arXiv subjects

Tonghua Tian

Publications and source records attributed to Tonghua Tian.

7 recordsLinked to original sources

arXiv2024

Local geometry of feasible regions via smooth paths

Variational analysis presents a unified theory encompassing in particular both smoothness and convexity. In a Euclidean space, convex sets and smooth manifolds both have straightforward local geometry. However, in the most basic hybrid case of feasible regions consisting of pre-images of convex sets under maps that are once (but not necessarily twice) continuously differentiable, the geometry is less transparent. We define a new approximate convexity property, that holds both for such feasible regions and also for all prox-regular sets. This new property requires that nearby points can always be joined by smooth feasible paths that are almost straight. In particular, in the terminology of real algebraic geometry, such feasible regions are locally normally embedded in the Euclidean space.

math.OC↗

arXiv2023

On the local convergence of the semismooth Newton method for composite optimization

In this paper, we consider a large class of nonlinear equations derived from first-order type methods for solving composite optimization problems. Traditional approaches to establishing superlinear convergence rates of semismooth Newton-type methods for solving nonlinear equations usually postulate either nonsingularity of the B-Jacobian or smoothness of the equation. We investigate the feasibility of both conditions. For the nonsingularity condition, we present equivalent characterizations in broad generality, and illustrate that they are easy-to-check criteria for some examples. For the smoothness condition, we show that it holds locally for a large class of residual mappings derived from composite optimization problems. Furthermore, we investigate a relaxed version of the smoothness condition - smoothness restricted to certain active manifolds. We present a conceptual algorithm utilizing such structures and prove that it has a superlinear convergence rate.

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arXiv2023

Automated Essay Scoring in Argumentative Writing: DeBERTeachingAssistant

Automated Essay scoring has been explored as a research and industry problem for over 50 years. It has drawn a lot of attention from the NLP community because of its clear educational value as a research area that can engender the creation of valuable time-saving tools for educators around the world. Yet, these tools are generally focused on detecting good grammar, spelling mistakes, and organization quality but tend to fail at incorporating persuasiveness features in their final assessment. The responsibility to give actionable feedback to the student to improve the strength of their arguments is left solely on the teacher's shoulders. In this work, we present a transformer-based architecture capable of achieving above-human accuracy in annotating argumentative writing discourse elements for their persuasiveness quality and we expand on planned future work investigating the explainability of our model so that actionable feedback can be offered to the student and thus potentially enable a partnership between the teacher's advice and the machine's advice.

cs.CL↗

arXiv2023

The complexity of first-order optimization methods from a metric perspective

A central tool for understanding first-order optimization algorithms is the Kurdyka-Lojasiewicz inequality. Standard approaches to such methods rely crucially on this inequality to leverage sufficient decrease conditions involving gradients or subgradients. However, the KL property fundamentally concerns not subgradients but rather "slope", a purely metric notion. By highlighting this view, and avoiding any use of subgradients, we present a simple and concise complexity analysis for first-order optimization algorithms on metric spaces. This subgradient-free perspective also frames a short and focused proof of the KL property for nonsmooth semi-algebraic functions.

math.OC↗

arXiv2022

Identifiability, the KL property in metric spaces, and subgradient curves

Identifiability, and the closely related idea of partial smoothness, unify classical active set methods and more general notions of solution structure. Diverse optimization algorithms generate iterates in discrete time that are eventually confined to identifiable sets. We present two fresh perspectives on identifiability. The first distills the notion to a simple metric property, applicable not just in Euclidean settings but to optimization over manifolds and beyond; the second reveals analogous continuous-time behavior for subgradient descent curves. The Kurdya-Lojasiewicz property typically governs convergence in both discrete and continuous time: we explore its interplay with identifiability.

math.OC↗

arXiv2021

The structure of conservative gradient fields

The classical Clarke subdifferential alone is inadequate for understanding automatic differentiation in nonsmooth contexts. Instead, we can sometimes rely on enlarged generalized gradients called "conservative fields", defined through the natural path-wise chain rule: one application is the convergence analysis of gradient-based deep learning algorithms. In the semi-algebraic case, we show that all conservative fields are in fact just Clarke subdifferentials plus normals of manifolds in underlying Whitney stratifications.

math.OC↗

arXiv2019

Ground states and their characterization of spin-$F$ Bose-Einstein condensates

The computation of the ground states of spin-$F$ Bose-Einstein condensates (BECs) can be formulated as an energy minimization problem with two quadratic constraints. We discretize the energy functional and constraints using the Fourier pseudospectral schemes and view the discretized problem as an optimization problem on manifold. Three different types of retractions to the manifold are designed. They enable us to apply various optimization methods on manifold to solve the problem. Specifically, an adaptive regularized Newton method is used together with a cascadic multigrid technique to accelerate the convergence. According to our limited knowledege, our method is the first applicable algorithm for BECs with an arbitrary integer spin, including the complicated spin-3 BECs. Extensive numerical results on ground states of spin-1, spin-2 and spin-3 BECs with diverse interaction and optical lattice potential in one/two/three dimensions are reported to show the efficiency of our method and to demonstrate some interesting physical phenomena.

math.NA↗