Zhe Li

Yihe Gong, Xue Jiang, Zhe Li et al.

In this paper we study the algebraic structure of error formulas for ideal interpolation. We introduce the so-called "normal" error formulas and prove that the lexicographic order reduced Gröbner basis admits such a formula for all ideal interpolation. This formula is a generalization of the "good" error formula proposed by Carl de Boor. Finally, we discuss a Shekhtman's example and give an explicit form of "normal" error formula for this example.

Zhe Li, Shugong Zhang, Tian Dong

In this paper, we verify Carl de Boor's conjecture on ideal projectors for real ideal projectors of type partial derivative by proving that there exists a positive $η\in \mathbb{R}$ such that a real ideal projector of type partial derivative $P$ is the pointwise limit of a sequence of Lagrange projectors which are perturbed from $P$ up to $η$ in magnitude. Furthermore, we present an algorithm for computing the value of such $η$ when the range of the Lagrange projectors is spanned by the Gröbner éscalier of their kernels w.r.t. lexicographic order.

Bicheng Ying, Zhe Li, Haibo Yang

This work tackles the fundamental challenges in Federated Learning (FL) posed by arbitrary client participation and data heterogeneity, prevalent characteristics in practical FL settings. It is well-established that popular FedAvg-style algorithms struggle with exact convergence and can suffer from slow convergence rates since a decaying learning rate is required to mitigate these scenarios. To address these issues, we introduce the concept of stochastic matrix and the corresponding time-varying graphs as a novel modeling tool to accurately capture the dynamics of arbitrary client participation and the local update procedure. Leveraging this approach, we offer a fresh decentralized perspective on designing FL algorithms and present FOCUS, Federated Optimization with Exact Convergence via Push-pull Strategy, a provably convergent algorithm designed to effectively overcome the previously mentioned two challenges. More specifically, we provide a rigorous proof demonstrating that FOCUS achieves exact convergence with a linear rate regardless of the arbitrary client participation, establishing it as the first work to demonstrate this significant result.