Francesco Bullo’s research while affiliated with University of California, Santa Barbara and other places

While monotone operator theory is often studied on Hilbert spaces, many interesting problems in machine learning and optimization arise naturally in finite-dimensional vector spaces endowed with non-Euclidean norms, such as diagonally-weighted 1 or ∞ norms. This paper provides a natural generalization of monotone operator theory to finite-dimensional non-Euclidean spaces. The key tools are weak pairings and logarithmic norms. We show that the resolvent and reflected resolvent operators of non-Euclidean monotone mappings exhibit similar properties to their counterparts in Hilbert spaces. Furthermore, classical iterative methods and splitting methods for finding zeros of monotone operators are shown to converge in the non-Euclidean case. We apply our theory to equilibrium computation and Lipschitz constant estimation of recurrent neural networks, obtaining novel iterations and tighter upper bounds via forward-backward splitting.

Firing rate models are dynamical systems widely used in applied and theoretical neuroscience to describe local cortical dynamics in neuronal populations. By providing a macroscopic perspective of neuronal activity, these models are essential for investigating oscillatory phenomena, chaotic behavior, and associative memory processes. Despite their widespread use, the application of firing rate models to associative memory networks has received limited mathematical exploration, and most existing studies are focused on specific models. Conversely, well-established associative memory designs, such as Hopfield networks, lack key biologically-relevant features intrinsic to firing rate models, including positivity and interpretable synaptic matrices that reflect excitatory and inhibitory interactions. To address this gap, we propose a general framework that ensures the emergence of re-scaled memory patterns as stable equilibria in the firing rate dynamics. Furthermore, we analyze the conditions under which the memories are locally and globally asymptotically stable, providing insights into constructing biologically-plausible and robust systems for associative memory retrieval.

The Hopfield model provides a mathematically idealized yet insightful framework for understanding the mechanisms of memory storage and retrieval in the human brain. This model has inspired four decades of extensive research on learning and retrieval dynamics, capacity estimates, and sequential transitions among memories. Notably, the role and impact of external inputs has been largely underexplored, from their effects on neural dynamics to how they facilitate effective memory retrieval. To bridge this gap, we propose a novel dynamical system framework in which the external input directly influences the neural synapses and shapes the energy landscape of the Hopfield model. This plasticity-based mechanism provides a clear energetic interpretation of the memory retrieval process and proves effective at correctly classifying highly mixed inputs. Furthermore, we integrate this model within the framework of modern Hopfield architectures, using this connection to elucidate how current and past information are combined during the retrieval process. Finally, we embed both the classic and the new model in an environment disrupted by noise and compare their robustness during memory retrieval.

Mechanistic and tractable mathematical models play a key role in understanding how social influence shapes public opinions. Recently, a weighted-median mechanism has been proposed as a new micro-foundation of opinion dynamics and validated via experimental data. Numerical studies indicate that this new mechanism recreates some non-trivial real-world features of opinion evolution. In this paper, we conduct a thorough analysis of the weighted-median opinion dynamics. We fully characterize the equilibria set, and establish the almost-sure convergence for any initial condition. Moreover, we prove a necessary and sufficient condition for the almost-sure convergence to consensus, as well as a sufficient condition for almost-sure dissensus. We related the rich dynamical bevaior of the weighted-median opinion dynamics to two delicate network structures: the cohesive sets and the decisive links. To complement our sufficient conditions for almost-sure dissensus, we further prove that, given the influence network, determining whether the system almost surely achieves persistent dissensus is NP-hard, which reflects the complexity the network topology contributes to opinion evolution.

In this letter, we study the proximal gradient dynamics. This recently-proposed continuous-time dynamics solves optimization problems whose cost functions are separable into a nonsmooth convex and a smooth component. First, we show that the cost function decreases monotonically along the trajectories of the proximal gradient dynamics. We then introduce a new condition that guarantees exponential convergence of the cost function to its optimal value, and show that this condition implies the proximal Polyak-{\L}ojasiewicz condition. We also show that the proximal Polyak-{\L}ojasiewicz condition guarantees exponential convergence of the cost function. Moreover, we extend these results to time-varying optimization problems, providing bounds for equilibrium tracking. Finally, we discuss applications of these findings, including the LASSO problem, quadratic optimization with polytopic constraints, and certain matrix based problems.

