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arXiv:1907.06840

The Quantum Version Of Classification Decision Tree Constructing Algorithm C5.0

Kamil Ravilevich Khadiev, Ilnaz Mannapov, Liliya Safina

Published Jul 16, 2019• 18 citations
arXivHTMLPDF

Abstract

In the paper, we focus on complexity of C5.0 algorithm for constructing decision tree classifier that is the models for the classification problem from machine learning. In classical case the decision tree is constructed in $O(hd(NM+N \log N))$ running time, where $M$ is a number of classes, $N$ is the size of a training data set, $d$ is a number of attributes of each element, $h$ is a tree height. Firstly, we improved the classical version, the running time of the new version is $O(h\cdot d\cdot N\log N)$. Secondly, we suggest a quantum version of this algorithm, which uses quantum subroutines like the amplitude amplification and the D{ü}rr-Høyer minimum search algorithms that are based on Grover's algorithm. The running time of the quantum algorithm is $O\big(h\cdot \sqrt{d}\log d \cdot N \log N\big)$ that is better than complexity of the classical algorithm.

DOI
10.48550/arxiv.1907.06840

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In the paper, we focus on complexity of C5.0 algorithm for constructing decision tree classifier that is the models for the classification problem from machine learning. In classical case the decision tree is constructed in O(hd(NM+Nlog⁡N))O(hd(NM+N \log N))O(hd(NM+NlogN)) running time, where MMM is a number of classes, NNN is the size of a training data set, ddd is a number of attributes of each element, hhh is a tree height. Firstly, we improved the classical version, the running time of the new version is O(h⋅d⋅Nlog⁡N)O(h\cdot d\cdot N\log N)O(h⋅d⋅NlogN). Secondly, we suggest a quantum version of this algorithm, which uses quantum subroutines like the amplitude amplification and the D{ü}rr-Høyer minimum search algorithms that are based on Grover's algorithm. The running time of the quantum algorithm is O(h⋅dlog⁡d⋅Nlog⁡N)O\big(h\cdot \sqrt{d}\log d \cdot N \log N\big)O(h⋅d​logd⋅NlogN) that is better than complexity of the classical algorithm.
Quantum computers and machine learning are on the forefront of research in both science and engineering. Quantum computers offer significant computational speed-ups in certain tasks. Machine learning on the other hand requires a lot of computational power and is heavily used by both governments and businesses. Thus it is no wonder that the possibility of leveraging the computational efficiency of quantum computers for machine learning tasks is attractive. This thesis focuses on decision trees models, due to their interpretability and wide use cases. This thesis tries to answer the question, how can quantum technology be leveraged to improve the performance and construction of decision trees. The thesis will explore the full implementation of decision trees in quantum circuits as quantum decision trees and their viability in modern hardware and its future relevance. Quantum algorithms for construction and retraining of classical decision trees will also be explored and compared to classical methods. The work found that quantum decision trees show comparable performance to classical decision trees when applied to simple data sets. However, due to hardware limitations quantum decision trees suffer when applied to more complex data sets. Quantum algorithms for tree construction show significant theoretical speed-ups at larger models when compared to classical methods of tree construction.
In this work, we consider the performance of using a quantum algorithm to predict the result of a binary classification problem when a machine learning model is an ensemble of any simple classifiers. This approach is faster than classical prediction and uses quantum and classical computing, but it is based on a probabilistic algorithm. Let N be the number of classifiers from an ensemble model and O(T) be the running time of prediction of one classifier. In classical case, the final result is obtained by ”averaging” outcomes of all ensemble model’s classifiers. The running time in classical case is O (N · T). We propose an algorithm that works in O (√N · T ).
