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

dist ⁡ ( x , Z ) α ≤ C | f ( x ) | . {\displaystyle \operatorname {dist} (x,Z)^{\alpha }\leq C|f(x)|.}

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

| f ( x ) − f ( p ) | θ ≤ c | ∇ f ( x ) | . {\displaystyle |f(x)-f(p)|^{\theta }\leq c|\nabla f(x)|.}

A special case of the Łojasiewicz inequality, due to Polyak [ru], 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].
  • Bierstone, Edward; Milman, Pierre D. (1988), "Semianalytic and subanalytic sets", Publications Mathématiques de l'IHÉS, 67 (67): 5–42, doi:10.1007/BF02699126, ISSN 1618-1913, MR 0972342, S2CID 56006439
  • Ji, Shanyu; Kollár, János; Shiffman, Bernard (1992), "A global Łojasiewicz inequality for algebraic varieties", Transactions of the American Mathematical Society, 329 (2): 813–818, doi:10.2307/2153965, ISSN 0002-9947, JSTOR 2153965, MR 1046016
  • "Lojasiewicz inequality", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
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