Path Length Bounds for Gradient Descent and Flow
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We provide path length bounds on gradient descent (GD) and flow (GF) curves for various classes of smooth convex and nonconvex functions. We make six distinct contributions: (a) we prove a meta-theorem that if GD has linear convergence towards an optimal set, then its path length is upper bounded by the distance to the optimal set multiplied by a function of the rate of convergence, (b) under the Polyak-Lojasiewicz (PL) condition (a generalization of strong convexity that allows for certain nonconvex functions), we show that the aforementioned multiplicative factor is at most √(κ), (c) we show an Ω(√(d)∧κ^1/4), times the length of the direct path, lower bound on the worst-case path length for PL functions, (d) for the special case of quadratics, we show that the bound is Θ({√(d),√(κ)}) and in some cases can be independent of κ, (e) under the weaker assumption of just convexity, where there is no natural notion of a condition number, we prove that the path length can be at most 2^10d^2 times the length of the direct path, (f) finally, for separable quasiconvex functions the path length is both upper and lower bounded by Θ(√(d)) times the length of the direct path.
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