On the global convergence rate of the gradient descent method for functions with Hölder continuous gradients

The gradient descent method minimizes an unconstrained nonlinear optimization problem with \({\mathcal {O}}(1/\sqrt{K})\), where K is the number of iterations performed by the gradient method. Traditionally, this analysis is obtained for smooth objective functions having Lipschitz continuous gradients. This paper aims to consider a more general class of nonlinear programming problems in which functions have Hölder continuous gradients. More precisely, for any function f in this class, denoted by \({{\mathcal {C}}}^{1,\nu }_L\), there is a \(\nu \in (0,1]\) and \(L>0\) such that for all \(\mathbf{x,y}\in {{\mathbb {R}}}^n\) the relation \(\Vert \nabla f(\mathbf{x})-\nabla f(\mathbf{y})\Vert \le L \Vert \mathbf{x}-\mathbf{y}\Vert ^{\nu }\) holds. We prove that the gradient descent method converges globally to a stationary point and exhibits a convergence rate of \({\mathcal {O}}(1/K^{\frac{\nu }{\nu +1}})\) when the step-size is chosen properly, i.e., less than \([\frac{\nu +1}{L}]^{\frac{1}{\nu }}\Vert \nabla f(\mathbf{x}_k)\Vert ^{\frac{1}{\nu }-1}\). Moreover, the algorithm employs \({\mathcal {O}}(1/\epsilon ^{\frac{1}{\nu }+1})\) number of calls to an oracle to find \({\bar{\mathbf{x}}}\) such that \(\Vert \nabla f({{\bar{\mathbf{x}}}})\Vert <\epsilon \).     

Extending Lipschitz and Hölder maps between metric spaces

We introduce a stochastic generalization of Lipschitz retracts, and apply it to the extension problems of Lipschitz, Hölder, large-scale Lipschitz and large-scale Hölder maps into barycentric metric spaces. Our discussion gives an appropriate interpretation of a work of Lee and Naor.     

Exponential convergence for functional SDEs with Hölder continuous drift

Abstract

Applying Zvonkin’s transform, the exponential convergence in Wasserstein distance for a class of functional SDEs with Hölder continuous drift is obtained. This combining with log-Harnack inequality implies the same convergence in the sense of entropy, which also yields the convergence in total variation norm by Pinsker’s inequality.     

Regularity results for deep-water waves with Hölder continuous vorticity

In this paper we study the regularity properties of periodic deep-water waves travelling under the influence of gravity. The flow beneath the wave surface is assumed to be rotational and the vorticity function is taken to be uniformly Hölder continuous. Excluding the presence of stagnation points, we transform the problem on a fixed reference half-plane and we use Schauder estimates to prove that the streamlines and the free surface of such waves are real-analytic graphs.     

Lipschitz and Hölder continuity results for some functions of cones

We prove the Lipschitz continuity of the maximal angle function on the set of closed convex cones in a Hilbert space. A similar result is obtained for the minimal angle function. On the other hand, we prove that the incenter of a solid cone and the circumcenter of a sharp cone behave in a locally Hölderian manner.     

The Weighted Fourier Inequality,Polarity, and Reverse Hölder Inequality

We examine the problem of the Fourier transform mapping one weighted Lebesgue space into another, by studying necessary conditions and sufficient conditions which expose an underlying geometry. In the necessary conditions, this geometry is connected to an old result of Mahler concerning the the measure of a convex body and its geometric polar being essentially reciprocal. An additional assumption, that the weights must belong to a reverse Hölder class, is used to formulate the sufficient condition.     

Improved regularity of harmonic map flows with Hölder continuous energy

For a smooth harmonic map flow with blow-up as , it has been asked [5,6,7] whether the weak limit is continuous. Recently, in [12], we showed that in general it need not be. Meanwhile, the energy function , being weakly positive, smooth and weakly decreasing, has a continuous extension to [0,T]. Here we show that if this extension is also Hölder continuous, then the weak limit u(T) must also be Hölder continuous.Received: 1 September 2003, Accepted: 7 October 2003, Published online: 25 February 2004Version of 19/9/03. Partly supported by an EPSRC Advanced Research Fellowship.     

