Guilherme França

Alessandro Barp, Lancelot Da Costa, Guilherme França et al.

In this chapter, we identify fundamental geometric structures that underlie the problems of sampling, optimisation, inference and adaptive decision-making. Based on this identification, we derive algorithms that exploit these geometric structures to solve these problems efficiently. We show that a wide range of geometric theories emerge naturally in these fields, ranging from measure-preserving processes, information divergences, Poisson geometry, and geometric integration. Specifically, we explain how (i) leveraging the symplectic geometry of Hamiltonian systems enable us to construct (accelerated) sampling and optimisation methods, (ii) the theory of Hilbertian subspaces and Stein operators provides a general methodology to obtain robust estimators, (iii) preserving the information geometry of decision-making yields adaptive agents that perform active inference. Throughout, we emphasise the rich connections between these fields; e.g., inference draws on sampling and optimisation, and adaptive decision-making assesses decisions by inferring their counterfactual consequences. Our exposition provides a conceptual overview of underlying ideas, rather than a technical discussion, which can be found in the references herein.

Guilherme França, Alessandro Barp, Mark Girolami et al.

Optimization tasks are crucial in statistical machine learning. Recently, there has been great interest in leveraging tools from dynamical systems to derive accelerated and robust optimization methods via suitable discretizations of continuous-time systems. However, these ideas have mostly been limited to Euclidean spaces and unconstrained settings, or to Riemannian gradient flows. In this work, we propose a dissipative extension of Dirac's theory of constrained Hamiltonian systems as a general framework for solving optimization problems over smooth manifolds, including problems with nonlinear constraints. We develop geometric/symplectic numerical integrators on manifolds that are "rate-matching," i.e., preserve the continuous-time rates of convergence. In particular, we introduce a dissipative RATTLE integrator able to achieve optimal convergence rate locally. Our class of (accelerated) algorithms are not only simple and efficient but also applicable to a broad range of contexts.

Guilherme França, José Bento

There has been an increasing necessity for scalable optimization methods, especially due to the explosion in the size of datasets and model complexity in modern machine learning applications. Scalable solvers often distribute the computation over a network of processing units. For simple algorithms such as gradient descent the dependency of the convergence time with the topology of this network is well-known. However, for more involved algorithms such as the Alternating Direction Methods of Multipliers (ADMM) much less is known. At the heart of many distributed optimization algorithms there exists a gossip subroutine which averages local information over the network, and whose efficiency is crucial for the overall performance of the method. In this paper we review recent research in this area and, with the goal of isolating such a communication exchange behaviour, we compare different algorithms when applied to a canonical distributed averaging consensus problem. We also show interesting connections between ADMM and lifted Markov chains besides providing an explicitly characterization of its convergence and optimal parameter tuning in terms of spectral properties of the network. Finally, we empirically study the connection between network topology and convergence rates for different algorithms on a real world problem of sensor localization.

Guilherme França, Michael I. Jordan, René Vidal

Recently, continuous-time dynamical systems have proved useful in providing conceptual and quantitative insights into gradient-based optimization, widely used in modern machine learning and statistics. An important question that arises in this line of work is how to discretize the system in such a way that its stability and rates of convergence are preserved. In this paper we propose a geometric framework in which such discretizations can be realized systematically, enabling the derivation of "rate-matching" algorithms without the need for a discrete convergence analysis. More specifically, we show that a generalization of symplectic integrators to nonconservative and in particular dissipative Hamiltonian systems is able to preserve rates of convergence up to a controlled error. Moreover, such methods preserve a shadow Hamiltonian despite the absence of a conservation law, extending key results of symplectic integrators to nonconservative cases. Our arguments rely on a combination of backward error analysis with fundamental results from symplectic geometry. We stress that although the original motivation for this work was the application to optimization, where dissipative systems play a natural role, they are fully general and not only provide a differential geometric framework for dissipative Hamiltonian systems but also substantially extend the theory of structure-preserving integration.

