Performance Estimation Problem Approach

• Performance Estimation Problem (PEP) approaches are formal methodologies that recast algorithm performance evaluation as an optimization problem, yielding tight worst-case guarantees.
• By leveraging semidefinite programming and saddle point formulations, PEP systematically derives quadratic Lyapunov functions to certify convergence rates.
• Integration with tools like PEPit automates constraint encoding and SDP solving, enabling practical analysis of complex, high-dimensional, or stochastic optimization algorithms.

The Performance Estimation Problem (PEP) approach refers to a class of methodologies that formalize the evaluation of algorithmic performance, typically through worst-case or expected measures, by posing a mathematical optimization or analysis problem whose solution yields explicit and often tight performance guarantees. In modern research, PEP frameworks are especially prevalent in optimization and signal processing, enabling the precise characterization of algorithms in complex scenarios where direct closed-form analysis is difficult or impossible.

1. Formalization of the Performance Estimation Problem

PEP recasts the analysis of the performance of iterative algorithms as an optimization problem—often, but not exclusively, a semidefinite program (SDP)—designed to compute the worst-case value of a target functional (e.g., optimality gap, error, or constraint violation) over all allowable input data and algorithmic iterates consistent with the assumptions on the function class (such as convexity, Lipschitz smoothness, strong convexity, or operator monotonicity).

More precisely, given:

• a functional class $\mathcal{F}$,
• an iterative algorithm $A$ with a set of update rules,
• a target performance measure $R(f, \{x_k\})$ (e.g., $f(x_N) - f(x^*)$),

the PEP seeks to solve: $$\sup_{f \in \mathcal{F}, \{x_k\}} R(f, \{x_k\}) \quad \text{subject to all interpolation conditions and algorithmic dynamics}.$$ This maximal value then represents the worst-case performance of algorithm $A$ over $\mathcal{F}$ for the specified measure.

A critical aspect is the encoding of function class properties and algorithmic iterates as constraints, usually represented via certain matrix inequalities (for example, interpolation inequalities for convex functions and the structure of updates encoded via Gram matrices).

2. Quadratic Lyapunov Functions via PEP Saddle Point Problems

A significant recent advancement is the recognition that quadratic Lyapunov functions—central to certifying linear (geometric) convergence rates—can be systematically discovered by solving a specific saddle point formulation of the PEP.

Consider an algorithm with update $x^+ = A(f, x)$, a candidate quadratic Lyapunov function $V(f, x) = x^\top Q x + q^\top N(f, x)$ (with $Q \succeq 0$ and $q \ge 0$, and $N(f, x)$ a vector of auxiliary nonnegative quantities dependent on iterates and $f$), and a residual term $R(f, x)$ representing, for instance, the cost function gap. The Lyapunov decrease inequality is: $V(f, A(f,x)) + R(f,x) \le V(f, x).$ PEP recasts the search for $V$ as a min-max problem: $$\min_{V \in \mathcal{Q}} \sup_{f \in \mathcal{F}, x} [V(f, A(f,x)) + R(f, x) - V(f, x)] \le 0.$$ This is a convex–concave saddle point problem. The Lyapunov function emerges as a variable in the outer minimization, and feasibility of the constraint (achievability of a nonpositive value) guarantees that the function $V$ certifies convergence. The construction of new points, their inner products, and linear combinations (e.g., for variable splits, multi-step dynamics, or auxiliary points encoding delayed or randomized updates) is handled algorithmically using tools like PEPit, which automates the building of the Gram matrix and the functional constraints (Fercoq, 19 Nov 2024).

3. Automation and Solver Integration

PEPit software enables expressively defining leaf points (iterates, gradients, auxiliary terms) and linear/nonlinear relationships among them, while automatically managing vector/matrix representations and all required interpolation and update constraints. The performance estimation saddle point problem is then solved using DSP-CVXPY, a library for disciplined convex–concave programming. This pipeline allows researchers to, for example:

• Define an algorithm’s update sequence (e.g., primal, dual, random coordinate, or primal–dual coordinate rules),
• Encode arbitrary function class constraints (e.g., convexity, strong convexity, smoothness, monotone operators, saddle-point structure),
• Encode target residuals (function value gap, duality gap, or normed iterates),
• Systematically search for (or verify) quadratic Lyapunov functions as solutions of the PEP saddle point problem.

