Related papers: Convergence problems along curves for generalized …
The convergence of stochastic gradient descent is highly dependent on the step-size, especially on non-convex problems such as neural network training. Step decay step-size schedules (constant and then cut) are widely used in practice…
Recently we found necessary and sufficient conditions for the convergence at a preassigned point of the spherical partial sums of the Fourier integral in a class of piecewise smooth functions in Euclidean space. These yield elementary…
Generalized self-concordance is a key property present in the objective function of many important learning problems. We establish the convergence rate of a simple Frank-Wolfe variant that uses the open-loop step size strategy $\gamma_t =…
We show that, under mild assumptions on the limiting curve, a sequence of simple chordal planar curves converges uniformly whenever certain Loewner driving functions converge. We extend this result to random curves. The random version…
It is proved that smooth closed curves of given length minimizing the principal eigenvalue of the Schr\"odinger operator $-\frac{d^2}{ds^2}+\kappa^2$ exist. Here $s$ denotes the arclength and $\kappa$ the curvature. These minimizers are…
Composite optimization problems, where the sum of a smooth and a merely lower semicontinuous function has to be minimized, are often tackled numerically by means of proximal gradient methods as soon as the lower semicontinuous part of the…
We develop new sub-optimality bounds for gradient descent (GD) that depend on the conditioning of the objective along the path of optimization rather than on global, worst-case constants. Key to our proofs is directional smoothness, a…
We consider the problem of optimizing the sum of a smooth convex function and a non-smooth convex function using proximal-gradient methods, where an error is present in the calculation of the gradient of the smooth term or in the proximity…
Quasi-convex optimization acts a pivotal part in many fields including economics and finance; the subgradient method is an effective iterative algorithm for solving large-scale quasi-convex optimization problems. In this paper, we…
We study the pointwise convergence of Landau type Schr\"odinger operators within the fractional Sobolev space $W^{s,p}(\mathbb R)$. Our results extend those established by Bailey (Rev. Mat. Iberoam., 29 (2): 531-546, 2013) and Yuan, Zhao…
We present a streamlined approach for generalized strong and norm convergence of self-adjoint operators in different Hilbert spaces. In particular, we establish convergence of associated (semi-)groups, (essential) spectra and spectral…
On the generalized tangent bundle of a smooth manifold, we study skew-symmetric endomorphism satisfying an arbitrary polynomial equation with real constant coefficients. We study the compatibility of these structures with the de Rham…
In two earlier papers, we designed a distributed deterministic asynchronous algorithm for minimizing the sum of subdifferentiable and proximable functions and a regularizing quadratic on time-varying graphs based on Dykstra's algorithm, or…
The strong convergence of Euler approximations of stochastic delay differential equations is proved under general conditions. The assumptions on drift and diffusion coefficients have been relaxed to include polynomial growth and only…
Shuffling-type gradient methods are favored in practice for their simplicity and rapid empirical performance. Despite extensive development of convergence guarantees under various assumptions in recent years, most require the Lipschitz…
In this paper, we issue an error analysis for integration over discrete surfaces using the surface parametrization presented in [PS22] as well as prove why even-degree polynomials exhibit a higher convergence rate than odd-degree…
Many practical optimization problems lack strong convexity. Fortunately, recent studies have revealed that first-order algorithms also enjoy linear convergences under various weaker regularity conditions. While the relationship among…
The article is devoted to the investigation of smoothness of functions $f(x_1,...,x_m)$ of variables $x_1,...,x_m$ in infinite fields with non-trivial multiplicative ultra-norms, where $m\ge 2$. Theorems about classes of smoothness $C^n$ or…
We consider optimizing a function smooth convex function $f$ that is the average of a set of differentiable functions $f_i$, under the assumption considered by Solodov [1998] and Tseng [1998] that the norm of each gradient $f_i'$ is bounded…
We clarify the question whether for a smooth curve of polynomials one can choose the roots smoothly and related questions. Applications to perturbation theory of operators are given.