Nesterov's accelerated gradient for unbounded convex functions finds the minimum-norm point in the dual space

We study the behavior of first-order methods applied to a lower-unbounded convex function $f$, i.e., $\inf f = -\infty$. Such a setting has received little attention since the trajectories of gradient descent and Nesterov’s accelerated gradient method diverge. In this paper, we establish quantitative convergence results describing their speeds and directions of divergence, with implications for unboundedness judgment. A key idea is a relation to a norm-minimization problem in the dual space: minimize $|p|^2/2$ over $p \in \mathrm{dom}f^\ast$, which can be naturally solved via mirror descent by taking the Legendre–Fenchel conjugate $f^\ast$ as the distance-generating function. It then turns out that gradient descent for $f$ coincides with mirror descent for this norm-minimization problem, and thus it simultaneously solves both problems at $\mathcal{O}(k^{-1})$. This result admits acceleration; Nesterov’s accelerated gradient method, without any modifications, simultaneously solves the original minimization and the dual norm-minimization problems at $\mathcal{O}(k^{-2})$, providing a quantitative characterization of divergence in unbounded convex optimization.

💡 Research Summary

The paper investigates the behavior of first‑order optimization methods when applied to convex functions that are not bounded below (i.e., inf f = −∞). While classical analyses focus on functions with a finite infimum, the authors ask a more subtle question: in which direction and at what speed do standard algorithms diverge when the objective is unbounded? Their answer hinges on a dual‑space norm‑minimization problem and reveals that both gradient descent (GD) and Nesterov’s accelerated gradient method (NAG) implicitly solve this auxiliary problem, thereby providing a quantitative certificate of unboundedness.

Key concepts and results

1. Dual norm‑minimization problem.
   For a proper, L‑smooth convex function f, the Legendre–Fenchel conjugate f* satisfies ∇f(ℝⁿ)=dom f*. The set dom f* is a closed convex set; consequently it possesses a unique minimum‑norm point
   \