In a previous publication we proposed discrete global optimization as a method to train a strong binary classifier constructed as a thresholded sum over weak classifiers. Our motivation was to cast the training of a classifier into a format amenable to solution by the quantum adiabatic algorithm. Applying adiabatic quantum computing (AQC) promises to yield solutions that are superior to those which can be achieved with classical heuristic solvers. Interestingly we found that by using heuristic solvers to obtain approximate solutions we could already gain an advantage over the standard method AdaBoost. In this communication we generalize the baseline method to large scale classifier training. By large scale we mean that either the cardinality of the dictionary of candidate weak classifiers or the number of weak learners used in the strong classifier exceed the number of variables that can be handled effectively in a single global optimization. For such situations we propose an iterative and piecewise approach in which a subset of weak classifiers is selected in each iteration via global optimization. The strong classifier is then constructed by concatenating the subsets of weak classifiers. We show in numerical studies that the generalized method again successfully competes with AdaBoost. We also provide theoretical arguments as to why the proposed optimization method, which does not only minimize the empirical loss but also adds L0-norm regularization, is superior to versions of boosting that only minimize the empirical loss. By conducting a Quantum Monte Carlo simulation we gather evidence that the quantum adiabatic algorithm is able to handle a generic training problem efficiently.
In a previous publication we proposed discrete global optimization as a method to train a strong binary classifier constructed as a thresholded sum over weak classifiers. Our motivation was to cast the training of a classifier into a format amenable to solution by the quantum adiabatic algorithm. Applying adiabatic quantum computing (AQC) promises to yield solutions that are superior to those which can be achieved with classical heuristic solvers. Interestingly we found that by using heuristic solvers to obtain approximate solutions we could already gain an advantage over the standard method AdaBoost. In this communication we generalize the baseline method to large scale classifier training. By large scale we mean that either the cardinality of the dictionary of candidate weak classifiers or the number of weak learners used in the strong classifier exceed the number of variables that can be handled effectively in a single global optimization. For such situations we propose an iterative and piecewise approach in which a subset of weak classifiers is selected in each iteration via global optimization. The strong classifier is then constructed by concatenating the subsets of weak classifiers. We show in numerical studies that the generalized method again successfully competes with AdaBoost. We also provide theoretical arguments as to why the proposed optimization method, which does not only minimize the empirical loss but also adds L0-norm regularization, is superior to versions of boosting that only minimize the empirical loss. By conducting a Quantum Monte Carlo simulation we gather evidence that the quantum adiabatic algorithm is able to handle a generic training problem efficiently.
We introduce a novel discrete optimization method for training in the context of a boosting framework for large scale binary classifiers. The motivation is to cast the training problem into the format required by existing adiabatic quantum hardware. First we provide theoretical arguments concerning the transformation of an originally continuous optimization problem into one with discrete variables of low bit depth. Next we propose QBoost as an iterative training algorithm in which a subset of weak classifiers is selected by solving a hard optimization problem in each iteration. A strong classifier is incrementally constructed by concatenating the subsets of weak classifiers. We supplement the findings with experiments on one synthetic and two natural data sets and compare against the performance of existing boosting algorithms. Finally, by conducting a quantum Monte Carlo simulation we gather evidence that adiabatic quantum optimization is able to handle the discrete optimization problems generated by QBoost.
This paper describes how to make the problem of binary classification amenable to quantum computing. A formulation is employed in which the binary classifier is constructed as a thresholded linear superposition of a set of weak classifiers. The weights in the superposition are optimized in a learning process that strives to minimize the training error as well as the number of weak classifiers used. No efficient solution to this problem is known. To bring it into a format that allows the application of adiabatic quantum computing (AQC), we first show that the bit-precision with which the weights need to be represented only grows logarithmically with the ratio of the number of training examples to the number of weak classifiers. This allows to effectively formulate the training process as a binary optimization problem. Solving it with heuristic solvers such as tabu search, we find that the resulting classifier outperforms a widely used state-of-the-art method, AdaBoost, on a variety of benchmark problems. Moreover, we discovered the interesting fact that bit-constrained learning machines often exhibit lower generalization error rates. Changing the loss function that measures the training error from 0-1 loss to least squares maps the training to quadratic unconstrained binary optimization. This corresponds to the format required by D-Wave's implementation of AQC. Simulations with heuristic solvers again yield results better than those obtained with boosting approaches. Since the resulting quadratic binary program is NP-hard, additional gains can be expected from applying the actual quantum processor.
