Convergence Rates of Evolutionary Algorithms for Quadratic Convex Functions with Rank-Deficient Hessian

Modularity is a system property of many natural and artificial adaptive systems. Evolutionary algorithms designed to produce modular solutions have increased convergence rates and improved generalization ability; however, their performance can be impacted if the task is inherently nonmodular. Previously, we have shown that some design variables can influence whether the task on the remaining variables is inherently modular. We investigate the possibility of exploiting that dependence to simplify optimization and arrive at a general design pattern that we use to show that evolutionary search can seek such modularity-inducing design variable values, thus easing subsequent search for highly fit, modular organization within the remaining design variables. We investigate this approach with embodied agents in which evolutionary discovery of morphology enables subsequent discovery of highly fit, modular controllers and show that it benefits from biasing search toward modular controllers and setting the mutation rate for control policies higher than that for morphology. This work also reinforces our previous finding that the relationship between modularity and evolvability that is well-studied in nonembodied systems can, under certain conditions, be generalized to include embodied systems as well and provides a practical approach to satisfying the conditions in question.

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Local convergence rates of simple evolutionary algorithms with Cauchy mutations

IEEE Transactions on Evolutionary Computation ◽

10.1109/4235.687885 ◽

1997 ◽

Vol 1 (4) ◽

pp. 249-258 ◽

Cited By ~ 80

Author(s):

G. Rudolph

Keyword(s):

Evolutionary Algorithms ◽

Local Convergence ◽

Convergence Rates

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Understanding the acceleration phenomenon via high-resolution differential equations

Mathematical Programming ◽

10.1007/s10107-021-01681-8 ◽

2021 ◽

Author(s):

Bin Shi ◽

Simon S. Du ◽

Michael I. Jordan ◽

Weijie J. Su

Keyword(s):

High Resolution ◽

Differential Equations ◽

Gradient Method ◽

Convex Functions ◽

Convergence Rates ◽

Qualitative Difference ◽

Optimization Methods ◽

Strongly Convex Functions ◽

Strongly Convex ◽

Accelerated Gradient

AbstractGradient-based optimization algorithms can be studied from the perspective of limiting ordinary differential equations (ODEs). Motivated by the fact that existing ODEs do not distinguish between two fundamentally different algorithms—Nesterov’s accelerated gradient method for strongly convex functions (NAG-) and Polyak’s heavy-ball method—we study an alternative limiting process that yields high-resolution ODEs. We show that these ODEs permit a general Lyapunov function framework for the analysis of convergence in both continuous and discrete time. We also show that these ODEs are more accurate surrogates for the underlying algorithms; in particular, they not only distinguish between NAG- and Polyak’s heavy-ball method, but they allow the identification of a term that we refer to as “gradient correction” that is present in NAG- but not in the heavy-ball method and is responsible for the qualitative difference in convergence of the two methods. We also use the high-resolution ODE framework to study Nesterov’s accelerated gradient method for (non-strongly) convex functions, uncovering a hitherto unknown result—that NAG- minimizes the squared gradient norm at an inverse cubic rate. Finally, by modifying the high-resolution ODE of NAG-, we obtain a family of new optimization methods that are shown to maintain the accelerated convergence rates of NAG- for smooth convex functions.

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