IYH Discussion of Doerfler et al (2024) Systems Theory of Algorithms http://dx.doi.org/10.1109/LCSYS.2024.3406943

This paper advocates viewing numerical algorithms as open dynamical systems that interact with their environment; addresses the challenge of understanding and managing the complexity of algorithms that dynamically engage with real-world scenarios, physical systems, social networks, or other algorithms in real-time. This includes but is not limited to algorithms used in optimization-based control, reinforcement learning, recommender systems, and distributed optimization.

## Key Terms

**Dynamical systems**: Mathematical models that describe how a system's state evolves over time.**Closed vs. open systems**: Isolated systems vs. systems that interact with their environment.**Offline vs. online algorithms**: Algorithms that operate on batch data vs. those that process streaming data in real-time.**Black-box vs. structured systems**: Algorithms that appear as monolithic entities vs. those with interpretable internal structure.**Lur'e problem**: The feedback interconnection of a linear dynamical system with a static nonlinearity.**Integral quadratic constraints**: A mathematical tool used to analyze the stability and performance of algorithms.**Small-gain theorem**: A system-theoretic result that can be used to certify the stability of interconnected systems.

## Technical and Mathematical Details

**Lyapunov Stability Analysis**: Lyapunov functions, a cornerstone of control theory, can be used to analyze the stability of both continuous-time and discrete-time algorithms. By finding a Lyapunov function that decreases along the algorithm's trajectories, we can establish convergence to an equilibrium point.**Integral Quadratic Constraints (IQCs)**: IQCs provide a powerful framework for characterizing the input-output properties of nonlinear systems and analyzing the stability of interconnections. By using IQCs to represent the behavior of nonlinear components in an algorithm, we can derive stability conditions and performance bounds.**Small-Gain Theorem**: The small-gain theorem, a fundamental result in control theory, provides conditions for the stability of interconnected systems based on their individual input-output gains. This theorem can be applied to analyze the stability of algorithms interconnected with each other or with physical plants, as illustrated in the analysis of sub-optimal MPC.**Singular Perturbation Analysis**: This technique is used to analyze systems with multiple time scales, such as those involving fast feedback control loops and slower optimization or learning algorithms. It allows us to separate the system's dynamics into different time scales and analyze their behavior independently.

## Approach

The paper showcases several successful applications of system-theoretic tools in the algorithmic domain. The examples cover optimization and learning algorithms, real-time algorithms in feedback loops, and decision-making architectures. - Using Lyapunov analysis, integral quadratic constraints, and small-gain theorems to characterize the stability, robustness, and performance of optimization algorithms and their interconnections. - Leveraging the feedback control structure of primal-dual optimization algorithms to draw connections to proportional-integral control. - Modeling gradient-based optimization algorithms as the feedback interconnection of a linear dynamical system and a static nonlinearity (the Lur'e problem), enabling the use of powerful analysis tools. - Applying small-gain and singular perturbation analysis to characterize the stability and performance of sub-optimal Model Predictive Control, where the optimization algorithm is interconnected with the physical plant. - Interpreting distributed optimization algorithms, such as gradient tracking, as the feedback interconnection of the optimization dynamics and a consensus protocol.

## System-Theoretic Design and Analysis of Real-Time Control Systems

The system-theoretic perspective on algorithms offers a powerful framework for enhancing the design and analysis of real-time control systems, particularly those involving complex optimization or learning components. Here's a detailed exploration of how this approach can be applied:

### Real-Time Optimization-Based Control

**Model Predictive Control (MPC)**: As exemplified in the paper, sub-optimal MPC can be analyzed as a feedback interconnection between the optimization algorithm and the physical plant. System-theoretic tools like the small-gain theorem provide conditions for stability and robustness, even when the optimization problem is not solved to optimality at each time step. This approach can be extended to more complex MPC formulations, such as nonlinear MPC or distributed MPC.**Feedback Optimization**: Online feedback optimization, where the optimization algorithm directly uses measurements from the plant instead of relying on an explicit model can be analyzed using input-to-state stability and small-gain theory. This approach is particularly useful for systems with complex or unknown dynamics, where obtaining accurate models is challenging.**Extremum Seeking**: Extremum seeking control, which uses a dither signal to probe the system and iteratively adjust control inputs to find an optimal operating point can also be viewed as a feedback interconnection between the optimization algorithm and the plant. System-theoretic analysis can help characterize convergence, stability, and robustness properties of extremum seeking controllers.

### Adaptive Control and System Identification

**Joint Convergence Analysis**: The system-theoretic perspective can be used to analyze the joint convergence of parameter estimation (system identification) and control adaptation in adaptive control systems. By modeling both the estimator and the controller as dynamical systems, we can establish conditions for stability and convergence of the overall system.**Robustness to Model Uncertainty**: System-theoretic tools can be employed to analyze the robustness of adaptive controllers to uncertainties in the identified model. Techniques like robust control and adaptive control with persistent excitation can be used to guarantee performance even when the model is not perfectly known.

### Learning-Based Control

**Reinforcement Learning (RL)**: Reinforcement learning algorithms, which learn optimal control policies by interacting with the environment and receiving rewards, can be analyzed as dynamical systems. This allows for the use of tools like Lyapunov stability analysis to characterize convergence and stability properties of RL algorithms.**Neural Network Controllers**: Neural networks used as controllers can be interpreted as nonlinear dynamical systems. By analyzing their stability and robustness using system-theoretic tools, we can gain insights into their behavior and provide guarantees on their performance.

### Distributed Control Systems

**Consensus and Coordination**: Distributed optimization algorithms, such as gradient tracking, rely on consensus protocols to ensure that agents in a network reach agreement on a common solution. System-theoretic analysis can be used to characterize the convergence rate of consensus protocols and their robustness to communication delays or failures.**Multi-Agent Control**: The system-theoretic perspective is well-suited for analyzing the behavior of multi-agent control systems, where agents interact with each other and their environment. Tools like graph theory and network control theory can be used to design controllers that achieve coordinated behavior and achieve global objectives.

## Potential Limitations of the System-Theoretic Approach

**Highly Nonlinear or Discontinuous Behavior**: Classical system-theoretic tools, often based on linearization or smoothness assumptions, can be challenging to apply to algorithms with highly nonlinear or discontinuous behavior. New techniques might be needed to analyze the stability and robustness of such algorithms.**Complexity of Analysis**: Applying system-theoretic tools to complex algorithms can be mathematically challenging and require specialized expertise.**Computational Costs**: The computational cost of performing system-theoretic analysis, such as finding Lyapunov functions or solving IQC problems, can be significant, especially for large-scale systems.

## Addressing Limitations

**Nonlinear Analysis Tools**: Expanding the use of nonlinear analysis tools, such as contraction theory, passivity theory, and Lyapunov-based methods for hybrid systems, to handle more complex algorithmic behaviors.**Numerical and Computational Methods**: Developing efficient numerical methods and leveraging computational tools like automatic differentiation to reduce the computational burden of system-theoretic analysis.**Approximation and Abstraction**: Using appropriate approximations and abstractions to simplify the analysis of complex algorithms while still capturing their essential behavior.

## Conflict of Interest

The paper does not mention any apparent conflicts of interest. The authors are affiliated with reputable academic institutions and research centers.