  With the rapid advancement of artificial intelligence technology, the usage
of machine learning models is gradually becoming part of our daily lives.
High-quality models rely not only on efficient optimization algorithms but also
on the training and learning processes built upon vast amounts of data and
computational power. However, in practice, due to various challenges such as
limited computational resources and data privacy concerns, users in need of
models often cannot train machine learning models locally. This has led them to
explore alternative approaches such as outsourced learning and federated
learning. While these methods address the feasibility of model training
effectively, they introduce concerns about the trustworthiness of the training
process since computations are not performed locally. Similarly, there are
trustworthiness issues associated with outsourced model inference. These two
problems can be summarized as the trustworthiness problem of model
computations: How can one verify that the results computed by other
participants are derived according to the specified algorithm, model, and input
data? To address this challenge, verifiable machine learning (VML) has emerged.
This paper presents a comprehensive survey of zero-knowledge proof-based
verifiable machine learning (ZKP-VML) technology. We first analyze the
potential verifiability issues that may exist in different machine learning
scenarios. Subsequently, we provide a formal definition of ZKP-VML. We then
conduct a detailed analysis and classification of existing works based on their
technical approaches. Finally, we discuss the key challenges and future
directions in the field of ZKP-based VML.
