Performance Estimation Problem Framework

• Performance Estimation Problem (PEP) is a formal framework that recasts worst-case analysis of first-order methods as a concrete optimization problem.
• The approach uses interpolation constraints and finite-dimensional reformulations to transform analytic bounds into tractable SDP relaxations and numerical estimates.
• PEP enables algorithmic design by optimizing hyperparameters such as step sizes, yielding performance bounds that outperform classical methods.

The Performance Estimation Problem (PEP) framework is a formal approach for computing the worst-case performance guarantees of optimization methods by explicitly recasting the worst-case analysis as an optimization problem itself. Rather than deriving analytic bounds for a given algorithm through functional inequalities and conservative analysis, PEP formulates (and often solves) an optimization problem whose optimum provides the tightest possible guarantee—typically as a bound on the error, residual, or other metric—extractable from the information available to the method within a given function class.

1. Formalization of the Performance Estimation Problem

The foundation of PEP is the observation that the worst-case value of a measure of interest (e.g., the accuracy $f(x_N) - f(x_*)$ after $N$ iterations for first-order convex minimization) over all functions $f$ in a prescribed class and all algorithm-generated sequences $\{x_0,...,x_N\}$ may itself be written as the optimal value of an explicitly defined optimization problem. For a generic first-order black-box method applied to $L$-smooth convex functions, the PEP is:

$$\begin{align*} \max_{f, x_0, ..., x_N, x_*}\quad & f(x_N) - f(x_*) \ \text{subject to} \quad & f \in C^{1,1}_{L}(\mathbb{R}^d) \ \text{and convex}, \ & x_0 \ \text{given},\ x_1,...,x_N\ \text{by algorithm}, \ & \|x_0 - x_*\| \leq R, \ & x_* \in \arg\min f. \end{align*}$$

This “lift” from analytical worst-case to a concrete optimization encapsulates all the available problem structure, the iterative rules of the method, the initial condition, and the function class. For the gradient method with step-size $h$:

$$x_{i+1} = x_i - \frac{h}{L} \nabla f(x_i), \text{ for } i = 0,...,N-1,$$

the PEP is transformed into a finite-dimensional, though generically nonconvex, program via normalization, variable elimination, and (crucially) the replacement of functional constraints with interpolation constraints.

2. Reduction via Interpolation Constraints and Finite-Dimensional Reformulation

A central insight in the PEP framework is the use of interpolation constraints: finite sets of inequalities that ensure a collection of iterates, gradients, and function values is compatible with some $f$ in the prescribed function class. For $L$-smooth convex $f$, a tight interpolation property is

$$\frac{1}{2L}\|\nabla f(x) - \nabla f(y)\|^2 \leq f(x) - f(y) - \langle \nabla f(y), x-y\rangle,\quad \forall x, y,$$

which, when restricted to the discrete set $\{x_0, ..., x_N, x_*\}$, permits f to be replaced by new variables: normalized gradients $g_i$, function value errors $\delta_i$, and their relationships.

In the PEP for the gradient method, after normalization, the optimization becomes

$$\begin{align*} \text{maximize} \quad & L R^2 \delta_N \ \text{subject to} \quad & \text{trace}(G^\top A_{i,j} G) \leq \delta_i - \delta_j, \ \forall i,j, \end{align*}$$

where $G$ collects the gradient vectors, $\delta$ the function errors, and the matrices $A_{i,j}$ derive from the algorithm and problem structure. Such reformulations lead to quadratic (often nonconvex) matrix programs. For more complex methods, including those with momentum or multi-step memory, the finite-dimensional reduction introduces additional variables representing, e.g., weighted sums and auxiliary sequences.

3. Analytical and Numerical Bounds via Duality and Semidefinite Relaxation

The PEP framework enables both the derivation of new, tight analytical upper and matching lower bounds and the computation of numerical performance guarantees via relaxed convex programs. For the basic gradient method, explicit bounds are obtained:

• For $0 < h \leq 1$:

$f(x_N) - f(x_*) \leq \frac{L R^2}{4Nh + 2}$

• For general $h > 0$, there exists a function (constructed using Moreau envelopes and quadratics) for which the lower bound matches:

$$f(x_N) - f(x_*) = \frac{L R^2}{2} \cdot \max\left\{ \frac{1}{2Nh+1}, (1-h)^{2N} \right\}$$

Optimal step-size selection—balancing these two limiting cases—improves worst-case guarantees by as much as a factor of four over classical conservative theory.

For algorithmic families beyond gradient descent (including heavy-ball and Nesterov's fast gradient methods, expressed via general step recurrence), closed-form analytical bounds become intractable. In such cases, the PEP is relaxed to a semidefinite program (SDP) in dual variables (e.g., Lagrange multipliers for each inequality), and solved numerically. The dual SDP has a canonical form:

$$\begin{align*} \min\quad & \frac{1}{2} L R^2 t \ \text{subject to} \quad & S(\lambda, t) \succeq 0,\ \lambda \in \Lambda, \end{align*}$$

with $S(\lambda, t)$ built from the dual certificates, the algorithmic parameters (including all step sizes), and problem constants.

Numerical solutions to these SDPs yield sharp, nonconservative worst-case bounds, often strictly improving over hand-derived analytical estimates.

4. Algorithmic Design: Step-Size Optimization via SDP

A distinguishing feature of the PEP approach is its use not only for analysis but for algorithmic design. By treating algorithmic hyperparameters (e.g., the set of step sizes $\{h_k^{(i)}\}$) as decision variables, the performance bound itself becomes a function to be minimized. The corresponding (in general, bilinear or even nonconvex) program can be further relaxed to a linear SDP by introducing auxiliary variables (e.g., $$r_{i,k} = \lambda_i h_k^{(i)} + \tau_i \sum_{t=k+1}^i h_k^{(t)}$$) for appropriate dual variables $\lambda$, $\tau$.

The optimized step-sizes are then reconstructed recursively from the optimal dual and auxiliary variables (i.e., by inverting the $r_{i,k}$ formula). Numerical experiments confirm that such PEP-optimized algorithms can outperform both classical and accelerated first-order methods (such as FGM and HBM), empirically attaining worst-case bounds as much as twofold better than previous best-in-class methods for moderate iteration counts.

5. Generalization and Impact on Understanding First-Order Methods

The PEP framework, as established in the foundational work (Drori et al., 2012), yields several enduring advances:

• It provides a systematic and constructive approach to measuring the sharpest possible (worst-case) performance for broad classes of first-order black-box optimization algorithms.
• By enabling the translation of infinite-dimensional worst-case problems into (sometimes purely analytical, often numerically tractable) finite-dimensional programs, it bridges the gap between traditional (often loose) complexity analyses and the true capabilities of algorithms.
• The extension to families of first-order methods—including those with momentum, multiple sequences, and adaptive memory—means the framework accommodates a wide array of practical schemes; SDP-based relaxations are tractable for a range of problem sizes and settings.
• The use of PEP as a design tool—allowing for principled, instance-agnostic tuning and invention of stepsizes or algorithmic modifications—introduces a new avenue for improving first-order methods beyond hand-crafted, intuition-driven exploration.

In summary, the PEP framework leverages the explicit formulation of worst-case performance as an optimization problem, reduces analytic intractability to finite sets of interpolation constraints, and utilizes both analytical and numerical optimization to yield tight, and sometimes optimal, performance bounds. Its influence extends to the design of new algorithms, explaining empirical observations (such as superior practical performance of gradient methods compared to classical analytic bounds), and offers a template for future advancement in first-order method analysis and synthesis.