# Adaptation and learning over networks under subspace constraints -- Part II: Performance Analysis

## TL;DR

This paper analyzes the steady-state performance of a distributed learning algorithm over networks with subspace constraints, showing it approaches centralized performance under small step-sizes and highlighting the effects of noise and data characteristics.

## Contribution

It provides explicit analysis of how gradient noise, data, and subspace constraints affect network performance, extending previous stability results to steady-state behavior.

## Key findings

- Distributed algorithm achieves centralized steady-state performance for small step-sizes.
- Gradient noise and data characteristics significantly influence network estimation errors.
- Explicit relationships between subspace constraints and network performance are established.

## Abstract

Part I of this paper considered optimization problems over networks where agents have individual objectives to meet, or individual parameter vectors to estimate, subject to subspace constraints that require the objectives across the network to lie in low-dimensional subspaces. Starting from the centralized projected gradient descent, an iterative and distributed solution was proposed that responds to streaming data and employs stochastic approximations in place of actual gradient vectors, which are generally unavailable. We examined the second-order stability of the learning algorithm and we showed that, for small step-sizes $\mu$, the proposed strategy leads to small estimation errors on the order of $\mu$. This Part II examines steady-state performance. The results reveal explicitly the influence of the gradient noise, data characteristics, and subspace constraints, on the network performance. The results also show that in the small step-size regime, the iterates generated by the distributed algorithm achieve the centralized steady-state performance.