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Many aspects of modern applied research rely on a crucial algorithm called gradient descent. This is a procedure generally used for finding the largest or smallest values of a particular mathematical function — a process known as optimizing the function. It can be used to calculate anything from the most profitable way to manufacture a product to the best way to assign shifts to workers.

Yet despite this widespread usefulness, researchers have never fully understood which situations the algorithm struggles with most. Now, new work explains it, establishing that gradient descent, at heart, tackles a fundamentally difficult computational problem. The new result places limits on the type of performance researchers can expect from the technique in particular applications.

“There is a kind of worst-case hardness to it that is worth knowing about,” said Paul Goldberg of the University of Oxford, co-author of the work along with John Fearnley and Rahul Savani of the University of Liverpool and Alexandros Hollender of Oxford. The result received a Best Paper Award in June at the annual Symposium on Theory of Computing.

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