Characterizing linear convergence in optimization: Polyak-Łojasiewicz inequality and weak-quasi-strong-convexity
Keywords: gradient descent, linear convergence, Polyak-Łojasiewicz inequality, weak-quasi-strong convexity
TL;DR: We completely characterise optimization problems that are solvable via gradient descent with linear convergence rate.
Abstract: We give a complete characterization of optimization problems that can be solved by gradient descent with a linear convergence rate. We show that the well-known Polyak-Łojasiewicz inequality is necessary and sufficient for linear convergence with respect to function values to the minimum, while a property that we call "weak-quasi-strong-convexity", or WQSC, is necessary and sufficient for linear convergence with respect to distances of the iterates to an optimum.
Primary Area: optimization
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Submission Number: 6537
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