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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 41 (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.
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
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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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.
Y.: Exact exploration and hanging algorithms
 Proceedings of the 19th EACSL Annual Conferences on Computer Science Logic (Brno, Czech Republic). Lecture Notes in Computer Science
, 2010
"... Abstract. Recent analysis of sequential algorithms resulted in their axiomatization and in a representation theorem stating that, for any sequential algorithm, there is an abstract state machine (ASM) with the same states, initial states and state transitions. That analysis, however, abstracted fro ..."
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Cited by 7 (7 self)
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Abstract. Recent analysis of sequential algorithms resulted in their axiomatization and in a representation theorem stating that, for any sequential algorithm, there is an abstract state machine (ASM) with the same states, initial states and state transitions. That analysis, however, abstracted from details of intrastep computation, and the ASM, produced in the proof of the representation theorem, may and often does explore parts of the state unexplored by the algorithm. We refine the analysis, the axiomatization and the representation theorem. Emulating a step of the given algorithm, the ASM, produced in the proof of the new representation theorem, explores exactly the part of the state explored by the algorithm. That frugality pays off when state exploration is costly. The algorithm may be a highlevel specification, and a simple function call on the abstraction level of the algorithm may hide expensive interaction with the environment. Furthermore, the original analysis presumed that state functions are total. Now we allow state functions, including equality, to be partial so that a function call may cause the algorithm as well as the ASM to hang. Since the emulating ASM does not make any superfluous function calls, it hangs only if the algorithm does. [T]he monotony of equality can only lead us to boredom. Francis Picabia
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.
Microsoft
"... Infons are statements viewed as containers of information (rather then representations of truth values). The logic of infons turns out to be a conservative extension of logic known as constructive or intuitionistic. Distributed Knowledge Authorization Language uses additional unary connectives “p sa ..."
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Infons are statements viewed as containers of information (rather then representations of truth values). The logic of infons turns out to be a conservative extension of logic known as constructive or intuitionistic. Distributed Knowledge Authorization Language uses additional unary connectives “p said ” and “p implied ” where p ranges over principals. Here we investigate infon logic and a narrow but useful primal fragment of it. In both cases, we develop model theory and analyze the derivability problem: Does the given query follow from the given hypotheses? Our more involved technical results are on primal infon logic. We construct an algorithm for the multiple derivability problem: Which of the given queries follow from the given hypotheses? Given a bound on the quotation depth of the hypotheses, the algorithm runs in linear time. We quickly discuss the significance of this result for access control.
WHAT IS AN ALGORITHM? (REVISED)
"... Abstract. We put the title problem and Church’s thesis into a proper perspective, and we address some common misconceptions about Turing’s analysis of computation. In addition, we comment on two approaches to the title problem, one well known among philosophers and another well known among logicians ..."
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Abstract. We put the title problem and Church’s thesis into a proper perspective, and we address some common misconceptions about Turing’s analysis of computation. In addition, we comment on two approaches to the title problem, one well known among philosophers and another well known among logicians.
Theses for Computation and Recursion on Concrete and Abstract Structures
"... Abstract: The main aim of this article is to examine proposed theses for computation and recursion on concrete and abstract structures. What is generally referred to as Church’s Thesis or the ChurchTuring Thesis (abbreviated CT here) must be restricted to concrete structures whose objects are finit ..."
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Abstract: The main aim of this article is to examine proposed theses for computation and recursion on concrete and abstract structures. What is generally referred to as Church’s Thesis or the ChurchTuring Thesis (abbreviated CT here) must be restricted to concrete structures whose objects are finite symbolic configurations of one sort or another. Informal and principled arguments for CT on concrete structures are reviewed. Next, it is argued that proposed generalizations of notions of computation to abstract structures must be considered instead under the general notion of algorithm. However, there is no clear general thesis in sight for that comparable to CT, though there are certain wide classes of algorithms for which plausible theses can be stated. The article concludes with a proposed thesis RT for recursion on abstract structures. 1. Introduction. The