This automation is essential for high-dimensional or stochastic algorithms, randomized protocols, or hybrid primal–dual splits, where manual analysis is intractable.

4. Applications: Functional Classes and Algorithmic Insights

The PEP approach with Lyapunov saddle point formulation has been successfully applied to several advanced scenarios:

• Convex–Concave Saddle Point Algorithms: By encoding the smoothed duality gap and error bound properties, the method yields sharp, sometimes previously unattainable, linear convergence rates for the Primal-Dual Hybrid Gradient algorithm. PEP-based numerical experiments can indicate empirically less conservative parameter regimes than existing theoretical results (e.g., convergence up to $\gamma = 4/3$ as opposed to previous $\gamma < 1$ requirements) (Fercoq, 19 Nov 2024).
• Randomized/Coordinate Descent: The framework handles expectation over the set of all random coordinate selections by encoding multiple transition matrices, one per possible random outcome, and aggregating via expectation. Numerical solutions of the SDP provide improved worst-case coefficients for proving accelerated rates or for conjecturing tighter Lyapunov inequalities.
• Primal-Dual Coordinate Descent and Other Complex Algorithms: High-complexity, hybrid update mechanisms (e.g., involving both primal and dual randomization, or blockwise updates) are encoded by introducing appropriate auxiliary points and tracking the necessary collection of Gram matrices and transition matrices, all handled in a scalable manner by the SDP solver.

5. Mathematical Formulation and Key Expressions

The critical mathematical structures in the PEP-Lyapunov saddle point framework include:

• Quadratic Lyapunov: $V(f, x) = x^\top Q x + q^\top N(f, x)$.
• Decrease Condition: $V(f, A(f,x)) + R(f,x) \le V(f, x)$, equivalently, $\max_{f, x} V(f, A(f,x)) + R(f,x) - V(f, x) \le 0$.
• Saddle Point Problem:

$$\min_{V \in \mathcal{Q}} \max_{f \in \mathcal{F}, x} \big[ V(f, A(f,x)) + R(f, x) - V(f, x) \big] \le 0.$$

• Transition Matrix Encoding: For each new point, linear combinations of previous leaves are encoded as $x^+ = \Sigma x$, leading to Gram matrix update $G^+ = \Sigma^\top G \Sigma$ and quadratic terms $x^+{}^\top Q x^+ = \operatorname{tr}(Q G^+)$.

6. Impact, Flexibility, and Computational Discovery

The PEP-Lyapunov framework delivers both a systematic verification tool and a discovery engine for convergence proofs. Once the functional class and algorithm are encoded, possible outcomes include:

• Certifying that a quadratic Lyapunov exists for given implementational parameters (e.g., step size),
• Discovering tight worst-case rates and new Lyapunov function structures,
• Revealing parameter regimes where classical proofs fail but numerical results suggest validity or tightness,
• Enabling the analysis and design of algorithms for which analytic Lyapunov construction is infeasible.

A notable implication is that the methodology can generalize beyond the classes of deterministic first-order methods to randomized, accelerated, or composite-structure algorithms, provided the appropriate algebraic relations and interpolability constraints are expressed.

7. Representative Use Cases and Limitations

The PEP-Lyapunov saddle point method is used to:

• Certify algorithmic convergence for convex–concave minimax optimization, coordinate descent, and primal–dual schemes (Fercoq, 19 Nov 2024),
• Numerically map out sharp rate domains, revealing latent dependencies on step size, problem conditioning, or algorithmic structure,
• Design auxiliary quantities or “new points” in the Gram matrix to paper functionals or dynamics beyond mere iterates.

However, the method’s limitations include: the scalability constraints inherent to large SDPs or high-dimensional Gram matrices, possible inapplicability to algorithms lacking linear or quadratic representations in the Gram framework, and the dependence of the Lyapunov search's scope on the expressivity of points and linear combinations defined in the PEP specification.

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In sum, the Performance Estimation Problem Approach, especially when extended via saddle point reformulation for quadratic Lyapunov search and implemented through tools such as PEPit and DSP-CVXPY, forms a foundational methodology for the rigorous, automated, and often numerically optimal certification of algorithmic convergence and performance in first-order optimization and related fields (Fercoq, 19 Nov 2024).