Łojasiewicz inequality

In real algebraic geometry, the Łojasiewicz inequality, named after Stanisław Łojasiewicz, gives an upper bound for the distance of a point to the nearest zero of a given real analytic function. Specifically, let ƒ : U  R be a real analytic function on an open set U in Rn, and let Z be the zero locus of ƒ. Assume that Z is not empty. Then for any compact set K in U, there exist positive constants α and C such that, for all x in K

Here α can be large.

The following form of this inequality is often seen in more analytic contexts: with the same assumptions on f, for every p  U there is a possibly smaller open neighborhood W of p and constants θ  (0,1) and c > 0 such that

A special case of the Łojasiewicz inequality, due to Polyak, is commonly used to prove linear convergence of gradient descent algorithms.[1]

References

  1. Karimi, Hamed; Nutini, Julie; Schmidt, Mark (2016). "Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak–Łojasiewicz Condition". arXiv:1608.04636 [cs.LG].
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