In Naples last month, at the European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases (ECML PKDD), Amazon senior applied scientist Hamed Karimi and colleagues won the 2026 ECML PKDD Test-of-Time Award for their 2016 paper “Linear convergence of gradient and proximal-gradient methods under the Polyak–Łojasiewicz condition”. The award recognizes research published a decade earlier that has had a lasting impact. The paper, written with Julie Nutini and Mark Schmidt while Karimi was a postdoctoral researcher at the University of British Columbia, has helped to organize a previously scattered field of machine learning optimization and offers useful approaches to analyzing new training methods. “When we wrote the paper in 2016, we genuinely didn’t expect it to make a splash,” Karimi says. “We did not know that the result would become part of the way researchers analyze optimization in machine learning.” Karimi and his coauthors showed that the Polyak-Łojasiewicz condition, a 1963 mathematical inequality that had been largely overlooked in machine learning, can prove that the standard method for training machine learning models closes in on the best solutions quickly, even when many different solutions could work equally well. Because modern machine learning models rarely have a single best solution, the result helped close a long-standing gap between what training methods do in practice and what theory could guarantee. Optimization landscape In the classic supervised-machine-learning setting, a model’s billions of settings, or parameters, are adjusted so that the model’s outputs match labeled training data as closely as possible. The distance between the model’s output and the labeled data is measured with a loss function; the goal is to tune the parameters to minimize the loss. This tuning is done through optimization processes, the most common approach being gradient descent or one of its variants. One way to understand gradient descent is to visualize a topographic map with peaks, valleys, cliffs, and plateaus. The lowest points on the map represent minimal loss, with each step downhill toward these minima representing parameter adjustments that optimize toward fewer errors. Ideally, researchers would like to know the quickest path to the bottom. Theory suggests that the fastest path shrinks the error by a constant fraction each step, a property known as linear convergence. The problem is that linear convergence is mathematically guaranteed only for a particular topographic shape: a steep bowl that uniformly curves upward, one that is strongly convex. A bowl like this has exactly one lowest point, so from any location, there is a single best move toward optimization. But many real problems don't work that way. When training data are redundant, as is the case with modern large language models, where the parameter count is disproportionately large compared to the amount of training data, the bottom of the landscape becomes a flat valley floor instead of a single point at the bottom of a bowl. These models — which exhibit weaker conditions than convexity — can be successfully trained with gradient descent, but the theory could neither explain why nor say how quickly, leaving a gap between what works in practice and what can be proven. By 2016, some 25 years’ worth of solutions to optimization problems with weaker conditions were in circulation, each with its own name, notation, and paper, but nobody had established how they related. Karimi and his colleagues’ paper settled those relationships. Researchers have since used variations of the paper’s method to rederive guarantees for essentially every standard optimization algorithm in machine learning. The paper has been cited over 2,000 times and has found a place in graduate textbooks. Tying it together At the center of the paper is the Polyak-Łojasiewicz condition, an inequality published in 1963 by Boris Polyak in a Russian-language paper on gradient methods and by Stanisław Łojasiewicz in French that same year, independently and for unrelated reasons; Polyak was studying optimization, Łojasiewicz real analytic geometry. In the paper, Karimi and coauthors named the condition after both. If the condition holds, then even though the problem doesn’t exhibit a strongly convex geometry, the optimization is still guaranteed to converge at a linear rate. The proof of this is only three lines long. Karimi and his colleagues’ paper made three contributions. The first was a map of existing solutions for problems with weaker conditions, all of which make different assumptions about problem topology. For smooth functions that need not be convex, the researchers worked out every implication of competing assumptions, from strongest to weakest. For most of these functions, linear convergence had been proved before, but only in the convex case. Before Karimi and his colleagues’ work, little was understood about how these competing approaches related to one another. The map gave researchers a clear picture of the existing methods, showing which assumptions imply which, and identified the Polyak-Łojasiewicz condition as the weakest of the known conditions that still guarantees a linear rate. The second contribution extended the results to landscapes that combine a smooth function — one that slopes gradually and predictably at all points — with nonsmooth ones — those with corners, kinks, or edges, points where the slope is undefined. The paper introduced a variant of the Polyak–Łojasiewicz condition that guarantees the same linear rate, bringing many widely used models under the same short analysis. The third contribution was a reanalysis of the standard optimization algorithms, variants of gradient descent that include coordinate descent, sign gradient descent, and stochastic gradient descent. In each case, the analysis recovered the rate of convergence previously known only for strongly convex functions. In other words, the same short argument gives researchers a ready-made template for analyzing new training methods as they appear. Rate of influence Two weeks after the preprint was posted, Polyak wrote the authors. “Actually, it was a surprise for me why the simple condition from my old publication attracted so little attention,” he said, “and now your work exhibits its importance.” He attached an English translation of his 1963 paper, largely unused in the five decades since. Karimi sees two reasons the condition has spread so far over the past decade. First, adopting it took almost no work. Many existing proofs that assumed the idealized bowl shape relied on just one consequence of it: loss, wherever it was measured in the landscape, would still be well above its minimum value. In other words, the ground slopes steeply enough to make real progress downhill. The Polyak–Łojasiewicz condition guarantees exactly that property, so those proofs still hold when the condition is swapped in. Second, and most consequentially, it made fast convergence on a nonconvex problem a provable statement, which is now part of how the field explains why the algorithms researchers run in practice work so well. The condition has also traveled well beyond the setting that Karimi and his colleagues studied. Variants now support convergence results in reinforcement learning, optimization over curved parameter spaces, and the sampling methods behind diffusion models. Ten years on, an inequality that sat unused for five decades has become one of the standard assumptions in the analysis of optimization methods for machine learning.