Guilherme França, Daniel P. Robinson, René Vidal

Optimization is at the heart of machine learning, statistics and many applied scientific disciplines. It also has a long history in physics, ranging from the minimal action principle to finding ground states of disordered systems such as spin glasses. Proximal algorithms form a class of methods that are broadly applicable and are particularly well-suited to nonsmooth, constrained, large-scale, and distributed optimization problems. There are essentially five proximal algorithms currently known: Forward-backward splitting, Tseng splitting, Douglas-Rachford, alternating direction method of multipliers, and the more recent Davis-Yin. These methods sit on a higher level of abstraction compared to gradient-based ones, with deep roots in nonlinear functional analysis. We show that all of these methods are actually different discretizations of a single differential equation, namely, the simple gradient flow which dates back to Cauchy (1847). An important aspect behind many of the success stories in machine learning relies on "accelerating" the convergence of first-order methods. We show that similar discretization schemes applied to Newton's equation with an additional dissipative force, which we refer to as accelerated gradient flow, allow us to obtain accelerated variants of all these proximal algorithms -- the majority of which are new although some recover known cases in the literature. Furthermore, we extend these methods to stochastic settings, allowing us to make connections with Langevin and Fokker-Planck equations. Similar ideas apply to gradient descent, heavy ball, and Nesterov's method which are simpler. Our results therefore provide a unified framework from which several important optimization methods are nothing but simulations of classical dissipative systems.

Guilherme França, Jeremias Sulam, Daniel P. Robinson et al.

Arguably, the two most popular accelerated or momentum-based optimization methods in machine learning are Nesterov's accelerated gradient and Polyaks's heavy ball, both corresponding to different discretizations of a particular second order differential equation with friction. Such connections with continuous-time dynamical systems have been instrumental in demystifying acceleration phenomena in optimization. Here we study structure-preserving discretizations for a certain class of dissipative (conformal) Hamiltonian systems, allowing us to analyze the symplectic structure of both Nesterov and heavy ball, besides providing several new insights into these methods. Moreover, we propose a new algorithm based on a dissipative relativistic system that normalizes the momentum and may result in more stable/faster optimization. Importantly, such a method generalizes both Nesterov and heavy ball, each being recovered as distinct limiting cases, and has potential advantages at no additional cost.

Guilherme França, Daniel P. Robinson, René Vidal

Recently, there has been great interest in connections between continuous-time dynamical systems and optimization methods, notably in the context of accelerated methods for smooth and unconstrained problems. In this paper we extend this perspective to nonsmooth and constrained problems by obtaining differential inclusions associated to novel accelerated variants of the alternating direction method of multipliers (ADMM). Through a Lyapunov analysis, we derive rates of convergence for these dynamical systems in different settings that illustrate an interesting tradeoff between decaying versus constant damping strategies. We also obtain modified equations capturing fine-grained details of these methods, which have improved stability and preserve the leading order convergence rates. An extension to general nonlinear equality and inequality constraints in connection with singular perturbation theory is provided.

Sam Safavi, Bikash Joshi, Guilherme França et al.

The framework of Integral Quadratic Constraints (IQC) introduced by Lessard et al. (2014) reduces the computation of upper bounds on the convergence rate of several optimization algorithms to semi-definite programming (SDP). In particular, this technique was applied to Nesterov's accelerated method (NAM). For quadratic functions, this SDP was explicitly solved leading to a new bound on the convergence rate of NAM, and for arbitrary strongly convex functions it was shown numerically that IQC can improve bounds from Nesterov (2004). Unfortunately, an explicit analytic solution to the SDP was not provided. In this paper, we provide such an analytical solution, obtaining a new general and explicit upper bound on the convergence rate of NAM, which we further optimize over its parameters. To the best of our knowledge, this is the best, and explicit, upper bound on the convergence rate of NAM for strongly convex functions.

