Modeling Algorithms as Open Dynamical Systems: Towards a Systems Theory of Algorithms

Core Concepts

Algorithms should be viewed as open dynamical systems that interact with their environment, including other algorithms, physical systems, humans, or databases, rather than as isolated pieces of code. This systems-theoretic perspective provides a powerful framework for analyzing, designing, and understanding the behavior of modern computational approaches in control, learning, optimization, and decision-making.

Abstract

The content advocates for a shift in perspective on how algorithms are viewed, from isolated pieces of code to open dynamical systems that interact with their environment. This systems-theoretic perspective is particularly relevant for modern computational approaches in control, learning, optimization, and decision-making, where algorithms often need to be reactive rather than isolated.
The paper presents several examples that demonstrate the advantages of the systems-theoretic perspective on algorithms:

Analysis, Design, and Interplay of Algorithms in Optimization and Learning:

Algorithms can be abstracted as feedback interconnections of dynamical systems, allowing the use of system-theoretic tools for analysis and design.
Examples include primal-dual algorithms, gradient-based optimization, and the use of feedforward for time-varying optimization.

Real-Time Algorithms in Feedback Loops:

Algorithms dynamically engage in real-time with real-world scenarios, such as physical plants, social networks, or other algorithms.
The systems-theoretic perspective, using tools like small-gain theory and singular perturbation analysis, can provide insights into the stability and robustness of these cyber-physical interconnections.
Examples include sub-optimal Model Predictive Control, distributed optimization via gradient tracking, and online feedback optimization.

Decision-Making Architectures:

Layered decision-making architectures, or control stacks, can be viewed as interconnected algorithms, where the systems-theoretic perspective can provide insights into propagation of uncertainty, (sub)optimality of architectures, and decomposition into sub-tasks.
Examples include internet congestion control and the separation and coupling of planning and tracking in control stacks.

The paper concludes by highlighting the potential of the systems-theoretic perspective in addressing emerging challenges, such as performative prediction and data-driven predictive control.

Stats

None.

Quotes

None.

Key Insights Distilled From

Towards a Systems Theory of Algorithms

by Flor... at arxiv.org 05-01-2024

https://arxiv.org/pdf/2401.14029.pdf

Deeper Inquiries

How can the systems-theoretic perspective on algorithms be extended to address the challenge of performative prediction, where algorithmic decisions affect the data-generating environment

The systems-theoretic perspective on algorithms can be extended to address the challenge of performative prediction by considering the feedback loop between the algorithm and the environment. In performative prediction, the decisions made by the algorithm impact the data it receives, leading to a dynamic and evolving data-generating process. By viewing this interaction as a feedback system, similar to control systems, we can analyze how the algorithm's actions influence the environment and how the environment, in turn, affects the algorithm's performance. This perspective allows for a deeper understanding of the interplay between the algorithm and the data it operates on, enabling the development of more robust and adaptive algorithms in scenarios where performative prediction is a concern.

What are the potential limitations or drawbacks of viewing algorithms as open dynamical systems, and how can these be addressed

Viewing algorithms as open dynamical systems can have potential limitations and drawbacks that need to be addressed. One limitation is the increased complexity introduced by considering algorithms as interconnected systems interacting with their environment. This complexity can make it challenging to analyze and predict the behavior of the algorithm under different conditions. Additionally, the openness of algorithms can lead to issues related to stability, robustness, and performance, especially in real-time applications where feedback loops are prevalent. To address these limitations, it is essential to develop robust control strategies, incorporate uncertainty quantification techniques, and implement adaptive mechanisms that can adjust the algorithm's behavior based on the feedback received from the environment. By integrating these approaches, the drawbacks of viewing algorithms as open dynamical systems can be mitigated, leading to more reliable and efficient algorithmic designs.

How can the systems-theoretic approach to algorithm design and analysis be integrated with other computational tools, such as automatic differentiation and parallelization, to create a more comprehensive framework for tackling complex algorithmic challenges

Integrating the systems-theoretic approach to algorithm design and analysis with other computational tools, such as automatic differentiation and parallelization, can create a more comprehensive framework for tackling complex algorithmic challenges. Automatic differentiation can be used to efficiently compute gradients and optimize algorithm parameters, enhancing the performance and convergence of iterative algorithms. Parallelization techniques can help distribute computational tasks across multiple processors or nodes, enabling faster execution of algorithms and scalability to handle large datasets or complex models. By combining systems theory principles with these computational tools, researchers and practitioners can leverage the strengths of each approach to address a wide range of algorithmic challenges effectively. This integration can lead to the development of more sophisticated and efficient algorithms that are capable of handling the complexities of modern computational tasks.