arXiv subjects

Bryan Van Scoy

Publications and source records attributed to Bryan Van Scoy.

At least 19 recordsLinked to original sources

arXiv2026

Convex Synthesis of First-Order Methods for Time-Varying Smooth Strongly Convex Optimization

Time-varying optimization is fundamental to decision-making in dynamic environments, where objectives evolve over time due to exogenous signals or data streams. However, algorithms designed for static problems yield suboptimal decisions in dynamic scenarios, even asymptotically. In this paper, we develop a robust control synthesis framework to systematically design first-order methods for smooth strongly convex problems that vary in time. Our approach leverages both convex robust control synthesis in the static setting and the internal model principle by directly embedding a model of the underlying variability into the designed algorithm.

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arXiv2025

The Speed-Robustness Trade-Off for First-Order Methods with Additive Gradient Noise

We study the trade-off between convergence rate and sensitivity to stochastic additive gradient noise for first-order optimization methods. Ordinary Gradient Descent (GD) can be made fast-and-sensitive or slow-and-robust by increasing or decreasing the stepsize, respectively. However, it is not clear how such a trade-off can be navigated when working with accelerated methods such as Polyak's Heavy Ball (HB) or Nesterov's Fast Gradient (FG) methods. We consider two classes of functions: (1) strongly convex quadratics and (2) smooth strongly convex functions. For each function class, we present a tractable way to compute the convergence rate and sensitivity to additive gradient noise for a broad family of first-order methods, and we present algorithm designs that trade off these competing performance metrics. Each design consists of a simple analytic update rule with two states of memory, similar to HB and FG. Moreover, each design has a scalar tuning parameter that explicitly trades off convergence rate and sensitivity to additive gradient noise. We numerically validate the performance of our designs by comparing their convergence rate and sensitivity to those of many other algorithms, and through simulations on Nesterov's "bad function".

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arXiv2025

The Internal Model Principle of Time-Varying Optimization

Time-varying optimization problems are central to many engineering applications, where performance metrics and system constraints evolve dynamically with time. Several algorithms have been proposed to address these problems; a common characteristic among them is their implicit reliance on knowledge of the optimizers' temporal variability. In this paper, we provide a fundamental characterization of this property: we show that an algorithm can track time-varying optimizers if and only if it incorporates a model of the temporal variability of the optimization problem. We refer to this concept as the internal model principle of time-varying optimization. Our analysis relies on showing that time-varying optimization problems can be recast as output regulation problems and, by using tools from center manifold theory, we establish necessary and sufficient conditions for exact asymptotic tracking. As a result, these findings enable the design of new algorithms for time-varying optimization. We demonstrate the effectiveness of the approach through numerical experiments on both synthetic problems and the dynamic traffic assignment problem from traffic control.

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arXiv2025

Temporal Variabilities Limit Convergence Rates in Gradient-Based Online Optimization

This paper investigates the fundamental performance limits of gradient-based algorithms for time-varying optimization. Leveraging the internal model principle and root locus techniques, we show that temporal variabilities impose intrinsic limits on the achievable rate of convergence. For a problem with condition ratio $κ$ and time variation whose model has degree $n$, we show that the worst-case convergence rate of any minimal-order gradient-based algorithm is $ρ_\text{TV} = (\frac{κ-1}{κ+1})^{1/n}$. This bound reveals a fundamental tradeoff between problem conditioning, temporal complexity, and rate of convergence. We further construct explicit controllers that attain the bound for low-degree models of time variation.

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arXiv2025

Feedback Optimization of Dynamical Systems in Time-Varying Environments: An Internal Model Principle Approach

Feedback optimization has emerged as a promising approach for regulating dynamical systems to optimal steady states that are implicitly defined by underlying optimization problems. Despite their effectiveness, existing methods face two key limitations: (i) reliable performance is restricted to time-invariant or slowly varying settings, and (ii) convergence rates are limited by the need for the controller to operate orders of magnitude slower than the plant. These limitations can be traced back to the reliance of existing techniques on numerical optimization algorithms. In this paper, we propose a novel perspective on the design of feedback optimization algorithms, by framing these objectives as an output regulation problem. We place particular emphasis on time-varying optimization problems, and show that an algorithm can track time-varying optimizers if and only if it incorporates a model of the temporal variability inherent to the optimization - a requirement that we term the internal model principle of feedback optimization. Building on this insight, we introduce a new design methodology that couples output-feedback stabilization with a control component that drives the system toward the critical points of the optimization problem. This framework enables feedback optimization algorithms to overcome the classical limitations of slow tracking and poor adaptability to time variations.

