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arXiv:2310.15128

Projected Stochastic Gradient Descent with Quantum Annealed Binary Gradients

Maximilian Krahn, Sasdelli, Michele, Fengyi Yang, Vladislav Golyanik, Juho Kannala, Tat-Jun Chin, Tolga Birdal

Published Oct 23, 2023• 0 citations
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Abstract

We present, QP-SBGD, a novel layer-wise stochastic optimiser tailored towards training neural networks with binary weights, known as binary neural networks (BNNs), on quantum hardware. BNNs reduce the computational requirements and energy consumption of deep learning models with minimal loss in accuracy. However, training them in practice remains to be an open challenge. Most known BNN-optimisers either rely on projected updates or binarise weights post-training. Instead, QP-SBGD approximately maps the gradient onto binary variables, by solving a quadratic constrained binary optimisation. Under practically reasonable assumptions, we show that this update rule converges with a rate of $\mathcal{O}(1 / \sqrt{T})$. Moreover, we show how the $\mathcal{NP}$-hard projection can be effectively executed on an adiabatic quantum annealer, harnessing recent advancements in quantum computation. We also introduce a projected version of this update rule and prove that if a fixed point exists in the binary variable space, the modified updates will converge to it. Last but not least, our algorithm is implemented layer-wise, making it suitable to train larger networks on resource-limited quantum hardware. Through extensive evaluations, we show that QP-SBGD outperforms or is on par with competitive and well-established baselines such as BinaryConnect, signSGD and ProxQuant when optimising the Rosenbrock function, training BNNs as well as binary graph neural networks.

DOI
10.48550/arxiv.2310.15128

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Quantum annealing is a promising paradigm for building practical quantum computers. Compared to other approaches, quantum annealing technology has been scaled up to a larger number of qubits. On the other hand, deep learning has been profoundly successful in pushing the boundaries of AI. It is thus natural to investigate potentially game changing technologies such as quantum annealers to augment the capabilities of deep learning. In this work, we explore binary neural networks, which are lightweight yet powerful models typically intended for resource constrained devices. Departing from current training regimes for binary networks that smooth/approximate the activation functions to make the network differentiable, we devise a quadratic unconstrained binary optimization formulation for the training problem. While the problem is intractable, i.e., the cost to estimate the binary weights scales exponentially with network size, we show how the problem can be optimized directly on a quantum annealer, thereby opening up to the potential gains of quantum computing. We experimentally validated our formulation via simulation and testing on an actual quantum annealer (D-Wave Advantage), the latter to the extent allowable by the capacity of current technology.
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Advances in artificial intelligence (AI) and deep learning have raised concerns about its increasing energy consumption, while demand for deploying AI in mobile devices and machines at the edge is growing. Binary neural networks (BNNs) have recently gained attention as energy and memory efficient models suitable for resource constrained environments; however, training BNNs exactly is computationally challenging because of its discrete characteristics. Recent work proposing a framework for training BNNs based on quadratic unconstrained binary optimisation (QUBO) and progress in the design of Ising machines for solving QUBO problems suggest a potential path to efficiently optimising discrete neural networks. In this work, we extend existing QUBO models for training BNNs to accommodate arbitrary network topologies and propose two novel methods for regularisation. The first method maximises neuron margins biasing the training process toward parameter configurations that yield larger pre-activation magnitudes. The second method employs a dropout-inspired iterative scheme in which reduced subnetworks are trained and used to adjust linear penalties on network parameters. We apply the proposed QUBO formulation to a small binary image classification problem and conduct computational experiments on a GPU-based Ising machine. The numerical results indicate that the proposed regularisation terms modify training behaviour and yield improvements in classification accuracy on data not present in the training set.
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