Guilherme França, Maria L. Rizzo, Joshua T. Vogelstein

Energy statistics was proposed by Sz\' ekely in the 80's inspired by Newton's gravitational potential in classical mechanics and it provides a model-free hypothesis test for equality of distributions. In its original form, energy statistics was formulated in Euclidean spaces. More recently, it was generalized to metric spaces of negative type. In this paper, we consider a formulation for the clustering problem using a weighted version of energy statistics in spaces of negative type. We show that this approach leads to a quadratically constrained quadratic program in the associated kernel space, establishing connections with graph partitioning problems and kernel methods in machine learning. To find local solutions of such an optimization problem, we propose kernel k-groups, which is an extension of Hartigan's method to kernel spaces. Kernel k-groups is cheaper than spectral clustering and has the same computational cost as kernel k-means (which is based on Lloyd's heuristic) but our numerical results show an improved performance, especially in higher dimensions. Moreover, we verify the efficiency of kernel k-groups in community detection in sparse stochastic block models which has fascinating applications in several areas of science.

Guilherme França, José Bento

When solving consensus optimization problems over a graph, there is often an explicit characterization of the convergence rate of Gradient Descent (GD) using the spectrum of the graph Laplacian. The same type of problems under the Alternating Direction Method of Multipliers (ADMM) are, however, poorly understood. For instance, simple but important non-strongly-convex consensus problems have not yet being analyzed, especially concerning the dependency of the convergence rate on the graph topology. Recently, for a non-strongly-convex consensus problem, a connection between distributed ADMM and lifted Markov chains was proposed, followed by a conjecture that ADMM is faster than GD by a square root factor in its convergence time, in close analogy to the mixing speedup achieved by lifting several Markov chains. Nevertheless, a proof of such a claim is is still lacking. Here we provide a full characterization of the convergence of distributed over-relaxed ADMM for the same type of consensus problem in terms of the topology of the underlying graph. Our results provide explicit formulas for optimal parameter selection in terms of the second largest eigenvalue of the transition matrix of the graph's random walk. Another consequence of our results is a proof of the aforementioned conjecture, which interestingly, we show it is valid for any graph, even the ones whose random walks cannot be accelerated via Markov chain lifting.

Guilherme França, José Bento

The framework of Integral Quadratic Constraints (IQC) reduces the computation of upper bounds on the convergence rate of several optimization algorithms to a semi-definite program (SDP). In the case of over-relaxed Alternating Direction Method of Multipliers (ADMM), an explicit and closed form solution to this SDP was derived in our recent work [1]. The purpose of this paper is twofold. First, we summarize these results. Second, we explore one of its consequences which allows us to obtain general and simple formulas for optimal parameter selection. These results are valid for arbitrary strongly convex objective functions.

Guilherme França, José Bento

The time to converge to the steady state of a finite Markov chain can be greatly reduced by a lifting operation, which creates a new Markov chain on an expanded state space. For a class of quadratic objectives, we show an analogous behavior where a distributed ADMM algorithm can be seen as a lifting of Gradient Descent algorithm. This provides a deep insight for its faster convergence rate under optimal parameter tuning. We conjecture that this gain is always present, as opposed to the lifting of a Markov chain which sometimes only provides a marginal speedup.

Guilherme França, José Bento

The framework of Integral Quadratic Constraints of Lessard et al. (2014) reduces the computation of upper bounds on the convergence rate of several optimization algorithms to semi-definite programming (SDP). Followup work by Nishihara et al. (2015) applies this technique to the entire family of over-relaxed Alternating Direction Method of Multipliers (ADMM). Unfortunately, they only provide an explicit error bound for sufficiently large values of some of the parameters of the problem, leaving the computation for the general case as a numerical optimization problem. In this paper we provide an exact analytical solution to this SDP and obtain a general and explicit upper bound on the convergence rate of the entire family of over-relaxed ADMM. Furthermore, we demonstrate that it is not possible to extract from this SDP a general bound better than ours. We end with a few numerical illustrations of our result and a comparison between the convergence rate we obtain for the ADMM with known convergence rates for the Gradient Descent.