Understanding the Curse of Unrolling

TL;DR

This paper analyzes the 'curse of unrolling' in machine learning, revealing how early divergence occurs in derivative computations and proposing truncation and warm-starting as practical solutions, supported by theoretical and experimental evidence.

Contribution

It provides a non-asymptotic analysis of the curse of unrolling, identifying factors that cause divergence and proposing truncation and warm-starting as effective remedies.

Findings

Early truncation reduces divergence and memory use.

Warm-starting induces implicit truncation in bilevel optimization.

Numerical experiments confirm theoretical predictions.

Abstract

Algorithm unrolling is ubiquitous in machine learning, particularly in hyperparameter optimization and meta-learning, where Jacobians of solution mappings are computed by differentiating through iterative algorithms. Although unrolling is known to yield asymptotically correct Jacobians under suitable conditions, recent work has shown that the derivative iterates may initially diverge from the true Jacobian, a phenomenon known as the curse of unrolling. In this work, we provide a non-asymptotic analysis that explains the origin of this behavior and identifies the algorithmic factors that govern it. We show that truncating early iterations of the derivative computation mitigates the curse while simultaneously reducing memory requirements. Finally, we demonstrate that warm-starting in bilevel optimization naturally induces an implicit form of truncation, providing a practical remedy. Our theoretical findings are supported by numerical experiments on representative examples.