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

This paper presents a fully specified model of long-run growth in which knowledge is assumed to be an input in production that has increasing marginal productivity. It is essentially a competitive equilibrium model with endogenous technological change. In contrast to models based on diminishing returns, growth rates can be increasing over time, the effects of small disturbances can be amplified by the actions of private agents, and large countries may always grow faster than small countries. Long-run evidence is offered in support of the empirical relevance of these possibilities.

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Increasing Returns and Long-Run Growth

(1977). Graph Theory with Applications. Journal of the Operational Research Society: Vol. 28, Volume 28, issue 1, pp. 237-238.

https://link.springer.com/content/pdf/10.1057%2Fjors.1977.45.pdf

Graph theory with applications

The use of randomness is now an accepted tool in Theoretical Computer Science but not everyone is aware of the underpinnings of this methodology in Combinatorics - particularly, in what is now called the probabilistic Method as developed primarily by Paul Erdoős over the past half century. Here I will explore a particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets. A central point will be the evolution of these problems from the purely existential proofs of Erdős to the algorithmic aspects of much interest to this audience.

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The Probabilistic Method

One strand of endogenous-growth models assumes constant returns to a broad concept of capital. I extend these models to include tax- financed government services that affect production or utility. Growth and saving rates fall with an increase in utility-type expenditures; the two rates rise initially with productive government expenditures but subsequently decline. With an income tax, the decentralized choices of growth and saving are "too low," but if the production function is Cobb-Douglas, the optimizing government still satisfies a natural condition for productive efficiency. Empirical evidence across countries supports some of the hypotheses about government and growth.

Government spending in a simple model of endogenous growth

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The mathematical theory of saving

TILE problem I propose to tackle is this: a given revenue is to be raised by proportionate taxes on some or all uses of income, the taxes on different uses being possibly at different rates; how should these rates be adjusted in order that the decrement of utility may be a minimum? I propose to neglect altogether questions of distribution and considerations arising from the differences in the marginal utility of money to different people; and I shall deal only with a purely competitive system with no foreign trade. Further I shall suppose that, in Professor Pigou's terminology, private and social net products are always equal or have been made so by State interference not included in the taxation we are considering. I thus exclude the case discussed in Marshall's Principles in which a bounty on increasing-return commodities is advisable. Nevertheless we shall find that the obvious solution that there should be no differentiation is entirely erroneous. The effect of taxation is to transfer income in the first place from individuals to the State and then, in part, back again to rentiers and pensioners. These transfers will slightly alter the demand schedules in a way depending on the incidence of the taxes and the manner of their expenditure. I neglect these alterations; 1 and I also suppose that \" a given revenue \" means a given money revenue, \" money \" being so adjusted that its marginal utility is constant. This problem was suggested to me by Professor Pigou, to whom I am also indebted for help and encouragement in its solution. In the first part I deal with the perfectly general utility function and establish a result which is valid for a sufficiently small revenue, and takes a peculiarly simple form if we can treat the revenue as an infinitesimal. I prove, in fact, that in raising an infinitesimal revenue by proportionate taxes on given commodities the taxes should be such as to diminish in the same proportion the production of each commodity taxed. In the second part I assume that the utility function is quadratic, which means roughly that the supply and demand

A Contribution to the Theory of Taxation

This paper is primarily concerned with a special case of one of the leading problems of mathematical logic, the problem of finding a regular procedure to determine the truth or falsity of any given logical formula*. But in the course of this investigation it is necessary to use certain theorems on combinations which have an independent interest and are most conveniently set out by themselves beforehand.

On a Problem of Formal Logic

This chapter is a reprint of Frank P. Ramsey’s seminal paper “Truth and Probability” written in 1926 and first published posthumous in the 1931 The Foundations of Mathematics and other Logical Essays, ed. R.B. Braithwaite, London: Routledge & Kegan Paul Ltd.

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Truth and Probability

Although not yet 27 years of age at the time of his death, Ramsey left contributions to mathematics, logic, and economics which were of the greatest value for contemporary philosophy. The present volume contains published and unpublished papers, ranging in date from 1923 to 1929, which show the development of his philosophic thought. – Published Papers : – I. «The Foundations of Mathematics», [i]Proceedings of the London Mathematical Society[/i], Ser. 2, Vol. 25, Part 5, pp. 338-384. Read 12th November. 1925; – II. «Mathematical Logic», [i]The Mathematical Gazette[/i], Vol. 13, N° 184, pp. 185-194. October, 1926. [A paper read before the British Association, Section A, Oxford, August, 1926]; — III. «On a Problem of Formal Logic», [i]Proceedings of the London Mathematical Society[/i], Ser. 2, Vol. 30, Part 4, pp. 338-384. Read 13th December, 1928; – IV. «Universals», [i]Mind[/i], N.S., Vol. 34, N° 136, pp. 401-417. October, 1925; – V. «Note on the Preceding Paper : Universals and the “Method of Analysis”», [i]Aristotelian Society[/i] Supplementary Volume VI, pp. 17-26. July, 1926. [Symposium with H. W. B. Joseph (i) and R. B. Braithwaite (iii)]; – VI. «Facts and Propositions», [i]Aristotelian Society[/i] Supplementary Volume VII, pp. 153-170. July, 1927. [Symposium with G. E. Moore (ii)]; – Appendix ; «Critical Notice of L. Wittgenstein’s [i]Tractatus Logico-Philosophicus[/i]», [i]Mind[/i], N.S., Vol. 32, N° 128, pp. 465-478. October, 1923. – Unpublished Papers : – VII. «Truth and Probability», 1926; – VIII. «Further Considerations : A. Reasonable Degree of Belief; B. Statistics; C. Chance», 1928; – IX. «Last Papers : A. Theories; B. General Propositions and Causality; C. Probability and Partial Belief; D. Knowledge; E. Causal Qualities; F. Philosophy», 1929. M.-M. V.

Frank Plumpton Ramsey

Papers

The mathematical theory of saving

A Contribution to the Theory of Taxation

On a Problem of Formal Logic

Truth and Probability

The Foundations of Mathematics and Other Logical Essays