The Deffuant-Weisbuch (DW) model is a well-known bounded-confidence opinion dynamics that has attracted wide interest. Although the heterogeneous DW model has been studied by simulations over 20 years, its convergence proof is open. Our previous paper \cite{GC-WS-WM-FB:20} solves the problem for the case of uniform weighting factors greater than or equal to 1/2, but the general case remains unresolved. This paper considers the DW model with heterogeneous confidence bounds and heterogeneous (unconstrained) weighting factors and shows that, with probability one, the opinion of each agent converges to a fixed vector. In other words, this paper resolves the convergence conjecture for the heterogeneous DW model. Our analysis also clarifies how the convergence speed may be arbitrarily slow under certain parameter conditions.

Recent studies on stability and contractivity have highlighted the importance of semi-inner products, which we refer to as ``pairings'', associated with general norms. A pairing is a binary operation that relates the derivative of a curve's norm to the radius-vector of the curve and its tangent. This relationship, known as the curve norm derivative formula, is crucial when using the norm as a Lyapunov function. Another important property of the pairing, used in stability and contraction criteria, is the so-called Lumer inequality, which relates the pairing to the induced logarithmic norm. We prove that the curve norm derivative formula and Lumer's inequality are, in fact, equivalent to each other and to several simpler properties. We then introduce and characterize regular pairings that satisfy all of these properties. Our results unify several independent theories of pairings (semi-inner products) developed in previous work on functional analysis and control theory. Additionally, we introduce the polyhedral max pairing and develop computational tools for polyhedral norms, advancing contraction theory in non-Euclidean spaces.

Multistationarity - the existence of multiple equilibrium points - is a common phenomenon in dynamical systems from a variety of fields, including neuroscience, opinion dynamics, systems biology, and power systems. A recently proposed generalization of contraction theory, called k-contraction, is a promising approach for analyzing the asymptotic behaviour of multistationary systems. In particular, all bounded trajectories of a time-invariant 2-contracting system converge to an equilibrium point, but the system may have multiple equilibrium points where more than one is locally stable. An important challenge is to study k-contraction in large-scale interconnected systems. Inspired by a recent small-gain theorem for 2-contraction by Angeli et al., we derive a new sufficient condition for 2-contraction of a feedback interconnection of two nonlinear dynamical systems. Our condition is based on (i) deriving new formulas for the 2-multiplicative [2-additive] compound of block matrices using block Kronecker products [sums], (ii) a hierarchical approach for proving standard contraction, and (iii) a network small-gain theorem for Metzler matrices. We demonstrate our results by deriving a simple sufficient condition for 2-contraction in a network of FitzHugh-Nagumo neurons.

In competitive resource allocation formulations multiple agents compete over different contests by committing their limited resources in them. For these settings, contest games offer a game-theoretic foundation to analyze how players can efficiently invest their resources. In this class of games the resulting behavior can be affected by external interactions among the players. In particular, players could be able to make coalitions that allow transferring resources among them, seeking to improve their outcomes. In this work, we study bilateral budgetary transfers in contest games played over networks. Particularly, we characterize the family of networks where there exist mutually beneficial bilateral transfer for some set of systems parameters. With this in mind, we provide sufficient conditions for the existence of mutually beneficial transfers. Moreover, we provide a constructive argument that guarantees that the benefit of making coalitions only depends on mild connectivity conditions of the graph structure. Lastly, we provide a characterization of the improvement of the utilities as a function of the transferred budget. Further, we demonstrate how gradient-based dynamics can be utilized to find desirable coalitional structures. Interestingly, our findings demonstrate that such collaborative opportunities extend well beyond the typical "enemy-of-my-enemy" alliances.