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arXiv2025

The Fastest Known First-Order Method for Minimizing Twice Continuously Differentiable Smooth Strongly Convex Functions

We consider iterative gradient-based optimization algorithms applied to functions that are smooth and strongly convex. The fastest globally convergent algorithm for this class of functions is the Triple Momentum (TM) method. We show that if the objective function is also twice continuously differentiable, a new, faster algorithm emerges, which we call $C^2$-Momentum (C2M). We prove that C2M is globally convergent and that its worst-case convergence rate is strictly faster than that of TM, with no additional computational cost. We validate our theoretical findings with numerical examples, demonstrating that C2M outperforms TM when the objective function is twice continuously differentiable.

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arXiv2023

Automated Lyapunov Analysis of Primal-Dual Optimization Algorithms: An Interpolation Approach

Primal-dual algorithms are frequently used for iteratively solving large-scale convex optimization problems. The analysis of such algorithms is usually done on a case-by-case basis, and the resulting guaranteed rates of convergence can be conservative. Here we consider a class of first-order algorithms for linearly constrained convex optimization problems, and provide a linear matrix inequality (LMI) analysis framework for certifying worst-case exponential convergence rates. Our approach builds on recent results for interpolation of convex functions and linear operators, and our LMI directly constructs a Lyapunov function certifying the guaranteed convergence rate. By comparing to rates established in the literature, we show that our approach can certify significantly faster convergence for this family of algorithms.

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arXiv2023

A Tutorial on a Lyapunov-Based Approach to the Analysis of Iterative Optimization Algorithms

Iterative gradient-based optimization algorithms are widely used to solve difficult or large-scale optimization problems. There are many algorithms to choose from, such as gradient descent and its accelerated variants such as Polyak's Heavy Ball method or Nesterov's Fast Gradient method. It has long been observed that iterative algorithms can be viewed as dynamical systems, and more recently, as robust controllers. Here, the "uncertainty" in the dynamics is the gradient of the function being optimized. Therefore, worst-case or average-case performance can be analyzed using tools from robust control theory, such as integral quadratic constraints (IQCs). In this tutorial paper, we show how such an analysis can be carried out using an alternative Lyapunov-based approach. This approach recovers the same performance bounds as with IQCs, but with the added benefit of constructing a Lyapunov function.

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arXiv2023

A Tutorial on the Structure of Distributed Optimization Algorithms

We consider the distributed optimization problem for a multi-agent system. Here, multiple agents cooperatively optimize an objective by sharing information through a communication network and performing computations. In this tutorial, we provide an overview of the problem, describe the structure of its algorithms, and use simulations to illustrate some algorithmic properties based on this structure.

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arXiv2022

Absolute Stability via Lifting and Interpolation

We revisit the classical problem of absolute stability; assessing the robust stability of a given linear time-invariant (LTI) plant in feedback with a nonlinearity belonging to some given function class. Standard results typically take the form of sufficient conditions on the LTI plant, the least conservative of which are based on O'Shea--Zames--Falb multipliers. We present an alternative analysis based on lifting and interpolation that directly constructs a Lyapunov function that certifies absolute stability without resorting to frequency-domain inequalities or integral quadratic constraints. In particular, we use linear matrix inequalities to search over Lyapunov functions that are quadratic in the iterates and linear in the corresponding function values of the system in a lifted space. We show that our approach recovers state-of-the-art results on a set of benchmark problems.

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arXiv2022

A Universal Decomposition for Distributed Optimization Algorithms

In the distributed optimization problem for a multi-agent system, each agent knows a local function and must find a minimizer of the sum of all agents' local functions by performing a combination of local gradient evaluations and communicating information with neighboring agents. We prove that every distributed optimization algorithm can be factored into a centralized optimization method and a second-order consensus estimator, effectively separating the "optimization" and "consensus" tasks. We illustrate this fact by providing the decomposition for many recently proposed distributed optimization algorithms. Conversely, we prove that any optimization method that converges in the centralized setting can be combined with any second-order consensus estimator to form a distributed optimization algorithm that converges in the multi-agent setting. Finally, we describe how our decomposition may lead to a more systematic algorithm design methodology.

