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ABSTRACT: This paper is based on the concept of the learning function, which represents the input-output relation of the learning algorithm. The learning process and the information compression process are formulated as the PAC learning function and the Occam function, respectively, and their equivalence is discussed. It is shown that the Occam function is always a consistent PAC learning function, while its converse is not always true.The weak Occam function which is obtained by weakening the condition concerning the information compression power of Occam function is defined anew and it is shown that the weak Occam function is always a consistent PAC learning function. Furthermore, a procedure is shown which derives the weak Occam function from the PAC learning function under a certain condition.
Systems and Computers in Japan 03/2007; 24(8):47 - 58.
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ABSTRACT: Several results on the monotone circuit complexity and the conjunctive complexity, i.e., the minimal number of AND gates in
monotone circuits, of quadratic Boolean functions are proved. We focus on the comparison between single level circuits, which
have only one level of AND gates, and arbitrary monotone circuits, and show that there is an exponential gap between the conjunctive
complexity of single level circuits and that of general monotone circuits for some explicit quadratic function. Nearly tight
upper bounds on the largest gap between the single level conjunctive complexity and the general conjunctive complexity over
all quadratic functions are also proved. Moreover, we describe the way of lower bounding
the single level circuit complexity and give a set of quadratic functions whose monotone complexity is strictly smaller than
its single level complexity.
Algorithmica 08/2006; 46(1):3-14. · 0.60 Impact Factor
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IEICE Transactions. 01/2006; 89-D:2340-2347.
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Computing and Combinatorics, 12th Annual International Conference, COCOON 2006, Taipei, Taiwan, August 15-18, 2006, Proceedings; 01/2006
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Computing and Combinatorics, 12th Annual International Conference, COCOON 2006, Taipei, Taiwan, August 15-18, 2006, Proceedings; 01/2006
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ABSTRACT: We consider the problem of dynamically apportioning resources among a set of options in a worst-case online framework. The
model we investigate is a generalization of the well studied online learning model. In particular, we allow the learner to
see as additional information how high the risk of each option is. This assumption is natural in many applications like horse-race
betting, where gamblers know odds for all options before placing bets. We apply the Aggregating Algorithm to this problem
and give a tight performance bound. The results support our intuition that we should bet more on low-risk options. Surprisingly,
however, the Hedge Algorithm without seeing risk information performs nearly as well as the Aggregating Algorithm. So the
risk information does not help much. Moreover, the loss bound does not depend on the values of relatively small risks.
10/2005: pages 343-355;
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Algorithmic Learning Theory, 16th International Conference, ALT 2005, Singapore, October 8-11, 2005, Proceedings; 01/2005
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Mathematical Foundations of Computer Science 2005, 30th International Symposium, MFCS 2005, Gdansk, Poland, August 29 - September 2, 2005, Proceedings; 01/2005
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[show abstract]
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ABSTRACT: Several results on the monotone circuit complexity and the conjunctive complexity, i.e., the minimal number of AND gates in monotone circuits, of quadratic Boolean functions are proved. We focus on the comparison between single level circuits, which have only one level of AND gates, and arbitrary monotone circuits, and show that there is a huge gap between the conjunctive complexity of single level circuits and that of general monotone circuits for some explicit quadratic function. Almost tight upper bounds on the largest gap between the single level conjunctive complexity and the general conjunctive complexity over all quadratic functions are also proved. Moreover, we describe the way of lower bounding the single level circuit complexity, and give a set of quadratic functions whose monotone complexity is strictly smaller than its single level complexity.
12/2004: pages 1715-1726;
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Algorithms and Computation, 15th International Symposium, ISAAC 2004, Hong Kong, China, December 20-22, 2004, Proceedings; 01/2004
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IEICE Transactions. 01/2004; 87-D:343-351.
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SIAM J. Comput. 01/2004; 33:433-447.
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Algorithmic Learning Theory, 15th International Conference, ALT 2004, Padova, Italy, October 2-5, 2004, Proceedings; 01/2004
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Electronic Colloquium on Computational Complexity (ECCC). 01/2004;
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ABSTRACT: We investigate the complexity of circuits consisting solely of modulo gates and obtain results which might be helpful to derive lower bounds on circuit complexity: (i) We describe a procedure that converts a circuit with only modulo 2p gates, where p is a prime number, into a depth two circuit with modulo 2 gates at the input level and a modulo p gate at the output. (ii) We show some properties of such depth two circuits computing symmetric functions. As a consequence we might think of the strategy for deriving lower bounds on modular circuits: Suppose that a polynomial size constant depth modulo 2p circuit C computes a symmetric function. If we can show that the circuit obtained by applying the procedure given in (i) to the circuit C cannot satisfy the properties described in (ii), then we have a super-polynomial lower bound on the size of a constant depth modulo 2p circuit computing a certain symmetric function.
06/2003;
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ABSTRACT: In this paper, we prove two general theorems on monotone Boolean functions which are useful for constructing an learning algorithm for monotone Boolean functions under the uniform distribution.
05/2003;
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ABSTRACT: In this paper, we investigate the lower bound on the number of gates in a Boolean circuit that computes the clique function with a limited number of negation gates. To derive strong lower bounds on the size of such a circuit we develop a new approach by combining the three approaches: the restriction applied to constant depth circuits due to Hastad the approximation method applied to monotone circuits due to Razborov and the boundary covering developed in the present paper.
05/2003;
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ABSTRACT: A negation-limited circuit is a combinational circuit that consists of AND, OR gates and a limited number of NOT gates. In this paper, we investigate the complexity of negation-limited circuits. The (n,n) merging function is a function that merges two presorted binary sequences x1⩽⋯⩽xn and y1⩽⋯⩽yn into a sequence z1⩽⋯⩽z2n. We prove that the size complexity of the (n,n) merging function with NOT gates is Θ(2an).
Discrete Applied Mathematics. 01/2003;
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Mathematical Foundations of Computer Science 2003, 28th International Symposium, MFCS 2003, Bratislava, Slovakia, August 25-29, 2003, Proceedings; 01/2003
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Algorithms and Complexity, 5th Italian Conference, CIAC 2003, Rome, Italy, May 28-30, 2003, Proceedings; 01/2003
