Veronica Centorrino

Veronica Centorrino, Francesca Rossi, Francesco Bullo et al.

This paper studies equality-constrained composite minimization problems. This class of problems, capturing regularization terms and inequality constraints, naturally arises in a wide range of engineering and machine learning applications. To tackle these optimization problems, inspired by recent results, we introduce the \emph{proportional--integral proximal gradient dynamics} (PI--PGD): a closed-loop system where the Lagrange multipliers are control inputs and states are the problem decision variables. First, we establish the equivalence between the stationary points of the minimization problem and the equilibria of the PI--PGD. Then for the case of affine constraints, by leveraging tools from contraction theory we give a comprehensive convergence analysis for the dynamics, showing linear--exponential convergence towards the equilibrium. That is, the distance between each solution and the equilibrium is upper bounded by a function that first decreases linearly and then exponentially. Our findings are illustrated numerically on a set of representative examples, which include an exploratory application to nonlinear equality constraints.

Francesca Rossi, Veronica Centorrino, Francesco Bullo et al.

The ability to compose acquired skills to plan and execute behaviors is a hallmark of natural intelligence. Yet, despite remarkable cross-disciplinary efforts, a principled account of how task structure shapes gating and how such computations could be delivered in neural circuits, remains elusive. Here we introduce GateMod, an interpretable theoretically grounded computational model linking the emergence of gating to the underlying decision-making task, and to a neural circuit architecture. We first develop GateFrame, a normative framework casting policy gating into the minimization of the free energy. This framework, relating gating rules to task, applies broadly across neuroscience, cognitive and computational sciences. We then derive GateFlow, a continuous-time energy based dynamics that provably converges to GateFrame optimal solution. Convergence, exponential and global, follows from a contractivity property that also yields robustness and other desirable properties. Finally, we derive a neural circuit from GateFlow, GateNet. This is a soft-competitive recurrent circuit whose components perform local and contextual computations consistent with known dendritic and neural processing motifs. We evaluate GateMod across two different settings: collective behaviors in multi-agent systems and human decision-making in multi-armed bandits. In all settings, GateMod provides interpretable mechanistic explanations of gating and quantitatively matches or outperforms established models. GateMod offers a unifying framework for neural policy gating, linking task objectives, dynamical computation, and circuit-level mechanisms. It provides a framework to understand gating in natural agents beyond current explanations and to equip machines with this ability.

Veronica Centorrino, Rawan Hoteit, Efe C. Balta et al.

This paper studies equality-constrained minimization problems through the lens of feedback control. We introduce a unified control-theoretic framework by showing that a PID feedback law acting on the dual variable induces the PID saddle-point flow (PID-SPF), a broad class of saddle-point dynamics associated with the augmented Lagrangian. This framework recovers several classical primal-dual flows as special cases. We prove that the equilibria of the proposed flow coincide with the stationary points of the original problem. Our analysis reveals how the feedback gains affect the optimization: integral action enforces constraint satisfaction, proportional action introduces the augmented Lagrangian structure, and derivative action modifies the geometry of the primal dynamics by inducing a state-dependent Riemannian metric. Moreover, for convex problems with affine constraints, we establish global exponential convergence by leveraging contraction theory for all admissible PID gains, providing in the process explicit bounds on the convergence rate. Finally, we validate our theoretical results on numerical examples including an application to bilevel optimization.