Random forests are a combination of tree predictors such that each tree depends on the values of a random vector sampled independently and with the same distribution for all trees in the forest. The generalization error for forests converges a.s. to a limit as the number of trees in the forest becomes large. The generalization error of a forest of tree classifiers depends on the strength of the individual trees in the forest and the correlation between them. Using a random selection of features to split each node yields error rates that compare favorably to Adaboost (Y. Freund & R. Schapire, Machine Learning: Proceedings of the Thirteenth International conference, aaa, 148–156), but are more robust with respect to noise. Internal estimates monitor error, strength, and correlation and these are used to show the response to increasing the number of features used in the splitting. Internal estimates are also used to measure variable importance. These ideas are also applicable to regression.

Random Forests

Deep learning is a form of machine learning that enables computers to learn from experience and understand the world in terms of a hierarchy of concepts. Because the computer gathers knowledge from experience, there is no need for a human computer operator to formally specify all the knowledge that the computer needs. The hierarchy of concepts allows the computer to learn complicated concepts by building them out of simpler ones; a graph of these hierarchies would be many layers deep. This book introduces a broad range of topics in deep learning. The text offers mathematical and conceptual background, covering relevant concepts in linear algebra, probability theory and information theory, numerical computation, and machine learning. It describes deep learning techniques used by practitioners in industry, including deep feedforward networks, regularization, optimization algorithms, convolutional networks, sequence modeling, and practical methodology; and it surveys such applications as natural language processing, speech recognition, computer vision, online recommendation systems, bioinformatics, and videogames. Finally, the book offers research perspectives, covering such theoretical topics as linear factor models, autoencoders, representation learning, structured probabilistic models, Monte Carlo methods, the partition function, approximate inference, and deep generative models. Deep Learning can be used by undergraduate or graduate students planning careers in either industry or research, and by software engineers who want to begin using deep learning in their products or platforms. A website offers supplementary material for both readers and instructors.

Deep Learning

Numerical Optimization presents a comprehensive and up-to-date description of the most effective methods in continuous optimization. It responds to the growing interest in optimization in engineering, science, and business by focusing on the methods that are best suited to practical problems. For this new edition the book has been thoroughly updated throughout. There are new chapters on nonlinear interior methods and derivative-free methods for optimization, both of which are used widely in practice and the focus of much current research. Because of the emphasis on practical methods, as well as the extensive illustrations and exercises, the book is accessible to a wide audience. It can be used as a graduate text in engineering, operations research, mathematics, computer science, and business. It also serves as a handbook for researchers and practitioners in the field. The authors have strived to produce a text that is pleasant to read, informative, and rigorous - one that reveals both the beautiful nature of the discipline and its practical side.

Numerical Optimization

Machine Learning is the study of methods for programming computers to learn. Computers are applied to a wide range of tasks, and for most of these it is relatively easy for programmers to design and implement the necessary software. However, there are many tasks for which this is difficult or impossible. These can be divided into four general categories. First, there are problems for which there exist no human experts. For example, in modern automated manufacturing facilities, there is a need to predict machine failures before they occur by analyzing sensor readings. Because the machines are new, there are no human experts who can be interviewed by a programmer to provide the knowledge necessary to build a computer system. A machine learning system can study recorded data and subsequent machine failures and learn prediction rules. Second, there are problems where human experts exist, but where they are unable to explain their expertise. This is the case in many perceptual tasks, such as speech recognition, hand-writing recognition, and natural language understanding. Virtually all humans exhibit expert-level abilities on these tasks, but none of them can describe the detailed steps that they follow as they perform them. Fortunately, humans can provide machines with examples of the inputs and correct outputs for these tasks, so machine learning algorithms can learn to map the inputs to the outputs. Third, there are problems where phenomena are changing rapidly. In finance, for example, people would like to predict the future behavior of the stock market, of consumer purchases, or of exchange rates. These behaviors change frequently, so that even if a programmer could construct a good predictive computer program, it would need to be rewritten frequently. A learning program can relieve the programmer of this burden by constantly modifying and tuning a set of learned prediction rules. Fourth, there are applications that need to be customized for each computer user separately. Consider, for example, a program to filter unwanted electronic mail messages. Different users will need different filters. It is unreasonable to expect each user to program his or her own rules, and it is infeasible to provide every user with a software engineer to keep the rules up-to-date. A machine learning system can learn which mail messages the user rejects and maintain the filtering rules automatically. Machine learning addresses many of the same research questions as the fields of statistics, data mining, and psychology, but with differences of emphasis. Statistics focuses on understanding the phenomena that have generated the data, often with the goal of testing different hypotheses about those phenomena. Data mining seeks to find patterns in the data that are understandable by people. Psychological studies of human learning aspire to understand the mechanisms underlying the various learning behaviors exhibited by people (concept learning, skill acquisition, strategy change, etc.).

Machine learning

A framework is developed to explore the connection between effective optimization algorithms and the problems they are solving. A number of "no free lunch" (NFL) theorems are presented which establish that for any algorithm, any elevated performance over one class of problems is offset by performance over another class. These theorems result in a geometric interpretation of what it means for an algorithm to be well suited to an optimization problem. Applications of the NFL theorems to information-theoretic aspects of optimization and benchmark measures of performance are also presented. Other issues addressed include time-varying optimization problems and a priori "head-to-head" minimax distinctions between optimization algorithms, distinctions that result despite the NFL theorems' enforcing of a type of uniformity over all algorithms.

