Einblick - Optimization algorithms - # Convergence analysis of Nesterov's accelerated gradient (NAG) and fast iterative shrinkage-thresholding algorithm (FISTA)

Linear Convergence of Forward-Backward Accelerated Algorithms for Strongly Convex Functions Without Knowledge of the Modulus of Strong Convexity

Q: What are the potential applications of the linear convergence results for NAG and FISTA in real-world optimization problems

The linear convergence results for NAG and FISTA have significant implications for real-world optimization problems, especially in scenarios where the objective functions are strongly convex. These results indicate that these algorithms can achieve convergence at a rate that is independent of the modulus of strong convexity, providing a more predictable and efficient optimization process. In practical applications, such as machine learning, image processing, and signal processing, where optimization plays a crucial role, the linear convergence of NAG and FISTA can lead to faster convergence to optimal solutions. This can result in reduced computational time and resources required for training models or solving complex optimization problems. Additionally, the ability to achieve linear convergence without prior knowledge of the modulus of strong convexity makes these algorithms more versatile and applicable to a wider range of optimization tasks.

Q: How do the convergence rates of NAG and FISTA compare to other accelerated first-order optimization methods for strongly convex functions

The convergence rates of NAG and FISTA for strongly convex functions are compared to other accelerated first-order optimization methods in terms of their efficiency and effectiveness. NAG and FISTA demonstrate linear convergence, which is a significant improvement over the traditional gradient descent methods that exhibit slower convergence rates. In the context of strongly convex functions, NAG and FISTA outperform other accelerated methods by achieving convergence without the need for prior knowledge of the modulus of strong convexity. This characteristic makes them more robust and easier to apply in various optimization problems. While other methods may also offer accelerated convergence, the linear convergence of NAG and FISTA sets them apart in terms of efficiency and simplicity.

Q: Can the techniques used in this paper be extended to analyze the convergence of other forward-backward accelerated algorithms in the presence of non-smooth terms

The techniques used in the paper can potentially be extended to analyze the convergence of other forward-backward accelerated algorithms in the presence of non-smooth terms. By leveraging the high-resolution ordinary differential equation (ODE) framework and the construction of Lyapunov functions, similar convergence analyses can be conducted for a wide range of optimization algorithms that incorporate forward-backward techniques. The key lies in adapting the methodology to suit the specific characteristics of the algorithms under consideration. By formulating appropriate Lyapunov functions and utilizing phase-space representations, it is possible to investigate the convergence properties of other accelerated algorithms in the presence of non-smooth terms. This extension can provide valuable insights into the convergence behavior of these algorithms and enhance our understanding of their performance in optimization tasks.

Kernkonzepte

The paper establishes the linear convergence of NAG and FISTA for strongly convex functions without requiring any prior knowledge of the modulus of strong convexity.

Zusammenfassung

The paper addresses the open question of whether NAG and FISTA exhibit linear convergence for strongly convex functions, without requiring any prior knowledge of the strongly convex modulus. The key contributions are:

The paper establishes the linear convergence of NAG for strongly convex functions by formulating an innovative Lyapunov function that incorporates a dynamically adapting coefficient for the kinetic energy. This linear convergence is shown to be independent of the parameter r.
The paper refines a key inequality associated with strong convexity to encompass the proximal setting, reconciling the theoretical delineation between smooth and composite optimization. Using the implicit-velocity phase-space representation, the Lyapunov function guarantees the linear convergence of function values within FISTA and the square of the proximal subgradient norm.
The paper provides an intuitive analysis on a quadratic function, demonstrating how both the function value and the square of the gradient norm converge linearly in NAG, with the linear convergence rate being rarely influenced by the choice of the parameter r.

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Statistiken

f(x) = 1/2 * μ * x^2, where μ > 0.
xk = yk-1 - μ * s * yk-1
yk = xk + (k-1)/(k+r) * (xk - xk-1)

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Wichtige Erkenntnisse aus

Linear convergence of forward-backward accelerated algorithms without knowledge of the modulus of strong convexity

by Bowen Li,Bin... um arxiv.org 04-10-2024

https://arxiv.org/pdf/2306.09694.pdf

Linear convergence of forward-backward accelerated algorithms without knowledge of the modulus of strong convexity

Tiefere Fragen

What are the potential applications of the linear convergence results for NAG and FISTA in real-world optimization problems

The linear convergence results for NAG and FISTA have significant implications for real-world optimization problems, especially in scenarios where the objective functions are strongly convex. These results indicate that these algorithms can achieve convergence at a rate that is independent of the modulus of strong convexity, providing a more predictable and efficient optimization process.
In practical applications, such as machine learning, image processing, and signal processing, where optimization plays a crucial role, the linear convergence of NAG and FISTA can lead to faster convergence to optimal solutions. This can result in reduced computational time and resources required for training models or solving complex optimization problems. Additionally, the ability to achieve linear convergence without prior knowledge of the modulus of strong convexity makes these algorithms more versatile and applicable to a wider range of optimization tasks.

How do the convergence rates of NAG and FISTA compare to other accelerated first-order optimization methods for strongly convex functions

The convergence rates of NAG and FISTA for strongly convex functions are compared to other accelerated first-order optimization methods in terms of their efficiency and effectiveness. NAG and FISTA demonstrate linear convergence, which is a significant improvement over the traditional gradient descent methods that exhibit slower convergence rates.
In the context of strongly convex functions, NAG and FISTA outperform other accelerated methods by achieving convergence without the need for prior knowledge of the modulus of strong convexity. This characteristic makes them more robust and easier to apply in various optimization problems. While other methods may also offer accelerated convergence, the linear convergence of NAG and FISTA sets them apart in terms of efficiency and simplicity.

Can the techniques used in this paper be extended to analyze the convergence of other forward-backward accelerated algorithms in the presence of non-smooth terms

The techniques used in the paper can potentially be extended to analyze the convergence of other forward-backward accelerated algorithms in the presence of non-smooth terms. By leveraging the high-resolution ordinary differential equation (ODE) framework and the construction of Lyapunov functions, similar convergence analyses can be conducted for a wide range of optimization algorithms that incorporate forward-backward techniques.
The key lies in adapting the methodology to suit the specific characteristics of the algorithms under consideration. By formulating appropriate Lyapunov functions and utilizing phase-space representations, it is possible to investigate the convergence properties of other accelerated algorithms in the presence of non-smooth terms. This extension can provide valuable insights into the convergence behavior of these algorithms and enhance our understanding of their performance in optimization tasks.