Paul Fischer and Hans Ulrich Simon. On learning ring-sumexpansions. SIAM Journal on Computing, 21(1):181--192, February 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the k term DNF) and k CNF can be easily learned by standard procedures. Suppose, however, that we wish to learn in the same manner another class of concepts C k;f (that is, other than k term DNF) for which learning C k;f by C k;f is NP hard. Our results and related results by Fischer and Simon [41] show that exclusive or (XOR) is one such function. In this case, an XOR of k monomials need not be representable as a k CNF or as a k DNF (for example, x 1 x 2 Phi x 3 written as a DNF requires one term of size 3, and written as a CNF requires one clause of size 3) In addition an XOR of k ....

Paul Fischer and Hans Ulrich Simon. On learning ring-sum-expansions. SIAM J. Computing, 21(1):181--192, February 1992.

....(pairs (x i ; y i ) whether or not there is some f 2 F that is consistent on S, i.e. such that for each i, f(x i ) y i ) This was used to show that if the consistency problem for F is NP hard, then F is not PAC learnable unless RP = NP. The technique has seen a number of applications [51, 15, 26, 14]. Because learning with equivalence queries implies PAC learning, it follows that for such classes F , F cannot be learned with equivalence queries unless RP = NP. A folklore extension of this observation, which we present in Section 4 for completeness, directly relates the consistency problem to ....

....with references to the associated NP hardness results. Some Classes not Learnable in Polytime with Equivalence Queries 1. For constant k 2, k term DNF formulas (dually, k clause CNF formulas) 51] 2. Various types of functions of k terms, including XORs of k terms, where k is constant [15, 26]. 3. Boolean threshold functions (halfspaces where each weight is in f0,1g) 51] 4. Read once Boolean formulas (each variable appears at most once) 51] 5. Certain types of existentially quantified conjunctive propositional formulas [31] 6. The intersection of k halfspaces, for constant k ....

P. Fischer and H. U. Simon, On learning ring-sum-expansions. SIAM Journal of Computing, 21(1):181--192, 1992.

....one of two values. Indeed, for small fields such as GF(2) most of the previous work has demonstrated that polynomials are hard to learn in various models, except in special cases. In the PAC model (in the absence of membership and equivalence queries) Blum and Singh [5] and Fischer and Simon [11] show that it is computationally hard to learn t sparse polynomials over GF(2) for any fixed t 2 (assuming RP 6= NP, and also assuming that the hypothesis must be expressed as a t sparse polynomial) Their result also implies that it is hard to learn t sparse GF(2) polynomials using a proper ....

Paul Fischer and Hans Ulrich Simon. On learning ring-sum-expansions. SIAM Journal on Computing, 21(1):181--192, February 1992.

....unless RP=NP (resp. P=NP) For example, it is known [23] that, for the class of (positive) read once functions and for the class of (positive) h term DNF functions, where h # 2, EXTENSION is NP hard. For some other classes such as neural networks, EXTENSION is known to be NP hard (e.g. [1, 14, 20, 23]) From the viewpoint of knowledge acquisition, we consider in this paper problem EXTENSION(C) for various classes C, which may not be polynomially size bounded or polynomially reasonable. Unfortunately, the real world data might contain errors. As for the above examples, 4 measurement or ....

Fischer, P., and Simon, H. U. (1992), On learning ring-sum-expansions, SIAM Journal on Computing 21, 181-192.

....of C) and m is the worst case mistake bound of the Closure Algorithm for learning C in the noise free model. For several concept classes our extension is efficient and in some cases it can tolerate a noise rate equal to the information theoretic upper bound for that class. Using the results of [7, 11] we show that the classes of monotone monomials, k CNF functions, parity functions, integer lattices, and k ring sum expansions can be efficiently learned on line with adversarial noise. We also propose a general technique for showing upper bounds on the noise rate tolerated by any on line learner ....

....is on line learnable in time polynomial in N and K. Furthermore MB(k CNFn ; K) N (N 1)K. For all n 1 let parityn be the class of parity functions over all subsets of fx1 ; xng. The following observation legitimates the use of the Extended Closure Algorithm to learn parityn . Lemma12 [7]. Each C 2 parityn is a linear subspace of f0; 1g n with respect to the addition modulo 2 and the usual scalar product over f0; 1g. Let subn be the class of all linear subspaces f0; 1g n with respect to the operations defined in the statement of Lemma 12. Corollary 13. For all K 0 the class ....

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P. Fischer and H.U. Simon. On learning ring-sum-expansions. SIAM Journal on Computing, 21:181--192, 1992.

