scholarly journals Robust Accelerated Gradient Methods for Smooth Strongly Convex Functions

2020 ◽  
Vol 30 (1) ◽  
pp. 717-751
Author(s):  
Necdet Serhat Aybat ◽  
Alireza Fallah ◽  
Mert Gürbüzbalaban ◽  
Asuman Ozdaglar
Keyword(s):  
Convex Functions ◽  
Gradient Methods ◽  
Strongly Convex Functions ◽  
Strongly Convex ◽  
Accelerated Gradient ◽  
Accelerated Gradient Methods

scholarly journals Understanding the acceleration phenomenon via high-resolution differential equations

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.

scholarly journals The second Hankel determinant for strongly convex and Ozaki close-to-convex functions

2021 ◽  
Author(s):  
Young Jae Sim ◽  
Adam Lecko ◽  
Derek K. Thomas
Keyword(s):  
Unit Disk ◽  
Convex Functions ◽  
Univalent Functions ◽  
Inverse Function ◽  
Invariance Property ◽  
Hankel Determinant ◽  
Sharp Bounds ◽  
Strongly Convex Functions ◽  
Strongly Convex ◽  
Second Hankel Determinant

AbstractLet f be analytic in the unit disk $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1 \}$$ D = { z ∈ C : | z | < 1 } , and $${{\mathcal {S}}}$$ S be the subclass of normalized univalent functions given by $$f(z)=z+\sum _{n=2}^{\infty }a_n z^n$$ f ( z ) = z + ∑ n = 2 ∞ a n z n for $$z\in {\mathbb {D}}$$ z ∈ D . We give sharp bounds for the modulus of the second Hankel determinant $$ H_2(2)(f)=a_2a_4-a_3^2$$ H 2 ( 2 ) ( f ) = a 2 a 4 - a 3 2 for the subclass $$ {\mathcal F_{O}}(\lambda ,\beta )$$ F O ( λ , β ) of strongly Ozaki close-to-convex functions, where $$1/2\le \lambda \le 1$$ 1 / 2 ≤ λ ≤ 1 , and $$0<\beta \le 1$$ 0 < β ≤ 1 . Sharp bounds are also given for $$|H_2(2)(f^{-1})|$$ | H 2 ( 2 ) ( f - 1 ) | , where $$f^{-1}$$ f - 1 is the inverse function of f. The results settle an invariance property of $$|H_2(2)(f)|$$ | H 2 ( 2 ) ( f ) | and $$|H_2(2)(f^{-1})|$$ | H 2 ( 2 ) ( f - 1 ) | for strongly convex functions.

Bounds having Riemann type quantum integrals via strongly convex functions

2017 ◽  
Vol 54 (2) ◽  
pp. 221-240 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
Gabriela Cristescu ◽  
Muhammad Uzair Awan
Keyword(s):  
Convex Functions ◽  
Strongly Convex Functions ◽  
Strongly Convex
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