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No free lunch theorems for optimization

Systems, methods and aspects, and embodiments thereof relate to unsupervised or semi-supervised features learning using a quantum processor. To achieve unsupervised or semi-supervised features learning, the quantum processor is programmed to achieve Hierarchal Deep Learning (referred to as HDL) over one or more data sets. Systems and methods search for, parse, and detect maximally repeating patterns in one or more data sets or across data or data sets. Embodiments and aspects regard using sparse coding to detect maximally repeating patterns in or across data. Examples of sparse coding include L0 and L1 sparse coding. Some implementations may involve appending, incorporating or attaching labels to dictionary elements, or constituent elements of one or more dictionaries. There may be a logical association between label and the element labeled such that the process of unsupervised or semi-supervised feature learning spans both the elements and the incorporated, attached or appended label.

Systems and methods for quantum processing of data

Computer experiments often require dense sweeps over input parameters to obtain a qualitative understanding of their response. Such sweeps can be prohibitively expensive, and are unnecessary in regions where the response is easy predicted; well-chosen designs could allow a mapping of the response with far fewer simulation runs. Thus, there is a need for computationally inexpensive surrogate models and an accompanying method for selecting small designs. We explore a general methodology for addressing this need that uses non-stationary Gaussian processes. Binary trees partition the input space to facilitate non-stationarity and a Bayesian interpretation provides an explicit measure of predictive uncertainty that can be used to guide sampling. Our methods are illustrated on several examples, including a motivating example involving computational fluid dynamics simulation of a NASA reentry vehicle.

Parameter space exploration with Gaussian process trees

In this article, we show how to map a sampling of the hardest artificial intelligence problems in space exploration onto equivalent Ising models that then can be attacked using quantum annealing implemented in D-Wave machine. We overview the existing results as well as propose new Ising model implementations for quantum annealing. We review supervised and unsupervised learning algorithms for classification and clustering with applications to feature identification and anomaly detection. We introduce algorithms for data fusion and image matching for remote sensing applications. We overview planning problems for space exploration mission applications and algorithms for diagnostics and recovery with applications to deep space missions. We describe combinatorial optimization algorithms for task assignment in the context of autonomous unmanned exploration. Finally, we discuss the ways to circumvent the limitation of the Ising mapping using a "blackbox" approach based on ideas from probabilistic computing. In this article we describe the architecture of the D-Wave One machine and report its benchmarks. Results on random ensemble of problems in the range of up to 96 qubits show improved scaling for median core quantum annealing time compared with classical algorithms; whether this scaling persists for larger problem sizes is an open question. We also review previous results of D-Wave One benchmarking studies for solving binary classification problems with a quantum boosting algorithm which is shown to outperform AdaBoost. We review quantum algorithms for structured learning for multi-label classification and introduce a hybrid classical/quantum approach for learning the weights. Results of D-Wave One benchmarking studies for learning structured labels on four different data sets show a better performance compared with an independent Support Vector Machine approach with linear kernel.

A Near-Term Quantum Computing Approach for Hard Computational Problems in Space Exploration

This note discusses the recent paper "Some technical remarks on the proof of the no free lunch theorem" by Koppen (2000). In that paper, some technical issues related to the formal proof of the no free lunch (NFL) theorem for search were given by Wolpert and Macready (1995, 1997). The present authors explore the issues raised in that paper including the presentation of a simpler version of the NFL proof in accord with a suggestion made explicitly by Koppen (2000) and implicitly by Wolpert and Macready (1997). They also includes the correction of an incorrect claim made by Koppen (2000) of a limitation of the NFL theorem. Finally, some thoughts on future research directions for research into algorithm performance are given.

Remarks on a recent paper on the "no free lunch" theorems

Current quantum annealing (QA) hardware suffers from practical limitations such as finite temperature, sparse connectivity, small qubit numbers, and control error. We propose new algorithms for mapping boolean constraint satisfaction problems (CSPs) onto QA hardware mitigating these limitations. In particular we develop a new embedding algorithm for mapping a CSP onto a hardware Ising model with a fixed sparse set of interactions, and propose two new decomposition algorithms for solving problems too large to map directly into hardware. 
The mapping technique is locally-structured, as hardware compatible Ising models are generated for each problem constraint, and variables appearing in different constraints are chained together using ferromagnetic couplings. In contrast, global embedding techniques generate a hardware independent Ising model for all the constraints, and then use a minor-embedding algorithm to generate a hardware compatible Ising model. We give an example of a class of CSPs for which the scaling performance of D-Wave's QA hardware using the local mapping technique is significantly better than global embedding. 
We validate the approach by applying D-Wave's hardware to circuit-based fault-diagnosis. For circuits that embed directly, we find that the hardware is typically able to find all solutions from a min-fault diagnosis set of size N using 1000N samples, using an annealing rate that is 25 times faster than a leading SAT-based sampling method. Further, we apply decomposition algorithms to find min-cardinality faults for circuits that are up to 5 times larger than can be solved directly on current hardware.

William G. Macready

Papers

Systems and methods for quantum processing of data

Parameter space exploration with Gaussian process trees

A Near-Term Quantum Computing Approach for Hard Computational Problems in Space Exploration

Remarks on a recent paper on the "no free lunch" theorems

Mapping constrained optimization problems to quantum annealing with application to fault diagnosis