Papers by A. C. B. de Oliveira and E.D. Sontag

The Lojasiewicz inequality characterizes objective-value convergence along gradient flows and, in special cases, yields exponential decay of the cost. However, such results do not directly imply convergence of the state. In this paper, we use contraction theory to derive state-space guarantees for gradient systems satisfying generalized Lojasiewicz inequalities. We first show that, when the objective has a unique strongly convex minimizer, the generalized Lojasiewicz inequality implies semi-global exponential stability; on arbitrary compact subsets, this yields exponential stability. We then give two curvature-based sufficient conditions, together with constraints on the Lojasiewicz rate, under which the nonconvex gradient flow is globally incrementally exponentially stable, a property strictly stronger than global exponential stability. A few examples are presented at the end of the paper to validate the proposed theory.

This note establishes a theoretical framework for finding (potentially overparameterized) approximations of a function on a compact set with a-priori bounds for the generalization error. The approximation method considered is to choose, among all functions that (approximately) interpolate a given data set, one with a minimal Lipschitz constant. The paper establishes rigorous generalization bounds over practically relevant classes of approximators, including deep neural networks. It also presents a neural network implementation based on Lipschitz-bounded network layers and an augmented Lagrangian method. The results are illustrated for a problem of learning the dynamics of an input-to-state stable system with certified bounds on simulation error.

This paper investigates how the compositional structure of neural networks shapes their optimization landscape and training dynamics. We analyze the gradient flow associated with overparameterized optimization problems, which can be interpreted as training a neural network with linear activations. Remarkably, we show that the global convergence properties can be derived for any cost function that is proper and real analytic. We then specialize the analysis to scalar cost functions, where the geometry of the landscape can be fully characterized. In this setting, we demonstrate that key structural features – such as the location and stability of saddle points – are universal across all admissible costs, depending solely on the overparameterized representation rather than on problem-specific details. Moreover, we show that convergence can be arbitrarily accelerated depending on the initialization, as measured by an imbalance metric introduced in this work. Finally, we discuss how these insights may generalize to neural networks with sigmoidal activations, showing through a simple example that certain geometric and dynamical properties persist beyond the linear case.

Data-driven control (DDC), that is the design of controllers directly from observed data, has attracted substantial attention in recent years due to its advantages over model-based control. DDC avoids a computationally expensive, potentially conservative model identification step and bypasses practically difficult questions such as model order/class selection. This tutorial paper seeks to offer a sampling of the different approaches that have been recently used to synthesize data driven controllers and filters, covering both analytic approaches and learning enabled ones, indicating the relative strengths of each. A second objective is to provide a key to the rapidly expanding literature in the subject, to help researchers newly interested in this field to quickly come up to speed.