Characterizations of Lojasiewicz inequalities: Subgradient flows, talweg, convexity

By Jérôme Bolte, Aris Daniilidis, Olivier Ley, Laurent Mazet. 2009

321 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Numerical Analysis

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Citations in the corpus

Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Łojasiewicz Condition (2016, background): “…analogous to the gradient vector in the smooth case, the negation of the quantity argmins∈∂F (x) ‖s‖ points in the direction of steepest descent [see Bertsekas et al., 2003, Section 8.4] We first derive an alternative representation of Dg(x, L) in terms of the so-called forward-backward…”
Learning in games with continuous action sets and unknown payoff functions (2016, result): “4 should be contrasted to classic results on the Kurdyka–Łojasiewicz inequality where having a “bounded length” property is crucial in establishing trajectory convergence (Bolte et al., 2010).”
Proximal alternating linearized minimization for nonconvex and nonsmooth problems (2013, background): “Then σ satisfies the KL property at the point x̄ for φ (s) = r c− 1 r s 1 r on the set U ∩ [min σ < σ < min σ + η] (see, for more details, [15,16]).”
Modern regularization methods for inverse problems (2018, methodology): “We refer the reader to [46] for a detailed investigation of the class of KL functions, and to [48] for more information on the Bregman proximal gradient.”
Convergence Analysis of a Momentum Algorithm with Adaptive Step Size for Non Convex Optimization (2019, background): “We refer to Bolte et al. (2010) for an in-depth presentation of this property.”
From error bounds to the complexity of first-order descent methods for convex functions (2015, background): “This inequality, also called KurdykaLojasiewicz (KL) inequality (see [18]), asserts that for any smooth semi-algebraic function f there is a smooth concave function φ such that ‖∇ (φ ◦ (f − min f)) (x)‖ ≥ 1 for all x in some neighborhood of the set argmin f .”
Gradient Descent Only Converges to Minimizers (2016, methodology): “More recently, Ge et al. (2015) give quantitative rates on the convergence of noise-added stochastic gradient descent to local minimizers, for strict saddle functions. The condition that the noise have large variance along all directions is often not satisfied by the randomness which arises in sampl”
Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods (2013, background): “For theoretical and geometrical developments concerning this inequality, see [18].”