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Differential estimates for fast first-order multilevel nonconvex optimisation

Neil Dizon, Tuomo Valkonen

TL;DR

This work develops a general single-loop framework for differential estimates in bilevel and multilevel optimization, enabling outer updates with inexact gradients computed from flexible inner and adjoint solvers. Central to the theory are tracking inequalities that govern inner/adjoint progress and a differential transformation bound, which together yield Lipschitz-type and descent properties for the approximate outer differential $\widetilde{F'}$. The authors extend convergence analysis to nonconvex forward-backward and primal-dual proximal splitting schemes under operator-relative regularity, proving subdifferential convergence, quasi-Féjer monotonicity, and weak/ergodic convergence in normed spaces. The framework accommodates mismatched adjoints, linear-system/PDE inner problems, and measure-space settings, and it provides constructive guidance for implementing efficient single-loop bilevel solvers with guaranteed stability and convergence.

Abstract

With a view on bilevel and PDE-constrained optimisation, we develop iterative estimates $\widetilde{F'}(x^k)$ of $F'(x^k)$ for composite functions $F :=J \circ S$, where $S$ is the solution mapping of the inner optimisation problem or PDE. The idea is to form a single-loop method by interweaving updates of the iterate $x^k$ by an outer optimisation method, with updates of the estimate by single steps of standard optimisation methods and linear system solvers. When the inner methods satisfy simple tracking inequalities, the differential estimates can almost directly be employed in standard convergence proofs for general forward-backward type methods. We adapt those proofs to a general inexact setting in normed spaces, that, besides our differential estimates, also covers mismatched adjoints and unreachable optimality conditions in measure spaces. As a side product of these efforts, we provide improved convergence results for nonconvex Primal-Dual Proximal Splitting (PDPS).

Differential estimates for fast first-order multilevel nonconvex optimisation

TL;DR

This work develops a general single-loop framework for differential estimates in bilevel and multilevel optimization, enabling outer updates with inexact gradients computed from flexible inner and adjoint solvers. Central to the theory are tracking inequalities that govern inner/adjoint progress and a differential transformation bound, which together yield Lipschitz-type and descent properties for the approximate outer differential . The authors extend convergence analysis to nonconvex forward-backward and primal-dual proximal splitting schemes under operator-relative regularity, proving subdifferential convergence, quasi-Féjer monotonicity, and weak/ergodic convergence in normed spaces. The framework accommodates mismatched adjoints, linear-system/PDE inner problems, and measure-space settings, and it provides constructive guidance for implementing efficient single-loop bilevel solvers with guaranteed stability and convergence.

Abstract

With a view on bilevel and PDE-constrained optimisation, we develop iterative estimates of for composite functions , where is the solution mapping of the inner optimisation problem or PDE. The idea is to form a single-loop method by interweaving updates of the iterate by an outer optimisation method, with updates of the estimate by single steps of standard optimisation methods and linear system solvers. When the inner methods satisfy simple tracking inequalities, the differential estimates can almost directly be employed in standard convergence proofs for general forward-backward type methods. We adapt those proofs to a general inexact setting in normed spaces, that, besides our differential estimates, also covers mismatched adjoints and unreachable optimality conditions in measure spaces. As a side product of these efforts, we provide improved convergence results for nonconvex Primal-Dual Proximal Splitting (PDPS).

Paper Structure

This paper contains 34 sections, 29 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Contributions
  4. Examples
  5. Further related work
  6. Notation and basic concepts
  7. Differential estimation from tracking inequalities
  8. The tracking assumption
  9. Smoothness of differential estimates
  10. Inner and adjoint algorithms
  11. Inner algorithms and the inner tracking inequality
  12. Forward-backward splitting
  13. Primal-dual proximal splitting
  14. Linear system splitting
  15. Adjoint algorithms and the adjoint tracking inequality; the differential transformation condition
  16. ...and 19 more sections

Key Result

Theorem 2.5

Suppose ass:tracking:main holds, $X$ is a normed space, $X^*$ its dual with ${d_{X^*}}$ Young to ${d_X}$. Also let $\{x^n\}_{n=0}^k \subset \Omega$ for a $k \ge 1$, and pick $p \in [1, \kappa)$. Then, for any $\tilde{\gamma}>0$ and $\bar{x} \in X$, we have for some $\varsigma_{p}\ge 0$ and $e_{p,k} \in \mathbb{R}$ that satisfies $\sup_{N \in \mathbb{N}} \sum_{k=0}^{N-1} p^k e_{p,k} < \infty$.

Theorems & Definitions (81)

  • Example 1.1: Forward-backward–linear system splitting–forward-backward
  • Example 1.2: A simple PDE-constrained optimisation problem
  • Example 2.1: Adjoint equations
  • Remark 2.2: Meaning of the conditions
  • Remark 2.3: Distance functions
  • Remark 2.4: Initial iterates and first-step errors
  • Theorem 2.5
  • proof
  • Remark 2.6
  • Theorem 2.7
  • ...and 71 more