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

We show that all algorithms that search for an extremum of a cost function perform exactly the same, when averaged over all possible cost functions. In particular, if algorithm A outperforms algorithm B on some cost functions, then loosely speaking there must exist exactly as many other functions where B outperforms A. Starting from this we analyze a number of the other a priori characteristics of the search problem, like its geometry and its information-theoretic aspects. This analysis allows us to derive mathematical benchmarks for assessing a particular search algorithm's performance. We also investigate minimax aspects of the search problem, the validity of using characteristics of a partial search over a cost function to predict future behavior of the search algorithm on that cost function, and time-varying cost functions. We conclude with some discussion of the justifiablility of biologically-inspired search methods.

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No Free Lunch Theorems for Search

We present a heuristic algorithm for finding a graph $H$ as a minor of a graph $G$ that is practical for sparse $G$ and $H$ with hundreds of vertices We also explain the practical importance of finding graph minors in mapping quadratic pseudo-boolean optimization problems onto an adiabatic quantum annealer

A practical heuristic for finding graph minors

We address the question of how a firm’s current location in the space of technological possibilities constrain its search for technological improvements. We formalize a quantitative notion of distance between technologies — encompassing the distinction between evolutionary changes (small distance) versus revolutionary change (large distance) — and introduce a technology landscape into an otherwise standard dynamic programming setting where the optimal strategy is to assign a reservation price to each possible technology. Technological search is modeled as movement, constrained by the cost of search, on a technology landscape. Simulations are presented on a stylized technology landscape while analytic results are derived using landscapes that are similar to Markov random fields. We find that early in the search for technological improvements, if the initial position is poor or average, it is optimal to search far away on the technology landscape; but as the firm succeeds in finding technological improvements it is optimal to confine search to a local region of the landscape. © 2000 Elsevier Science B.V. All rights reserved.

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Optimal search on a technology landscape

Recent work on the foundational underpinnings of black-box optimization has begun to uncover a rich mathematical structure. In particular, it is now known that an inner product between the optimization algorithm and the distribution of optimization problems likely to be encountered fixes the distribution over likely performances in running that algorithm. One ramification of this is the "No Free Lunch" (NFL) theorems, which state that any two algorithms are equivalent when their performance is averaged across all possible problems. This highlights the need for exploiting problem-specific knowledge to achieve better than random performance. In this paper, we present a general framework covering most optimization scenarios. In addition to the optimization scenarios addressed in the NFL results, this framework covers multiarmed bandit problems and evolution of multiple coevolving players. As a particular instance of the latter, it covers "self-play" problems. In these problems, the set of players work together to produce a champion, who then engages one or more antagonists in a subsequent multiplayer game. In contrast to the traditional optimization case where the NFL results hold, we show that in self-play there are free lunches: in coevolution some algorithms have better performance than other algorithms, averaged across all possible problems. However, in the typical coevolutionary scenarios encountered in biology, where there is no champion, the NFL theorems still hold.

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William G. Macready

Papers

No free lunch theorems for optimization

No Free Lunch Theorems for Search

A practical heuristic for finding graph minors

Optimal search on a technology landscape

Coevolutionary free lunches