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.
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.
Binary neural networks (BNNs) are increasingly deployed in edge computing applications due to their low hardware complexity and high energy efficiency. However, verifying the robustness of BNNs against input perturbations, including adversarial attacks, remains computationally challenging because the underlying decision problem is inherently combinatorial. In this paper, we propose an Ising- and quantum-inspired framework for BNN robustness verification. We show that, for a broad class of BNN architectures, robustness verification can be formulated as a Quadratic Constrained Boolean Optimization (QCBO) problem and subsequently transformed into a Quadratic Unconstrained Boolean Optimization (QUBO) instance amenable to Ising and quantum-inspired solvers. We demonstrate the feasibility of this formulation on binarized MNIST by solving the resulting QUBOs with a free energy machine (FEM) solver and simulated annealing. We also show the deployment of this framework on quantum annealing and digital annealing platforms. Our results highlight the potential of quantum-inspired computing and Ising computing as a pathway toward trustworthy AI systems.
As a dedicated quantum device, Ising machines could solve large-scale binary optimization problems in milliseconds. There is emerging interest in utilizing Ising machines to train feedforward neural networks due to the prosperity of generative artificial intelligence. However, existing methods can only train single-layer feedforward networks because of the complex nonlinear network topology. This paper proposes an Ising learning algorithm to train quantized neural network (QNN), by incorporating two essential techinques, namely binary representation of topological network and order reduction of loss function. As far as we know, this is the first algorithm to train multi-layer feedforward networks on Ising machines, providing an alternative to gradient-based backpropagation. Firstly, training QNN is formulated as a quadratic constrained binary optimization (QCBO) problem by representing neuron connection and activation function as equality constraints. All quantized variables are encoded by binary bits based on binary encoding protocol. Secondly, QCBO is converted to a quadratic unconstrained binary optimization (QUBO) problem, that can be efficiently solved on Ising machines. The conversion leverages both penalty function and Rosenberg order reduction, who together eliminate equality constraints and reduce high-order loss function into a quadratic one. With some assumptions, theoretical analysis shows the space complexity of our algorithm is O(H2L+HLNlogH), quantifying the required number of Ising spins. Finally, the algorithm effectiveness is validated with a simulated Ising machine on MNIST dataset. After annealing 700 ms, the classification accuracy achieves 98.3%. Among 100 runs, the success probability of finding the optimal solution is 72%. Along with the increasing number of spins on Ising machine, our algorithm has the potential to train deeper neural networks.
A constantly increasing number of parameters in modern machine learning models has become a significant bottleneck in their efficient training, resulting in a disproportionate consumption of resources, expressed in terms of energy and computational time. The vast majority of such resources are allocated to the training phase rather than the inference stage, leading to an inefficient process [1] , [2] , [3] . This study addresses this problem by introducing a quantum alternative for training machine learning models, formalized as a Quadratic Unconstrained Binary Optimization ( QUBO ) problem. Adiabatic quantum computers have shown promise in approximately solving certain NP -hard problems, such as QUBO , potentially offering speed advantages over classical computers in specific scenarios [ 4 , 5, [6] , [7] ]. Given that many machine learning problems are also NP -hard, adiabatic quantum computers could play a pivotal role in efficiently training machine learning models. Furthermore, inspired by the ones’ complement binary numeral system, this work includes the proposal of an efficient binary-to-decimal encoding, which improves data representation in the quantum context. A concrete implementation of a regression training problem on specific quantum hardware is also presented here, the results of which suggest that the adoption of quantum techniques in model training allows not only to address the problem of parameter explosion but also to offer promising prospects for significantly accelerate the training process by exploiting the unique properties of quantum computation.
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.
We investigate a link between Graph Neural Networks (GNNs) and Quadratic Unconstrained Binary Optimization (QUBO) problems, laying the groundwork for GNNs to approximate solutions for these computationally challenging tasks. By analyzing the sensitivity of QUBO formulations, we frame the solution of QUBO problems as a heterophilic node classification task. We then propose QUBO-GNN, an architecture that integrates graph representation learning techniques with QUBO-aware features to approximate solutions efficiently. Additionally, we introduce a self-supervised data generation mechanism to enable efficient and scalable training data acquisition even for large-scale QUBO instances. Experimental evaluations of QUBO-GNN across diverse QUBO problem sizes demonstrate its superior performance compared to exhaustive search and heuristic methods. Finally, we discuss open challenges in the emerging intersection between QUBO optimization and GNN-based learning.
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 O(1/T). Moreover, we show how the 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.