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Information, Divergence and Risk for Binary Experiments
 JOURNAL OF MACHINE LEARNING RESEARCH
, 2009
"... We unify fdivergences, Bregman divergences, surrogate regret bounds, proper scoring rules, cost curves, ROCcurves and statistical information. We do this by systematically studying integral and variational representations of these various objects and in so doing identify their primitives which all ..."
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Cited by 37 (8 self)
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We unify fdivergences, Bregman divergences, surrogate regret bounds, proper scoring rules, cost curves, ROCcurves and statistical information. We do this by systematically studying integral and variational representations of these various objects and in so doing identify their primitives which all are related to costsensitive binary classification. As well as developing relationships between generative and discriminative views of learning, the new machinery leads to tight and more general surrogate regret bounds and generalised Pinsker inequalities relating fdivergences to variational divergence. The new viewpoint also illuminates existing algorithms: it provides a new derivation of Support Vector Machines in terms of divergences and relates Maximum Mean Discrepancy to Fisher Linear Discriminants.
Towards a Definition of an Algorithm
, 2005
"... We define an algorithm to be the set of programs that implement or express that algorithm. The set of all programs is partitioned into equivalence classes. Two programs are equivalent if they are “essentially ” the same program. The set of all equivalence classes is the category of all algorithms. I ..."
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Cited by 9 (0 self)
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We define an algorithm to be the set of programs that implement or express that algorithm. The set of all programs is partitioned into equivalence classes. Two programs are equivalent if they are “essentially ” the same program. The set of all equivalence classes is the category of all algorithms. In order to explore these ideas, the set of primitive recursive functions is considered. Each primitive recursive function can be described by many labeled binary trees that show how the function is built up. Each tree is like a program that shows how to compute a function. We give relations that say when two such trees are “essentially” the same. An equivalence class of such trees will be called an algorithm.
In Some Curved Spaces, One Can Solve NPHard Problems in Polynomial Time
"... In the late 1970s and the early 1980s, Yuri Matiyasevich actively used his knowledge of engineering and physical phenomena to come up with parallelized schemes for solving NPhard problems in polynomial time. In this paper, we describe one such scheme in which we use parallel computation in curved s ..."
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Cited by 8 (8 self)
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In the late 1970s and the early 1980s, Yuri Matiyasevich actively used his knowledge of engineering and physical phenomena to come up with parallelized schemes for solving NPhard problems in polynomial time. In this paper, we describe one such scheme in which we use parallel computation in curved spaces. 1 Introduction and Formulation of the Problem Many practical problems are NPhard. It is well known that many important practical problems are NPhard; see, e.g., [11, 14, 27]. Under the usual hypothesis that P̸=NP, NPhardness has the following intuitive meaning: every algorithm which solves all instances of the corresponding problem requires, for
What Is an Algorithm
 SOFSEM, Lecture Notes in
"... We present a twopart exposition on the notion of algorithm and foundational analyses of computation. The first part is below, and the second is here: ..."
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Cited by 6 (4 self)
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We present a twopart exposition on the notion of algorithm and foundational analyses of computation. The first part is below, and the second is here:
Interactive smallstep algorithms I: Axiomatization,
, 2006
"... In earlier work, the Abstract State Machine Thesis — that arbitrary algorithms are behaviorally equivalent to abstract state machines — was established for several classes of algorithms, including ordinary, interactive, smallstep algorithms. This was accomplished on the basis of axiomatizations o ..."
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In earlier work, the Abstract State Machine Thesis — that arbitrary algorithms are behaviorally equivalent to abstract state machines — was established for several classes of algorithms, including ordinary, interactive, smallstep algorithms. This was accomplished on the basis of axiomatizations of these classes of algorithms. Here we extend the axiomatization and, in a companion paper, the proof, to cover interactive smallstep algorithms that are not necessarily ordinary. This means that the algorithms (1) can complete a step without necessarily waiting for replies to all queries from that step and (2) can use not only the environment’s replies but also the order in which the replies were received.
A natural axiomatization of Church’s thesis
, 2007
"... The Abstract State Machine Thesis asserts that every classical algorithm is behaviorally equivalent to an abstract state machine. This thesis has been shown to follow from three natural postulates about algorithmic computation. Here, we prove that augmenting those postulates with an additional requ ..."
