arXiv subjects

Liyuan Liang

Publications and source records attributed to Liyuan Liang.

5 recordsLinked to original sources

arXiv2026

On the Linear Speedup of the Push-Pull Method for Decentralized Optimization over Digraphs

The linear speedup property is essential for demonstrating the advantage of distributed algorithms over their single-node counterparts. In this paper, we study the stochastic Push-Pull method, a widely adopted decentralized optimization algorithm over directed graphs (digraphs). Unlike methods that rely solely on row-stochastic or column-stochastic mixing matrices, Push-Pull avoids nonlinear correction and has shown superior empirical performance across a variety of settings. However, its theoretical analysis remains challenging, and the linear speedup property has not been generally establishe--revealing a significant gap between empirical success and limited theoretical understanding. To bridge this gap, we propose a novel analysis framework and prove that Push-Pull achieves linear speedup over arbitrary strongly connected digraphs. Our results provide the comprehensive theoretical understanding for stochastic Push-Pull, aligning its theory with empirical performance. Code: https://github.com/pkumelon/PushPull.

math.OC↗

arXiv2026

Decentralized Optimization over Time-Varying Row-Stochastic Digraphs

Decentralized optimization over directed graphs is essential for applications such as robotic swarms, sensor networks, and distributed learning. In many practical scenarios, the underlying network takes the form of a Time-Varying Broadcast Network (TVBN), where only row-stochastic mixing matrices can be constructed due to the unavailability of out-degree information. Achieving exact convergence for decentralized optimization over TVBNs has remained a long-standing open problem, as the limiting distribution of time-varying row-stochastic mixing matrices depends on unpredictable future graph realizations, rendering standard bias-correction techniques infeasible. This paper develops the first decentralized optimization algorithm that achieves exact convergence using only time-varying row-stochastic matrices. We first propose PULM (Pull-with-Memory), a gossip protocol that achieves average consensus with exponential convergence by alternating between row-stochastic mixing and local adjustment steps. Building on PULM, we develop PULM-DGD, which converges to a stationary solution at a rate of $\mathcal{O}(\ln(T)/T)$ for smooth nonconvex objectives, where $T$ denotes the communication round. Our results significantly broaden the applicability of decentralized optimization to highly dynamic communication environments.

math.OC↗

arXiv2025

Achieving Linear Speedup and Near-Optimal Complexity for Decentralized Optimization over Row-stochastic Networks

A key challenge in decentralized optimization is determining the optimal convergence rate and designing algorithms to achieve it. While this problem has been extensively addressed for doubly-stochastic and column-stochastic mixing matrices, the row-stochastic scenario remains unexplored. This paper bridges this gap by introducing effective metrics to capture the influence of row-stochastic mixing matrices and establishing the first convergence lower bound for decentralized learning over row-stochastic networks. However, existing algorithms fail to attain this lower bound due to two key issues: deviation in the descent direction caused by the adapted gradient tracking (GT) and instability introduced by the Pull-Diag protocol. To address descent deviation, we propose a novel analysis framework demonstrating that Pull-Diag-GT achieves linear speedup, the first such result for row-stochastic decentralized optimization. Moreover, by incorporating a multi-step gossip (MG) protocol, we resolve the instability issue and attain the lower bound, achieving near-optimal complexity for decentralized optimization over row-stochastic networks.

math.OC↗

arXiv2024

Understanding the Influence of Digraphs on Decentralized Optimization: Effective Metrics, Lower Bound, and Optimal Algorithm

This paper investigates the influence of directed networks on decentralized stochastic non-convex optimization associated with column-stochastic mixing matrices. Surprisingly, we find that the canonical spectral gap, a widely used metric in undirected networks, is insufficient to characterize the impact of directed topology on decentralized algorithms. To overcome this limitation, we introduce a novel metric termed equilibrium skewness. This metric, together with the spectral gap, accurately and comprehensively captures the influence of column-stochastic mixing matrices on decentralized stochastic algorithms. With these two metrics, we clarify, for the first time, how the directed network topology influences the performance of prevalent algorithms such as Push-Sum and Push-Diging. Furthermore, we establish the first lower bound of the convergence rate for decentralized stochastic non-convex algorithms over directed networks. Since existing algorithms cannot match our lower bound, we further propose the MG-Push-Diging algorithm, which integrates Push-Diging with a multi-round gossip technique. MG-Push-Diging attains our lower bound up to logarithmic factors, demonstrating its near-optimal performance and the tightness of the lower bound. Numerical experiments verify our theoretical results.

math.OC↗

arXiv2015

A method for accurate electron-atom resonances: The complex-scaled multiconfigurational spin-tensor electron propagator method for the $^2P\, \mbox{Be}^{-}$ shape resonance problem

We propose and develop the complex scaled multiconfigurational spin-tensor electron propagator (CMCSTEP) technique for theoretical determination of resonance parameters with electron-atom/molecule systems including open-shell and highly correlated atoms and molecules. The multiconfigurational spin-tensor electron propagator method (MCSTEP) developed and implemented by Yeager his coworkers in real space gives very accurate and reliable ionization potentials and attachment energies. The CMCSTEP method uses a complex scaled multiconfigurational self-consistent field (CMCSCF) state as an initial state along with a dilated Hamiltonian where all of the electronic coordinates are scaled by a complex factor. CMCSCF was developed and applied successfully to resonance problems earlier. We apply the CMCSTEP method to get $^2 P\,\mbox{Be}^{-}$ shape resonance parameters using $14s11p5d$, $14s14p2d$, and $14s14p5d$ basis sets with a $2s2p3d$\,CAS. The obtained value of the resonance parameters are compared to previous results. This is the first time CMCSTEP has been developed and used for a resonance problem. It will be among the most accurate and reliable techniques. Vertical ionization potentials and attachment energies in real space are typically within $\pm 0.2\,eV$ or better of excellent experiments and full configuration interaction calculations with a good basis set. We expect the same sort of agreement in complex space.

physics.atom-ph↗