages 7356-7361, 2023. Note: Extended version is On the ISS property of the gradient flow for single hidden-layer neural networks with linear activations, arXiv https://arxiv.org/abs/2305.09904. [PDF] [doi:10.1109/CDC49753.2023.10383478] Keyword(s): neural networks, overparametrization, gradient descent, input to state stability, gradient systems. Abstract:

   Recent research in neural networks and machine learning suggests that using many more parameters than strictly required by the initial complexity of a regression problem can result in more accurate or faster-converging models -- contrary to classical statistical belief. This phenomenon, sometimes known as ``benign overfitting'', raises questions regarding in what other ways might overparameterization affect the properties of a learning problem. In this work, we investigate the effects of overfitting on the robustness of gradient-descent training when subject to uncertainty on the gradient estimation. This uncertainty arises naturally if the gradient is estimated from noisy data or directly measured. Our object of study is a linear neural network with a single, arbitrarily wide, hidden layer and an arbitrary number of inputs and outputs. In this paper we solve the problem for the case where the input and output of our neural-network are one-dimensional, deriving sufficient conditions for robustness of our system based on necessary and sufficient conditions for convergence in the undisturbed case. We then show that the general overparametrized formulation introduces a set of spurious equilibria which lay outside the set where the loss function is minimized, and discuss directions of future work that might extend our current results for more general formulations.

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