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arXiv2020

Systematic Analysis of Distributed Optimization Algorithms over Jointly-Connected Networks

We consider the distributed optimization problem, where a group of agents work together to optimize a common objective by communicating with neighboring agents and performing local computations. For a given algorithm, we use tools from robust control to systematically analyze the performance in the case where the communication network is time-varying. In particular, we assume only that the network is jointly connected over a finite time horizon (commonly referred to as B-connectivity), which does not require connectivity at each time instant. When applied to the distributed algorithm DIGing, our bounds are orders of magnitude tighter than those available in the literature.

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arXiv2020

Analysis and Design of First-Order Distributed Optimization Algorithms over Time-Varying Graphs

This work concerns the analysis and design of distributed first-order optimization algorithms over time-varying graphs. The goal of such algorithms is to optimize a global function that is the average of local functions using only local computations and communications. Several different algorithms have been proposed that achieve linear convergence to the global optimum when the local functions are strongly convex. We provide a unified analysis that yields the worst-case linear convergence rate as a function of the condition number of the local functions, the spectral gap of the graph, and the parameters of the algorithm. The framework requires solving a small semidefinite program whose size is fixed; it does not depend on the number of local functions or the dimension of their domain. The result is a computationally efficient method for distributed algorithm analysis that enables the rapid comparison, selection, and tuning of algorithms. Finally, we propose a new algorithm, which we call SVL, that is easily implementable and achieves a faster worst-case convergence rate than all other known algorithms.

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arXiv2020

A Distributed Optimization Algorithm over Time-Varying Graphs with Efficient Gradient Evaluations

We propose an algorithm for distributed optimization over time-varying communication networks. Our algorithm uses an optimized ratio between the number of rounds of communication and gradient evaluations to achieve fast convergence. The iterates converge to the global optimizer at the same rate as centralized gradient descent when measured in terms of the number of gradient evaluations while using the minimum number of communications to do so. Furthermore, the iterates converge at a near-optimal rate when measured in terms of the number of communication rounds. We compare our algorithm with several other known algorithms on a distributed target localization problem.

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arXiv2019

Integral Quadratic Constraints: Exact Convergence Rates and Worst-Case Trajectories

We consider a linear time-invariant system in discrete time where the state and input signals satisfy a set of integral quadratic constraints (IQCs). Analogous to the autonomous linear systems case, we define a new notion of spectral radius that exactly characterizes stability of this system. In particular, (i) when the spectral radius is less than one, we show that the system is asymptotically stable for all trajectories that satisfy the IQCs, and (ii) when the spectral radius is equal to one, we construct an unstable trajectory that satisfies the IQCs. Furthermore, we connect our new definition of the spectral radius to the existing literature on IQCs.

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arXiv2018

A Canonical Form for First-Order Distributed Optimization Algorithms

We consider the distributed optimization problem in which a network of agents aims to minimize the average of local functions. To solve this problem, several algorithms have recently been proposed where agents perform various combinations of communication with neighbors, local gradient computations, and updates to local state variables. In this paper, we present a canonical form that characterizes any first-order distributed algorithm that can be implemented using a single round of communication and gradient computation per iteration, and where each agent stores up to two state variables. The canonical form features a minimal set of parameters that are both unique and expressive enough to capture any distributed algorithm in this class. The generic nature of our canonical form enables the systematic analysis and design of distributed optimization algorithms.

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arXiv2018

Lyapunov Functions for First-Order Methods: Tight Automated Convergence Guarantees

We present a novel way of generating Lyapunov functions for proving linear convergence rates of first-order optimization methods. Our approach provably obtains the fastest linear convergence rate that can be verified by a quadratic Lyapunov function (with given states), and only relies on solving a small-sized semidefinite program. Our approach combines the advantages of performance estimation problems (PEP, due to Drori & Teboulle (2014)) and integral quadratic constraints (IQC, due to Lessard et al. (2016)), and relies on convex interpolation (due to Taylor et al. (2017c;b)).

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arXiv2018

Tutorial on dynamic average consensus: the problem, its applications, and the algorithms

This paper considers the problem of dynamic average consensus algorithm design for a group of communicating agents. This problem consists of designing a distributed algorithm that enables a group of agents with communication and computation capabilities to use local interactions to track the average of locally time-varying reference signals at each agent. The objective of this article is to provide an overview of the dynamic average consensus problem that serves as a comprehensive introduction to the problem definition, its applications, and the distributed methods available to solve them. Our primary intention, rather than providing a full account of all the available literature, is to introduce the reader, in a tutorial fashion, to the main ideas behind dynamic average consensus algorithms, the performance trade-offs considered in their design, and the requirements needed for their analysis and convergence guarantees.

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