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The Abstract State Machine Thesis asserts that every classical algorithm is behaviorally equivalent to an abstract state machine. This thesis has been shown to follow from three natural postulates about algorithmic computation. Here, we prove that augmenting those postulates with an additional requirement regarding basic operations implies Church’s Thesis, namely, that the only numeric functions that can be calculated by effective means are the recursive ones (which are the same, extensionally, as the Turingcomputable numeric functions). In particular, this gives a natural axiomatization of Church’s Thesis, as Gödel and others suggested may be possible.
IntraStep Interaction
"... For a while it seemed possible to pretend that all interaction between an algorithm and its environment occurs interstep, but not anymore. Andreas Blass, ..."
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For a while it seemed possible to pretend that all interaction between an algorithm and its environment occurs interstep, but not anymore. Andreas Blass,
Chapter 19 Algorithms and Arguments: The Foundational Role of the ATAIquestion
"... Argumentation theory underwent a significant development in the Fifties and Sixties: its revival is usually connected to Perelman's criticism of formal logic and the development of informal logic. Interestingly enough it was during this period that Artificial Intelligence was developed, which d ..."
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Argumentation theory underwent a significant development in the Fifties and Sixties: its revival is usually connected to Perelman's criticism of formal logic and the development of informal logic. Interestingly enough it was during this period that Artificial Intelligence was developed, which defended the following thesis (from now on referred
RECURSION IN COGNITION: A COMPUTATIONAL INVESTIGATION INTO THE REPRESENTATION AND PROCESSING OF LANGUAGE
"... ADVERTIMENT. L'accés als continguts d'aquesta tesi doctoral i la seva utilització ha de respectar els drets de la persona autora. Pot ser utilitzada per a consulta o estudi personal, així com en activitats o materials d'investigació i docència en els termes establerts a l'art. 32 ..."
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ADVERTIMENT. L'accés als continguts d'aquesta tesi doctoral i la seva utilització ha de respectar els drets de la persona autora. Pot ser utilitzada per a consulta o estudi personal, així com en activitats o materials d'investigació i docència en els termes establerts a l'art. 32 del Text Refós de la Llei de Propietat Intel·lectual (RDL 1/1996). Per altres utilitzacions es requereix l'autorització prèvia i expressa de la persona autora. En qualsevol cas, en la utilització dels seus continguts caldrà indicar de forma clara el nom i cognoms de la persona autora i el títol de la tesi doctoral. No s'autoritza la seva reproducció o altres formes d'explotació efectuades amb finalitats de lucre ni la seva comunicació pública des d'un lloc aliè al servei TDX. Tampoc s'autoritza la presentació del seu contingut en una finestra o marc aliè a TDX (framing). Aquesta reserva de drets afecta tant als continguts de la tesi com als seus resums i índexs. ADVERTENCIA. El acceso a los contenidos de esta tesis doctoral y su utilización debe respetar los derechos de la persona autora. Puede ser utilizada para consulta o estudio personal, así como en actividades o materiales de investigación y docencia en los términos establecidos en el art. 32 del Texto Refundido de la Ley de Propiedad Intelectual (RDL 1/1996). Para otros usos se requiere la autorización previa y expresa de la persona autora. En cualquier caso, en la utilización de sus contenidos se deberá indicar de forma clara el nombre y apellidos de la persona autora y el título de la tesis doctoral. No se
Categories and Subject Descriptors: F.1.2 [Computation by Abstract Devices]: Modes of ComputationInteractive and Reactive Computation
"... We indicate how a great many sorts of interaction meet these requirements. We also discuss in detail the structure of queries and replies and the appropriate definition of equivalence of algorithms. Finally, motivated by our considerations concerning queries, we discuss a generalization of firstord ..."
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We indicate how a great many sorts of interaction meet these requirements. We also discuss in detail the structure of queries and replies and the appropriate definition of equivalence of algorithms. Finally, motivated by our considerations concerning queries, we discuss a generalization of firstorder logic in which the arguments of function and relation symbols are not merely tuples of elements but orbits of such tuples under groups of permutations of the argument places.