A powerful way to improve performance in machine learning is to construct an ensemble that combines the predictions of multiple models. Ensemble methods are often much more accurate and lower variance than the individual classifiers that make them up but have high requirements in terms of memory and computational time. In fact, a large number of alternative algorithms is usually adopted, each requiring to query all available data. We propose a new quantum algorithm that exploits quantum superposition, entanglement and interference to build an ensemble of classification models. Thanks to the generation of the several quantum trajectories in superposition, we obtain BBB transformations of the quantum state which encodes the training set in only log(B)log\left(B\right)log(B) operations. This implies exponential growth of the ensemble size while increasing linearly the depth of the correspondent circuit. Furthermore, when considering the overall cost of the algorithm, we show that the training of a single weak classifier impacts additively the overall time complexity rather than multiplicatively, as it usually happens in classical ensemble methods. We also present small-scale experiments on real-world datasets, defining a quantum version of the cosine classifier and using the IBM qiskit environment to show how the algorithms work.
Abstract A powerful way to improve performance in machine learning is to construct an ensemble that combines the predictions of multiple models. Ensemble methods are often much more accurate and have a lower variance than the individual classifiers that make them up, but have high requirements in terms of memory and computational time. In fact, a large number of alternative algorithms is usually adopted, each requiring to query all available data. We propose a new quantum algorithm that exploits quantum superposition, entanglement and interference to build an ensemble of classification models. Thanks to the generation of the several quantum trajectories in superposition, we obtain B transformations of the quantum state which encodes the training set in only log ( B ) operations. This implies an exponential growth of the ensemble size while increasing linearly the depth of the correspondent circuit. Furthermore, when considering the overall cost of the algorithm, we show that the training of a single weak classifier impacts additively to the overall time complexity rather than multiplicatively, as it usually happens in classical ensemble methods. We also present small-scale experiments on real-world datasets, defining a quantum version of the cosine classifier and using the IBM qiskit environment to show how the algorithms work.
Perceptrons, which perform binary classification, are the fundamental building blocks of neural networks. Given a data set of size NNN and margin $\ensuremath{\gamma}$ (how well the given data are separated), the query complexity of the best-known quantum training algorithm scales as either $(\sqrt{N}/{\ensuremath{\gamma}}^{2})log(1/{\ensuremath{\gamma}}^{2})$ or $N/\sqrt{\ensuremath{\gamma}}$, which is achieved by a hybrid of classical and quantum search. In this paper, we improve the version space quantum training method for perceptrons such that the query complexity of our algorithm scales as $\sqrt{N/\ensuremath{\gamma}}$. This is achieved by constructing an oracle for the perceptrons using quantum counting of the number of data elements that are correctly classified. Once such an oracle is constructed, bounded-error quantum search can be used to search over the hyperplane instances. The optimality of our algorithm is proven by reducing the evaluation of a two-level and-or tree (for which the query complexity lower bound is known) to a multicriterion search. Our quantum training algorithm can be generalized to train more complex machine learning models such as neural networks, which are built on a large number of perceptrons.
In this work, we investigate quantum tree search via amplitude amplification. Amplitude amplification generalizes Grover's algorithm by replacing the Hadamard initialization with an arbitrary unitary AAA, with Grover's algorithm recovered as the special case of uniform initialization. We demonstrate the construction of a dynamic search tree of depth mmm with query complexity (bavg)m\sqrt{\left(b_{avg}\right)^m}(bavg​)m​ where bavgb_{avg}bavg​ denotes the average branching factor, improving upon the commonly assumed (bmax)m\sqrt{\left(b_{max}\right)^m}(bmax​)m​, where bmaxb_{max}bmax​ is the maximum branching factor. We further challenge the widespread assumption that amplitude amplification is inferior to quantum backtracking. In fact, quantum backtracking is unsuitable for problems that do not naturally admit a backtracking structure; in such cases, amplitude amplification yields improved query complexity. We observe that amplitude amplification constructs the search tree dynamically, rendering its internal structure inaccessible, a constraint that applies equally to quantum backtracking. To address this, we propose sampling-based methods to estimate the tree structure, under the assumption that it approximates a normal distribution with increasing depth. Finally, we introduce a quantum greedy search based on a lookahead heuristic inspired by the classical cognitive architecture Soar, which models human-like problem-solving strategies.