[Paper Review] Towards provably efficient quantum algorithms for large-scale machine-learning models

Junyu Liu, Minzhao Liu+6 more|arXiv (Cornell University)|Mar 6, 2023

Stochastic Gradient Optimization Techniques92 references4 citations

TL;DR

This paper proposes a provably efficient fault-tolerant quantum algorithm for large-scale machine learning, leveraging quantum-enhanced gradient descent via a modified HHL-type algorithm. It achieves a theoretical scaling of 𝒪(T² × polylog(n)) for training sparse, dissipative models, demonstrating potential quantum speedup in early-stage training after pruning, with numerical evidence up to 103 million parameters.

ABSTRACT

Large machine learning models are revolutionary technologies of artificial intelligence whose bottlenecks include huge computational expenses, power, and time used both in the pre-training and fine-tuning process. In this work, we show that fault-tolerant quantum computing could possibly provide provably efficient resolutions for generic (stochastic) gradient descent algorithms, scaling as O(T^2 polylog(n)), where n is the size of the models and T is the number of iterations in the training, as long as the models are both sufficiently dissipative and sparse, with small learning rates. Based on earlier efficient quantum algorithms for dissipative differential equations, we find and prove that similar algorithms work for (stochastic) gradient descent, the primary algorithm for machine learning. In practice, we benchmark instances of large machine learning models from 7 million to 103 million parameters. We find that, in the context of sparse training, a quantum enhancement is possible at the early stage of learning after model pruning, motivating a sparse parameter download and re-upload scheme. Our work shows solidly that fault-tolerant quantum algorithms could potentially contribute to most state-of-the-art, large-scale machine-learning problems.

Motivation & Objective

• To develop a theoretically grounded, provably efficient quantum algorithm for training large-scale machine learning models.
• To address the high computational cost and carbon footprint of training large models like GPT-3.
• To identify conditions under which quantum algorithms can outperform classical counterparts in standard ML workflows.
• To shift focus from variational quantum algorithms toward end-to-end quantum acceleration of classical training bottlenecks.
• To establish theoretical guarantees for quantum speedup in gradient descent, particularly in sparse and dissipative settings.

Proposed method

• Adapts efficient quantum algorithms for dissipative differential equations to solve (stochastic) gradient descent problems.
• Applies a modified HHL algorithm for sparse matrix inversion to accelerate weight updates in neural network training.
• Imposes constraints on model sparsity and dissipation to ensure quantum speedup is theoretically possible.
• Employs linearization of the training dynamics to map the non-linear optimization problem into a form solvable by quantum differential equation solvers.
• Uses a parameter download and re-upload scheme after pruning to enable quantum processing of sparse model states.
• Relies on fault-tolerant quantum computing assumptions to ensure error resilience and provable efficiency.

Experimental results

Research questions

• RQ1 Can fault-tolerant quantum algorithms provide provably efficient solutions to large-scale gradient descent in machine learning?
• RQ2 Under what conditions—specifically sparsity and dissipation—can quantum algorithms outperform classical training in practice?
• RQ3 Is there a quantum advantage in the early stages of training after model pruning, before error growth becomes exponential?
• RQ4 Can quantum-enhanced gradient descent be integrated into standard machine learning pipelines with theoretical guarantees?
• RQ5 What are the formal criteria for dissipation and sparsity that preserve quantum speedup in training large models?

Key findings

• The proposed quantum algorithm achieves a theoretical runtime scaling of 𝒪(T² × polylog(n)) for training large, sparse, and dissipative models.
• Numerical benchmarks show a potential quantum enhancement in the early training phase after pruning, particularly in sparse parameter regimes.
• The algorithm is robust to model size, with successful simulations up to 103 million parameters, suggesting scalability.
• The quantum speedup is not dequantizable by classical methods like those of Tang, as it relies on solving BQP-hard problems via non-linear dissipative dynamics.
• The method is consistent with the lottery ticket hypothesis, as it maintains performance under constant or polynomially scaling sparsity.
• The work provides a framework for end-to-end quantum acceleration of classical training, shifting focus from variational algorithms to core optimization bottlenecks.