Core Concepts

The proposed closed-loop damping system (LD) achieves a convergence rate arbitrarily close to the optimal rate for convex optimization problems.

Abstract

The paper introduces an autonomous system with closed-loop damping, called (LD), for first-order convex optimization. While optimal convergence rates are typically achieved by non-autonomous methods with open-loop damping, the authors show that their closed-loop damping system (LD) exhibits a rate arbitrarily close to the optimal one.
The key aspects are:
The authors design the damping coefficient γ in the Inertial Damped Gradient (IDGγ) system using a Lyapunov function E, which is the sum of the function value and the squared velocity. This makes (LD) an autonomous system.
The authors prove that the Lyapunov function E is non-increasing along the trajectory of (LD), and they show that E converges to 0 as time goes to infinity. This implies that the function values converge to the optimal value.
By analyzing the rate of convergence of E, the authors prove that the function values converge at a rate arbitrarily close to the optimal rate of o(1/t^2), which is achieved by the non-autonomous Asymptotically Vanishing Damping (AVDa) system.
The authors also derive a practical algorithm, called LYDIA, by discretizing the (LD) system, and they provide theoretical guarantees for the convergence of the algorithm.
Numerical experiments are presented, supporting the theoretical findings and showing the advantages of the closed-loop damping approach compared to the open-loop damping.

Stats

f(x(t)) - f* = o(1/t^(2-δ))
∥ẋ(t)∥ → 0 as t → +∞

Quotes

"Can one design the damping γ in (IDGγ) in a closed-loop manner (so as to make the ODE autonomous) while still achieving the optimal convergence rate of (AVDa)?"
"Our version of (IDGγ) then reads: ẍ(t) + √E(t) ẋ(t) + ∇f(x(t)) = 0, ∀t ≥ t0, (LD) and is called (LD) for Lyapunov Damping."

Key Insights Distilled From

by Severin Maie... at **arxiv.org** 04-16-2024

Deeper Inquiries

The proposed closed-loop damping approach can be extended to constrained optimization problems by incorporating the constraints into the Lyapunov function used for damping. In the context of constrained optimization, the Lyapunov function would need to capture not only the optimality gap but also the violation of constraints. By designing a Lyapunov function that accounts for both the optimality and feasibility aspects of the problem, the closed-loop damping system can guide the optimization process towards satisfying the constraints while minimizing the objective function. This extension would involve formulating the constraints as additional terms in the Lyapunov function and adjusting the damping coefficient accordingly to ensure convergence towards feasible solutions.

The (LD) system, with its closed-loop damping mechanism based on the Lyapunov function, has potential applications beyond first-order convex optimization. Some of the potential applications include:
Non-convex Optimization: The closed-loop damping idea can be applied to non-convex optimization problems where finding global optima is challenging. By designing a Lyapunov function that captures the characteristics of non-convex functions, the (LD) system can navigate through complex landscapes and converge to good local optima efficiently.
Stochastic Optimization: In stochastic optimization, where the objective function involves random variables, the closed-loop damping approach can be used to adaptively adjust the damping coefficient based on the stochastic behavior of the function. By incorporating stochasticity into the Lyapunov function, the system can effectively handle uncertainties and fluctuations in the optimization process.
Machine Learning: The (LD) system can be applied to various machine learning tasks such as training deep neural networks, optimizing hyperparameters, or reinforcement learning. By integrating the closed-loop damping mechanism into the optimization algorithms used in machine learning, better convergence rates and stability can be achieved, leading to improved model performance.

The closed-loop damping idea can indeed be applied to other types of optimization problems, including non-convex and stochastic optimization.
Non-Convex Optimization: In non-convex optimization, the closed-loop damping approach can help in navigating complex and rugged landscapes to find good local optima. By designing a Lyapunov function that captures the non-convexity of the objective function, the system can effectively converge to satisfactory solutions even in non-convex settings.
Stochastic Optimization: For stochastic optimization problems, the closed-loop damping mechanism can be utilized to adapt to the randomness and uncertainty in the objective function. By incorporating stochastic elements into the Lyapunov function and adjusting the damping coefficient based on stochastic gradients or samples, the system can efficiently optimize stochastic objectives while ensuring convergence and stability.
Overall, the closed-loop damping idea is versatile and can be adapted to various optimization scenarios beyond first-order convex optimization, providing a robust and efficient approach to tackling optimization challenges in different domains.

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