Quantum advantage for learning shallow neural networks with natural data distributions (2503.20879v1)

Abstract: The application of quantum computers to machine learning tasks is an exciting potential direction to explore in search of quantum advantage. In the absence of large quantum computers to empirically evaluate performance, theoretical frameworks such as the quantum probably approximately correct (PAC) and quantum statistical query (QSQ) models have been proposed to study quantum algorithms for learning classical functions. Despite numerous works investigating quantum advantage in these models, we nevertheless only understand it at two extremes: either exponential quantum advantages for uniform input distributions or no advantage for potentially adversarial distributions. In this work, we study the gap between these two regimes by designing an efficient quantum algorithm for learning periodic neurons in the QSQ model over a broad range of non-uniform distributions, which includes Gaussian, generalized Gaussian, and logistic distributions. To our knowledge, our work is also the first result in quantum learning theory for classical functions that explicitly considers real-valued functions. Recent advances in classical learning theory prove that learning periodic neurons is hard for any classical gradient-based algorithm, giving us an exponential quantum advantage over such algorithms, which are the standard workhorses of machine learning. Moreover, in some parameter regimes, the problem remains hard for classical statistical query algorithms and even general classical algorithms learning under small amounts of noise.

Quantum Advantage in Learning Shallow Neural Networks with Natural Data Distributions

The paper "Quantum advantage for learning shallow neural networks with natural data distributions" proposes a novel approach to explore quantum algorithms in the field of machine learning, particularly with an emphasis on shallow neural networks. Leveraging recent advancements in quantum computing, the authors explore the intersection between quantum computation and classical complex function learning, elucidating scenarios where quantum algorithms offer significant performance improvements.

Contextual Framework

The exploration of quantum computing's potential for machine learning has primarily revolved around theoretical frameworks such as the quantum Probably Approximately Correct (PAC) and Quantum Statistical Query (QSQ) models. These frameworks have enabled researchers to forecast and analyze possible quantum advantages in learning classical functions, extending the capability of quantum computers beyond the advances achieved by classical counterparts. This paper addresses a noticeable gap within the current understanding: the application of quantum algorithms to non-uniform distributions. Classical learning theories often highlight instances where uniform distributions present inherent challenges for classical systems but remain unresolved on a quantum plane. Therefore, non-uniform distributions, representing a more practical scenario in natural data distributions, become a focal point of interest in this paper.

Primary Contributions

The authors develop a quantum algorithm within the QSQ model to efficiently learn periodic neurons from non-uniform distributions such as Gaussian, generalized Gaussian, and logistic distributions. These types of distribution are frequently encountered across various real-world applications, including image processing and simulated population dynamics. A significant outcome of their work is demonstrating an exponential quantum advantage over classical gradient-based methods, traditionally recognized as successful in classical machine learning environments.

A key aspect of their paper is the consideration of real-valued functions, enhancing the application scope; historically, research has been heavily slanted towards Boolean functions, leaving practical function representation underexplored. By demonstrating how quantum algorithms can optimize learning tasks involving continuous real functions, this work leverages quantum Fourier transform methodologies, circumventing challenges created by classical sparse Fourier conditions that lead to barren plateaus difficult for gradient-based methods to negotiate.

Methodology

The research hinges on innovatively integrating quantum period finding, a component of quantum Fourier sampling, into the QSQ model. This integration assists in accurately retrieving neuron periodicity components, which translates into identifying model weights. Noteworthy advances include:

1. Adaptation of the Quantum Fourier Transform to accommodate real-valued, pseudoperiodic functions through precise discretization and truncation techniques.
2. Period Finding Algorithms adapted for cases where non-uniform distributions prevail, thereby ensuring that the empirical utility of quantum algorithms aligns with theoretical constructs.

These methods allow for uncovering significant quantum advantages, particularly in distributions where conventional gradient descent demands an infeasibly high number of iterations and resources.

Implications and Future Directions

The implications of this work underscore the feasibility of harnessing quantum computing for advanced machine learning tasks, especially those constrained by classical methodologies. By establishing exponential advantages in specific parameter spaces, the paper suggests pathways for practical applications of quantum machine learning, encouraging developments in quantum hardware to leverage these theoretical benefits.

The research alludes to future investigations that could bridge quantum learning theory with practical data distributions encountered in machine learning. Specifically, exploration into generalizing results across a broader range of non-uniform distributions, and optimizing quantum state preparation, remain as promising avenues for further research. As the realization of quantum technologies progresses, these findings lay foundational principles for the advanced integration of quantum and classical learning paradigms, with the ultimate aim of optimizing computational efficiency in real-world applications.