We study the problem of supervised binary classification from the perspective of deploying adiabatic quantum optimization in training. A vast body of prior academic work consisting of both theoretical and numerical studies has indicated that quantum technology promises to provide computational power that may be fundamentally superior to any classical computing methods. Given the abundance of NP-hard optimization problems that naturally arise in learning, it is clear that machine learning can immensely benefit from such an optimization tool. We describe a series of increasingly complex designs that result in computationally hard training problems of combinatorial nature. In return for accepting classical computational hardness, we retain theoretical properties such as maximal sparsity and robustness to label noise, which are otherwise sacrificed by convex methods for the sake of computational efficiency and sound theoretical footing. In order to be compatible with emerging quantum hardware technology, we formalize the training problem as quadratic unconstrained binary optimization. Our initial investigations focus on a simple training formulation with non-convex regularization that conforms to the architecture of existing quantum hardware and makes frugal use of a limited number of available physical qubits. Next, we extend this baseline formulation to a scalable algorithm, QBoost, which is able to train incrementally large-scale classifiers on data sets of practical interest. Further, we derive another algorithm, TotalQBoost, as a theoretically motivated totally corrective boosting algorithm with cardinality penalization that also makes use of quantum optimization. Both QBoost and TotalQBoost perform explicit cardinality regularization, which is the only known way of achieving maximal sparsity in the trained classifiers. We apply QBoost and TotalQBoost to three different real-world computer vision problems and make use of a quantum processor for solving the sequence of discrete optimization problems generated by one of them. Finally, we study a learning formulation with convex regularization and a non-convex loss function, q-loss, specifically designed for robust supervised learning in the presence of label noise as it occurs in practice. For compatibility with quantum hardware we derive the corresponding quadratic binary problem via variational approximation. For all proposed algorithms we compare results on a variety of popular synthetic and natural data sets against a rich selection of existing rival learning formulations.
In the noisy intermediate-scale quantum (NISQ) era, the capabilities of variational quantum algorithms are greatly constrained due to a limited number of qubits and the shallow depth of quantum circuits. We may view these variational quantum algorithms as weak learners in supervised learning. Ensemble methods are general approaches to combining weak learners to construct a strong one in machine learning. In this paper, by focusing on classification, we theoretically establish and numerically verify a learning guarantee for quantum adaptive boosting (AdaBoost). The supervised-learning risk bound describes how the prediction error of quantum AdaBoost on binary classification decreases as the number of boosting rounds and sample size increase. We further empirically demonstrate the advantages of quantum AdaBoost by focusing on a 4-class classification. The quantum AdaBoost not only outperforms several other ensemble methods, but in the presence of noise it can also surpass the ideally noiseless but unboosted primitive classifier after only a few boosting rounds. Our work indicates that in the current NISQ era, introducing appropriate ensemble methods is particularly valuable in improving the performance of quantum machine learning algorithms.
The training of neural networks (NNs) is a computationally intensive task requiring significant time and resources. This article presents a novel approach to NN training using adiabatic quantum computing (AQC), a paradigm that leverages the principles of adiabatic evolution to solve optimization problems. We propose a universal AQC method that can be implemented on gate quantum computers, allowing for a broad range of Hamiltonians and thus enabling the training of expressive neural networks. We apply this approach to various neural networks with continuous, discrete, and binary weights. The study results indicate that AQC can very efficiently evaluate the global minimum of the loss function, offering a promising alternative to classical training methods.
Abstract Boosting is an ensemble learning method that converts a weak learner into a strong learner in the PAC learning framework. The AdaBoost algorithm is a well-known classical boosting algorithm for weak learners with binary hypotheses. Recently, Arunachalam and Maity presented the first quantum boosting algorithm by quantizing AdaBoost. Their algorithm, which we refer to as QAdaBoost hereafter, is a quantum adaptation of AdaBoost and only works for the binary hypothesis case. QAdaBoost is quadratically faster than AdaBoost in terms of the VC dimension of the hypothesis class of the weak learner. However, QAdaBoost is polynomially worse in the bias of the weak learner. In this work, we address an open question by Izdebski et al. on the existence of boosting algorithms for quantum weak learners with non-binary hypotheses. We answer this question in the affirmative by developing the QRealBoost algorithm motivated by the classical RealBoost algorithm. The main technical challenge was to provide provable guarantees for convergence, generalization bounds, and quantum speedup, given that quantum subroutines are noisy and probabilistic. We prove that QRealBoost retains the quadratic speedup of QAdaBoost over AdaBoost and further achieves a polynomial speedup over QAdaBoost in terms of both the bias of the learner and the time taken by the learner to learn the target concept class. Finally, we perform empirical evaluations on QRealBoost and report encouraging observations on quantum simulators by benchmarking the convergence performance of QRealBoost against QAdaBoost, AdaBoost, and RealBoost on a subset of the MNIST dataset and Breast Cancer Wisconsin dataset.