The Presence and Absence of Barren Plateaus in Tensor-network Based Machine Learning

TL;DR

This paper investigates the trainability of tensor-network based machine learning models, demonstrating that barren plateaus occur with global loss functions but are absent with local loss functions, thus guiding efficient training strategies.

Contribution

It provides a rigorous analysis distinguishing conditions under which barren plateaus appear or are avoided in tensor-network machine learning models.

Findings

Barren plateaus occur with global loss functions.

Local loss functions avoid vanishing gradients.

Results guide practical training of tensor-network models.

Abstract

Tensor networks are efficient representations of high-dimensional tensors with widespread applications in quantum many-body physics. Recently, they have been adapted to the field of machine learning, giving rise to an emergent research frontier that has attracted considerable attention. Here, we study the trainability of tensor-network based machine learning models by exploring the landscapes of different loss functions, with a focus on the matrix product states (also called tensor trains) architecture. In particular, we rigorously prove that barren plateaus (i.e., exponentially vanishing gradients) prevail in the training process of the machine learning algorithms with global loss functions. Whereas, for local loss functions the gradients with respect to variational parameters near the local observables do not vanish as the system size increases. Therefore, the barren plateaus are absent in this case and the corresponding models could be efficiently trainable. Our results reveal a crucial aspect of tensor-network based machine learning in a rigorous fashion, which provide a valuable guide for both practical applications and theoretical studies in the future.