arXiv research

Improved uniform convergence bound with fat-shattering dimension reduces sample complexity gap.

New convergence bounds for shuffling-based SGD methods in distributed learning.

The article introduces a new convergence concept for Lorentzian spaces and applies it to generalized cones.

DCDC calculates convergence rates for Markov chains using neural networks.

We show that for a noncollapsing sequence of closed, connected, oriented Riemannian manifolds with Ricci curvature uniformly bounded from below and diameter uniformly bounded above, Gromov-Hausdorff convergence essentially agrees with intrinsic flat convergence.

Study shows gap between uniform convergence and test error in random feature models.

New bounds on scalar curvature for metric sequences.

Sharp bounds on weak convergence rate for rough volatility models.

The overall performance or expected excess risk of an iterative machine learning algorithm can be decomposed into training error and generalization error. While the former is controlled by its convergence analysis, the latter can be tightly handled by algorithmic stability. The machine learning community has a rich his…

Proves convergence of gradient Ricci shrinkers with uniform bounds.

New method estimates convergence bounds for nonlinear Markov chains.

The study extends convergence theorems for Ricci-limit spaces with bounded curvature.

Paper establishes convergence rates and concentration bounds for stochastic approximation and reinforcement learning with Markovian noise.

We provide a simple convergence proof for Adam and Adagrad.

New bounds show diffusion models converge nearly linearly in data dimension.

Uniform convergence of metrics on surfaces with bounded curvature measures proved.

In this paper we prove convergence and compactness results for Ricci flows with bounded scalar curvature and entropy. More specifically, we show that Ricci flows with bounded scalar curvature converge smoothly away from a singular set of codimension $\geq 4$. We also establish a general form of the Hamilton-Tian Conjec…

Ricci flow stability on manifolds with bounded geometry ensures convergence to hyperbolic metrics.

Paper revisits set membership estimation for linear systems with relaxed disturbance bounds.

We provide non-asymptotic bounds for the well-known temporal difference learning algorithm TD(0) with linear function approximators. These include high-probability bounds as well as bounds in expectation. Our analysis suggests that a step-size inversely proportional to the number of iterations cannot guarantee optimal …

The objective of this paper is to introduce the notion of generalized almost statistical (briefly, GAS) convergence of bounded real sequences, which generalizes the notion of almost convergence as well as statistical convergence of bounded real sequences. As a special kind of Banach limit functional, we also introduce …

The paper converts metric bounds to distance function Hölder bounds and proves compactness theorems.

Quantifies scalar curvature under $C^0$ convergence, proving a refined version in all dimensions.

pHMC converges on infinite-dimensional spaces with bounds.

The paper explores convergence and structure of spaces with scalar curvature and entropy bounds, introducing new $d_p$ convergence.

Dropout speeds up convergence in shallow linear NNs, with a rate bound.

The equivariant Gromov--Hausdorff convergence of metric spaces is studied. Where all isometry groups under consideration are compact Lie, it is shown that an upper bound on the dimension of the group guarantees that the convergence is by Lie homomorphisms. Additional lower bounds on curvature and volume strengthen this…

In this paper, we present the Bennett-type generalization bounds of the learning process for i.i.d. samples, and then show that the generalization bounds have a faster rate of convergence than the traditional results. In particular, we first develop two types of Bennett-type deviation inequality for the i.i.d. learning…

Adam achieves optimal convergence in deep ReLU networks via novel Kakeya bounds.

Stochastic gradient descent (SGD) is the optimization algorithm of choice in many machine learning applications such as regularized empirical risk minimization and training deep neural networks. The classical convergence analysis of SGD is carried out under the assumption that the norm of the stochastic gradient is uni…

ADOPT optimizes Adam to converge with any β2 without bounded noise.

Study on CMC hypersurfaces with bounded index and area, proving multiplicity one convergence and bounds on genus.

Optimization rates improved for manifolds with bounded geometry.

Improved BO algorithms reduce prediction error under Gaussian noise.

We explore the distinctions between $L^p$ convergence of metric tensors on a fixed Riemannian manifold versus Gromov-Hausdorff, uniform, and intrinsic flat convergence of the corresponding sequence of metric spaces. We provide a number of examples which demonstrate these notions of convergence do not agree even for two…

The paper analyzes kNN density estimation's convergence rates under different conditions.

The paper analyzes convergence rates of Gaussian process approximations for scalable regression.

A comparison theorem for the isoperimetric profile on the universal cover of surfaces evolving by normalised Ricci flow is proven. For any initial metric, a model comparison is constructed that initially lies below the profile of the initial metric and which converges to the profile of the constant curvature metric. Th…

Many practitioners who use the EM algorithm complain that it is sometimes slow. When does this happen, and what can be done about it? In this paper, we study the general class of bound optimization algorithms - including Expectation-Maximization, Iterative Scaling and CCCP - and their relationship to direct optimizatio…

Proves curvature tensor convergence for smoothable spaces.

Paper analyzes convergence of two time-scale stochastic approximation using martingale approach.

New bounds on SGD's final iterate convergence rate in constant dimension.

New method shows hidden state can significantly improve differential privacy in SGD.

Paper tightens lower bounds on decentralized training complexity.

Study uses SGD to learn operators in Hilbert spaces with convergence analysis.

The paper shows how MMD metrizes weak convergence for certain kernels.

The paper studies Ricci flow with specific curvature and volume constraints, proving convergence and curvature bounds.

Study limits of manifolds with Kato bound Ricci curvature, proving volume convergence.