# Enhancing Convergence of Decentralized Gradient Tracking under the KL Property

arXiv: https://arxiv.org/abs/2412.09556 · HTML: https://mlsanity.com/paper/2412.09556/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 54 references that are in the map.
- First posted on arXiv 12 Dec 2024.

## Builds on (most cited first, 6 of 6)

- [A Proximal Block Coordinate Descent Algorithm for Deep Neural Network Training](https://mlsanity.com/paper/1803.09082/) (2018)
- [Characterization of Gradient Dominance and Regularity Conditions for Neural Networks](https://mlsanity.com/paper/1710.06910/) (2017)
- [Convergence of Descent Optimization Algorithms under Polyak-L ojasiewicz-Kurdyka Conditions](https://mlsanity.com/paper/2407.00812/) (2024)
- [Asynchronous Decentralized Successive Convex Approximation](https://mlsanity.com/paper/1909.10144/) (2019)
- [Subregularity of subdifferential mappings relative to the critical set and KL property of exponent 1/2](https://mlsanity.com/paper/1812.00558/) (2018)
- [A KL-based Analysis Framework with Applications to Non-Descent Optimization Methods](https://mlsanity.com/paper/2406.02273/) (2024)

## Built on by (most cited first, 1 of 1)

- [A Unified Fractional Regularization Framework for Sparse Recovery](https://mlsanity.com/paper/2604.23184/) (2026)

## Abstract (arXiv)

We study decentralized multiagent optimization over networks, modeled as undirected graphs. The optimization problem consists of minimizing a nonconvex smooth function plus a convex extended-value function, which enforces constraints or extra structure on the solution (e.g., sparsity, low-rank). We further assume that the objective function satisfies the Kurdyka-Lojasiewicz (KL) property, with given exponent θ∈ [0,1). The KL property is satisfied by several (nonconvex) functions of practical interest, e.g., arising from machine learning applications; in the centralized setting, it permits to achieve strong convergence guarantees. Here we establish convergence of the same type for the notorious decentralized gradient-tracking-based algorithm SONATA. Specifically, (i) when θ∈ (0,1/2], the sequence generated by SONATA converges to a stationary solution of the problem at R-linear rate; (ii) when θ∈ (1/2,1), sublinear rate is certified; and finally (iii) when θ=0, the iterates will either converge in a finite number of steps or converges at R-linear rate. This matches the convergence behavior of centralized proximal-gradient algorithms except when θ=0. Numerical results validate our theoretical findings.