....of n variables have degree less than n=2. Hence, most Boolean functions of n arguments require at least (n) Gamma 1 steps to be computed by the ROBUST PRAM with single bit memory cells. Any nonconstant Boolean function f satisfies jf Gamma1 (0)j; jf Gamma1 (1)j 2 n Gammadeg 2 (f) [6]. Thus, if 0 jf Gamma1 (0)j 2 n Gammad or 0 jf Gamma1 (1)j 2 n Gammad , then deg 2 (f) d, so the ROBUST PRAM with 1 bit memory cells requires at least (d) steps to compute f . The PARITY function of n variables has degree 1 over IF 2 . However, over IF 3 , it has degree n. It ....

P. Fischer, and H. U. Simon, "On Learning Ring-Sum-Expansions", Proc. Third Workshop on Computational Learning Theory, 1990, to appear in SIAM J. Comput.

....address any queries directly to the expert. Instead, it always queries the corrector. A simple example of an application of the above observation is a learning algorithm (using membership queries) for noisy parity functions. This algorithm is composed of an algorithm for learning parity functions [13], 19] by solving a system of linear equations over the field of integers modulo 2, and a self correcting algorithm [9] 26] for the same family of functions. We do not know of any other self correcting algorithm that has been directly applied to a related learning problem, but the possibility ....

Paul Fischer and Hans Ulrich Simon. (1992) On learning ring-sum-expansions. SIAM Journal on Computing, 21 , 181--192.

....of all polynomials of n variables have degree less than n=2. Hence, most Boolean functions of n arguments require at least OE(n) Gamma 1 steps to be computed by the ROBUST PRAM of wordsize 1. Any nonconstant Boolean function f satisfies jf Gamma1 (0)j; jf Gamma1 (1)j 2 n Gammadeg 2 (f) [7]. Thus, if 0 jf Gamma1 (0)j 2 n Gammak or 0 jf Gamma1 (1)j 2 n Gammak , then these PRAMs require at least OE(k) steps to compute f . The PARITY function of n variables has degree 1 over F 2 . However, over F 3 , it has degree n. It follows from Theorem 1 that a ROBUST PRAM whose ....

P. Fischer, and H. U. Simon, On Learning Ring-Sum-Expansions, in "Proc. 3rd Workshop on Computational Learning Theory", Morgan Kaufmann, 1990, to appear in SIAM J. Comp..

....learning problem can be solved using an algorithm for a related PAC learning problem. Theorem 8 The class PARITY n is efficiently exactly learnable with a generator and evaluator. Proof: The learning algorithm uses as a subroutine an algorithm for learning parity functions in the PAC model [10, 15] by solving a system of linear equations over the field of integers modulo 2. In the current context, this subroutine receives random examples of the form h x; fS ( x)i, where x 2 f0; 1g n is chosen uniformly, and fS computes the parity of the vector x on the subset S fx1 ; xng. ....

Paul Fischer and Hans Ulrich Simon. On learning ring-sumexpansions. SIAM Journal on Computing, 21(1):181--192, February 1992.

....walks W n , and n NBTFn;Wn n 1. 5 Exact mistake bound learning of 2 Term RSE A 2 term RSE is the parity of two monotone monomials, e.g. x 1 x 3 ) Phi(x 3 x 4 x 5 ) It is known that this class is not properly learnable in the PAC model but learnable using a larger hypothesis class (see [5]) Our algorithm will use intermediate hypotheses which are not 2 term RSE. We will omit an explicit definition of this hypothesis class. Instead the hypotheses will be implicitly described in the algorithms. Basically they are nested case distinctions based on previously gathered information. We ....

P. Fischer and H. Simon. On learning ring-sum expansions. SIAM J. Comput., 21:181--192, 1992.

....walks Wn , and n NBTFn;Wn n 1. 5 EXACT MISTAKE BOUND LEARNING 2 TERM RSE A 2 term RSE is the parity of two monotone monomials, e.g. x 1 x 3 ) Phi (x 3 x 4 x 5 ) It is known that this class is not properly learnable in the PAC model but learnable using a larger hypothesis class (see [6]) Our algorithm will use intermediate hypotheses which are not 2 term RSE. We will omit an explicit definition of this hypothesis class. Instead the hypotheses will be implicitly described in the algorithms. Basically they are nested case distinctions based on previously gathered information. We ....

P. Fischer and H. Simon. On learning ring-sum expansions. SIAM J. Comput., 21:181--192, 1992.

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Paul Fischer and Hans Ulrich Simon. On learning ring-sumexpansions. SIAM Journal on Computing, 21(1):181--192, February 1992.

No context found.

Paul Fischer and Hans Ulrich Simon. On learning ring-sum expansions. SIAM J. Computing, 21(1):181--192, 1992.

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Fischer, P. and Simon, H. U., "On Learning Ring-Sum Expansions", SIAM J. Computing, 21, 1(1992): 181-192.

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Fischer, P. and Simon, H. U., "On Learning Ring-Sum Expansions", SIAM J. Computing, 21, 1(1992): 181-192.

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