Hölder continuous periodic solution of Boussinesq equation with partial viscosity

We show the existence of Hölder continuous periodic solution with compact support in time of the Boussinesq equations with partial viscosity. The Hölder regularity of the solution we constructed is anisotropic which is compatible with partial viscosity of the equations.     

Ergodicity of scalar stochastic differential equations with Hölder continuous coefficients

It is well-known that for a one dimensional stochastic differential equation driven by Brownian noise, with coefficient functions satisfying the assumptions of the Yamada–Watanabe theorem (Yamada and Watanabe, 1971, [31,32]) and the Feller test for explosions (Feller, 1951, 1954), there exists a unique stationary distribution with respect to the Markov semigroup of transition probabilities. We consider systems on a restricted domain $D$ of the phase space $\mathbb{R}$ and study the rate of convergence to the stationary distribution. Using a geometrical approach that uses the so called free energy function on the density function space, we prove that the density functions, which are solutions of the Fokker–Planck equation, converge to the stationary density function exponentially under the Kullback–Leibler divergence, thus also in the total variation norm. The results show that there is a relation between the Bakry–Émery curvature dimension condition and the dissipativity condition of the transformed system under the Fisher–Lamperti transformation. Several applications are discussed, including the Cox–Ingersoll–Ross model and the Ait-Sahalia model in finance and the Wright–Fisher model in population genetics.     

The Hölder exponent of some Fourier series

In this paper we study the local regularity of fractional integrals of Fourier series using several definitions of the Hölder exponent. We especially consider series coming from fractional integrals of modular forms. Our results show that in general, cusp forms give rise to pure fractals (as opposed to multifractals). We include explicit examples and computer plots.     

Eigenfunction statistics for Anderson model with Hölder continuous single site potential

We consider random Schrödinger operators on \(\ell ^{2}(\mathbb {Z}^{d})\) with α-Hölder continuous (0<α≤1) single site distribution. In localized regime, we study the distribution of eigenfunctions in space and energy simultaneously. In a certain scaling limit, we prove limit points are Poisson.     

Area formulas for classes of Hölder continuous mappings of Carnot groups

We prove area formulas for classes of the mappings that are Hölder continuous in the sub-Riemannian sense and defined on nilpotent graded groups. Moreover, in one of the model cases, we establish an area formula for calculating the initial measure and a measure close to it.     

Second-order subelliptic operators on Lie groups III: Hölder continuous coefficients

Let G be a connected Lie group with Lie algebra and an algebraic basis of . Further let denote the generators of left translations, acting on the -spaces formed with left Haar measure dg, in the directions . We consider second-order operators corresponding to a quadratic form with complex coefficients , , , . The principal coefficients are assumed to be H?lder continuous and the matrix is assumed to satisfy the (sub)ellipticity condition uniformly over G. We discuss the hierarchy relating smoothness properties of the coefficients of H with smoothness of the kernel. Moreover, we establish Gaussian type bounds for the kernel and its derivatives. Similar theorems are proved for operators in nondivergence form for which the principal coefficients are at least once differentiable. Received January 24, 1997 / Accepted June 5, 1998     

Sharp nonremovability examples for Hölder continuous quasiregular mappings in the plane

Let α ∈ 2 (0, 1), K ≥ 1, and \(d = 2\frac{{1 + \alpha K}}{{1 + K}}\). Given a compact set E ? ?, it is known that if \(\mathcal{H}^d (E) = 0\), then E is removable for α-Hölder continuous K-quasiregular mappings in the plane. The sharpness of the index d is shown with the construction, for any t > d, of a set E of Hausdorff dimension dim(E) = t which is not removable. In this paper, we improve this result and construct compact nonremovable sets E such that \(0 < \mathcal{H}^d (E) < \infty \). For the proof, we give a precise planar K-quasiconformal mapping whose Hölder exponent is strictly bigger than \(\frac{1}{K}\) and which exhibits extremal distortion properties.     

Infinite entropy is generic in Hölder and Sobolev spaces

In 1980, Yano showed that on smooth compact manifolds, for endomorphisms in dimension one or above and homeomorphisms in dimensions greater than one, topological entropy is generically infinite. It had earlier been shown that, for Lipschitz endomorphisms on such spaces, topological entropy is always finite. In this article, we investigate what occurs between $C^0$-regularity and Lipschitz regularity, focussing on two cases: Hölder mappings and Sobolev mappings.