Hamed Karimi Wins ECML PKDD Test-of-Time Award
A 2016 paper on machine learning optimization receives recognition for its lasting theoretical impact.
1Prestigious Award for Optimization
Amazon senior applied scientist Hamed Karimi received the 2026 ECML PKDD Test-of-Time Award in Naples. He was honored alongside coauthors Julie Nutini and Mark Schmidt for their 2016 paper. The award recognizes research published a decade ago that has significantly influenced the field of machine learning optimization and model training analysis.
2Bridging Theory and Practice
The research addressed a long-standing gap between theoretical guarantees and practical model performance. While standard theory required strongly convex landscapes, modern models often feature flat valleys. Karimi’s work proved that gradient descent methods can still achieve linear convergence in these complex, non-convex settings, aligning mathematical theory with real-world training outcomes.
3The Polyak-Łojasiewicz Condition
Central to the paper is the Polyak-Łojasiewicz condition, a 1963 mathematical inequality that remained largely overlooked for five decades. By demonstrating that this condition guarantees linear convergence even without strict convexity, the authors provided a concise, three-line proof. This discovery transformed how researchers analyze the speed and efficiency of optimization algorithms.
4Mapping Scattered Optimization Methods
Before 2016, the field of optimization under weaker conditions was fragmented, with numerous papers using different notations and assumptions. Karimi’s team systematically mapped these relationships, identifying the Polyak-Łojasiewicz condition as the weakest requirement for linear convergence. This synthesis allowed researchers to rederive guarantees for standard algorithms using a unified, consistent framework.
5Extending to Complex Landscapes
The authors extended their analysis to include non-smooth functions, which contain kinks or undefined slopes. By introducing a variant of the Polyak-Łojasiewicz condition, they ensured that linear convergence rates apply to these complex models as well. This provided a template for analyzing new training methods as they emerge in the industry.
6Widespread Academic and Practical Adoption
The paper has been cited over 2,000 times and is now featured in graduate-level textbooks. Its influence extends beyond standard supervised learning into reinforcement learning and diffusion models. By replacing restrictive convexity assumptions with this condition, researchers have successfully validated the performance of algorithms used in modern large language models.
7Historical Context of the Inequality
The condition was independently published in 1963 by Boris Polyak and Stanisław Łojasiewicz for unrelated research purposes. Polyak later expressed surprise at the lack of attention his original work received before Karimi’s team highlighted its importance. The authors honored both mathematicians by naming the condition after them in their 2016 publication.
8Lasting Impact on Machine Learning
Ten years after its publication, the paper remains a cornerstone for understanding why gradient descent works so effectively in practice. It transformed a forgotten inequality into a standard assumption for optimization analysis. This work continues to guide researchers in developing and verifying the training methods that power today’s most advanced artificial intelligence.
Original article: Amazon Science — Hamed Karimi Wins ECML PKDD Test-of-Time Award