Hamed Karimi Wins 2026 ECML PKDD Test-of-Time Award for 2016 Optimization Paper

Hamed Karimi and colleagues won the 2026 ECML PKDD Test-of-Time Award for their 2016 paper on the linear convergence of gradient and proximal-gradient methods under the Polyak–Łojasiewicz condition.
The paper demonstrated that the Polyak–Łojasiewicz condition, a 1963 mathematical inequality, guarantees linear convergence for optimization even when the problem is not strongly convex, a property previously guaranteed only for single-minimum landscapes.
The work organized a scattered field of optimization solutions by mapping the implications of competing assumptions, identifying the Polyak–Łojasiewicz condition as the weakest known condition that still ensures a linear rate.
The paper has been cited over 2,000 times and is included in graduate textbooks, with its methods now applied to rederive guarantees for standard algorithms like stochastic gradient descent and to support convergence results in reinforcement learning and diffusion models.

Theoretical Contributions

The authors proved that the Polyak–Łojasiewicz condition allows standard training methods to close in on best solutions quickly, even when many different solutions work equally well.
The proof of the linear convergence guarantee is only three lines long, providing a ready-made template for analyzing new training methods.
The condition resolves the gap between practical success in training large models with redundant data and the theoretical inability to explain why gradient descent works in non-convex settings.

Boris Polyak, one of the original authors of the 1963 inequality, contacted the researchers two weeks after the preprint was posted, noting the surprise that his old work had attracted little attention until this paper.
Adoption of the condition required minimal effort because many existing proofs relying on strongly convex assumptions could be adapted by swapping in the Polyak–Łojasiewicz condition.
The condition has expanded beyond its original setting to support convergence results in reinforcement learning, optimization over curved parameter spaces, and sampling methods for diffusion models.