Learning to Optimize with Convergence Guarantees Using Nonlinear System Theory

Core Concepts

Automated synthesis of reliable, efficient algorithms for smooth non-convex optimization.

Abstract

The article discusses the need for algorithms that efficiently navigate complex optimization landscapes. Traditional methods require meticulous hyperparameter tuning for non-convex problems. The emerging paradigm of Learning to Optimize (L2O) automates algorithm discovery but lacks a theoretical framework for analyzing convergence and robustness. By leveraging nonlinear system theory, the authors propose an unconstrained parametrization for convergent algorithms. This framework is compatible with automatic differentiation tools, ensuring convergence while learning to optimize. The paper establishes methods to learn high-performance optimization algorithms that are inherently convergent for smooth non-convex functions. It combines control system theory's emphasis on convergence and robustness guarantees with machine learning's ability to tackle user-defined performance metrics through automatic differentiation.

Stats

Classical gradient descent methods offer strong theoretical guarantees for convex problems.
Widely-used optimization algorithms include vanilla gradient descent, the heavy-ball method, and Nesterov’s accelerated method.
For training deep neural networks, numerical optimization algorithms like gradient descent are relied upon.
The L2O approach parametrizes algorithms in a general way and performs meta-training over these parameters.
Learned optimizers may lack convergence guarantees on unseen tasks.
A mitigation proposed in reinforcement meta-learning helps avoid compounding errors.
For convex objective functions, provable convergence guarantees of learned optimizers were considered by exploiting a conservative fall-back mechanism.

Quotes

"Learning to optimize automates the discovery of algorithms with optimized performance leveraging learning models and data."
"Our key contribution is reformulating the problem of learning optimal convergent algorithms into an equivalent, unconstrained one."
"Our methodology ensures convergence even when dealing with incomplete gradient measurements."

Key Insights Distilled From

Learning to optimize with convergence guarantees using nonlinear system theory

by Andrea Marti... at arxiv.org 03-15-2024

https://arxiv.org/pdf/2403.09389.pdf

Deeper Inquiries

How can the integration of closed-loop cyber-physical systems benefit from this automated synthesis of reliable optimization algorithms

The integration of closed-loop cyber-physical systems can greatly benefit from the automated synthesis of reliable optimization algorithms in several ways. Firstly, by leveraging learned optimizers that guarantee convergence for smooth non-convex functions, these systems can operate more efficiently and effectively. The ability to automate the discovery of algorithms with optimized performance ensures that the systems can navigate complex optimization landscapes reliably. This is crucial for tasks in machine learning and optimal control where numerical methods are heavily relied upon.
Moreover, the use of nonlinear system theory provides a solid theoretical framework to analyze convergence and robustness guarantees of the learned algorithms. By incorporating feedback optimization principles into closed-loop systems, it enables self-regulation and convergence towards desired solutions for nonlinear optimization problems. This not only enhances the overall performance but also increases the adaptability and resilience of cyber-physical systems in dynamic environments.
Furthermore, by parametrizing convergent update rules through automatic differentiation tools as proposed in this context, it allows for customizable performance metrics tailored to specific system requirements. This level of customization ensures that the synthesized algorithms align closely with the objectives and constraints of each individual cyber-physical system, leading to improved stability, efficiency, and reliability in their operation.

What counterarguments exist against relying on learned optimizers lacking convergence guarantees on unseen tasks

While relying on learned optimizers without convergence guarantees on unseen tasks may present certain challenges or limitations, there are potential counterarguments against dismissing them outright:

Empirical Performance: Despite lacking formal theoretical guarantees on all unseen tasks, learned optimizers have demonstrated outstanding empirical performance across various applications. Their ability to discover shortcuts or novel strategies that traditional hand-crafted algorithms might overlook can lead to significant improvements in efficiency and effectiveness.

Adaptability: Learned optimizers are often designed with flexibility in mind – they can adapt quickly to new problem instances or changing environments due to their data-driven nature. This adaptability makes them well-suited for scenarios where traditional approaches struggle due to rigid design constraints.

Reinforcement Learning Techniques: Techniques like reinforcement meta-learning offer ways to mitigate errors or lack of convergence guarantees by continuously refining learned policies based on feedback from task outcomes over time. These iterative processes help improve generalization capabilities even without explicit theoretical assurances.

Hybrid Approaches: Combining elements from both hand-crafted algorithms with proven theoretical foundations and learned optimizers could provide a balanced approach that leverages strengths from both sides while mitigating weaknesses.

How can the principles of nonlinear system theory be applied beyond optimization landscapes

The principles of nonlinear system theory extend far beyond just optimizing landscapes; they offer a versatile framework applicable across various domains:

Control Systems Design: Nonlinear system theory plays a vital role in designing robust control strategies for complex dynamical systems characterized by nonlinearity or uncertainty.

2Modeling Biological Systems: In fields like neuroscience or physiology where biological processes exhibit intricate nonlinear dynamics,
nonlinear system theory helps model neural networks' behavior accurately.
3Signal Processing: Understanding how signals interact within nonlinear frameworks aids signal processing techniques such as filtering,
compression,and feature extraction.
4Robotics: Applying concepts like stability analysis helps develop controllers ensuring robots' precise movements despite environmental uncertainties
and disturbances
5Economic Modeling: Economic models often involve nonlinear relationships between variables; utilizing nonlinear system theory improves forecasting accuracy
and policy decision-making
By applying these principles beyond optimization landscapes,theoretical insights gained contribute significantlyto advancing research,influencing technological innovations,and addressing real-world challenges across diverse disciplines