Smooth Strongly Convex Interpolation and Exact Worst-case Performance of First-order Methods (1502.05666v6)

Abstract: We show that the exact worst-case performance of fixed-step first-order methods for unconstrained optimization of smooth (possibly strongly) convex functions can be obtained by solving convex programs. Finding the worst-case performance of a black-box first-order method is formulated as an optimization problem over a set of smooth (strongly) convex functions and initial conditions. We develop closed-form necessary and sufficient conditions for smooth (strongly) convex interpolation, which provide a finite representation for those functions. This allows us to reformulate the worst-case performance estimation problem as an equivalent finite dimension-independent semidefinite optimization problem, whose exact solution can be recovered up to numerical precision. Optimal solutions to this performance estimation problem provide both worst-case performance bounds and explicit functions matching them, as our smooth (strongly) convex interpolation procedure is constructive. Our works build on those of Drori and Teboulle in [Math. Prog. 145 (1-2), 2014] who introduced and solved relaxations of the performance estimation problem for smooth convex functions. We apply our approach to different fixed-step first-order methods with several performance criteria, including objective function accuracy and gradient norm. We conjecture several numerically supported worst-case bounds on the performance of the fixed-step gradient, fast gradient and optimized gradient methods, both in the smooth convex and the smooth strongly convex cases, and deduce tight estimates of the optimal step size for the gradient method.

• The paper proposes necessary and sufficient conditions for smooth strongly convex interpolation, enabling the exact analysis of first-order optimization methods.
• It formulates the worst-case performance estimation problem as a semidefinite program (SDP), allowing for accurate and practical computation of bounds and worst-case function instances.
• The framework applies to functions of arbitrary dimensions and has been used to validate known optimal step sizes and provide new insights into algorithm performance.

Smooth Strongly Convex Interpolation and Exact Worst-case Performance of First-order Methods

The paper by Taylor, Hendrickx, and Glineur addresses the problem of determining the exact worst-case performance of fixed-step first-order methods when applied to the optimization of unconstrained smooth (potentially strongly) convex functions. By formulating the problem as an optimization task over a class of convex functions, the authors derive necessary and sufficient conditions for what they term "smooth strongly convex interpolation." This allows them to reformulate the worst-case performance estimation as a semidefinite program (SDP), facilitating efficient computation of both the worst-case bounds and the corresponding function instances.

The paper builds upon earlier efforts by Drori and Teboulle who tackled the performance estimation problem with relaxations that resulted in less precise bounds. Here, the authors extend this line of work by demonstrating a more exact formulation through the use of constructive interpolation procedures.

Main Contributions

1. Interpolation Conditions: The authors propose both necessary and sufficient conditions for a set of function values and gradients to be interpolable by a smooth strongly convex function. This is crucial for translating the original infinite-dimensional performance estimation problem into a more manageable finite-dimensional one.
2. SDP Formulation: The reformulated performance estimation problem is presented as an SDP, which enables the practical computation of worst-case bounds. This is a notable improvement over previous approaches that often relied on duality gaps or approximations, making the current method more accurate.
3. Dimensionality Independence: The results apply to functions of arbitrary dimensions, though the worst-case bound is typically achieved in a space of dimension N+2 for a method with N iterations. This makes the approach practical for large-scale problems where the dimensionality can be a limiting factor in performance analysis.
4. Validation and Insights: The paper provides compelling numerical support for the proposed method. It verifies known conjectures regarding optimal step sizes for the gradient method and extends this to strongly convex cases, highlighting the known optimal behavior and providing new insights.

Implications and Future Directions

This work provides a powerful tool for exact performance analysis of first-order methods and offers a framework that could be extended to any fixed-step method, potentially including those with line search or projected iterations. The constructive manner in which the worst-case instances are derived enhances our theoretical understanding while offering practical techniques for algorithm design and analysis.

Future developments might include extending these principles to constrained optimization problems or dynamic programs where the problem landscape evolves over time. Investigating methods for optimizing fixed-step algorithms within this framework, as well as deepening the exploration into less structured function classes and multi-objective criteria, stands as a promising avenue of inquiry. Furthermore, addressing computational efficiency for higher iteration numbers or higher-dimensional function spaces by leveraging advanced numerical techniques would enhance the practical applicability of this framework.

Overall, this work anchors a significant advance in the analysis of first-order optimization methods, offering both a rigorous mathematical framework and a practical toolkit for performance guarantees in convex optimization scenarios.