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
- [math]\displaystyle{ \operatorname{dist}(x,Z)^\alpha \le C|f(x)|. }[/math]
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
- [math]\displaystyle{ |f(x)-f(p)|^\theta\le c|\nabla f(x)|. }[/math]
A special case of the Łojasiewicz inequality, due to Boris Polyak (ru), is commonly used to prove linear convergence of gradient descent algorithms.[1]
References
- ↑ 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, http://www.numdam.org/item?id=PMIHES_1988__67__5_0
- 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, http://www.ams.org/journals/tran/1992-329-02/S0002-9947-1992-1046016-6/
External links
- Hazewinkel, Michiel, ed. (2001), "Lojasiewicz inequality", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page
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