R. Beigel, N. Reingold, and D. A. Spielman, The perceptron strikes back, in Proc. 6th Annual Conference on Structure in Complexity Theory, IEEE Comp.Soc. Press, 1991.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....and because of some similarity they have to real neurons. In some papers a generalization of these gates is considered: instead of taking a weighted sum of the input variables, we are allowed weighted sums of arbitrary functions of at most d variables each [MP88, Bru90, BS92, GHR92, HG91, Bei92, BRS91] These generalized thresholds are sometimes called threshold gates of degree d or perceptrons of order d. It is not difficult to see that the following definition captures this generalization. Definition 1 A boolean function is called a threshold gate of degree d (or in short d threshold gate) ....

R. Beigel, N. Reingold, and D. Spielman. The perceptron strikes back. In Proceedings of the 6th structures, pages 286--291, 1991.

.... 1g n can be expressed as a polynomial threshold function of degree O(n 1=3 log s) Theorem 2 essentially closes the gap which was left open by the O( p n log n) upper bounds implicit in [11, 36] it shows that the Minsky Papert lower bound is in fact tight, up 1 Beigel et al. stated in [5] that Minsky and Papert gave an p n) lower bound for DNF but this was in error [4] to a logarithmic factor, for all polynomial size DNF. Theorem 3 also yields a 2 O(n 1=3 log 2 n) time algorithm for learning polynomial size DNF, which improves on the algorithms of Bshouty and ....

R. Beigel, N. Reingold and D. Spielman. The perceptron strikes back, in \Proc. 6th Conf. on Structure in Complexity Theory" (1991), 286-291.

....provide an oracle A such that P NP A 6 PP A . Some interesting results for perceptrons were also obtained by Beigel, Reingold and Spielman [3] as a consequence of the closure under intersection of PP. Further related results and extensions have been obtained by Beigel, Reingold and Spielman [4], Aspnes, Beigel, Furst and Rudich [2] and Tarui [11] An interesting open question left by the work of [6] is whether there is an oracle separating the hierarchy PP PH . More precisely, is there an oracle A such that the following is true: 8d) PP Sigma p;A d 1 6 PP Sigma p;A d ) ....

R. Beigel, N. Reingold, and D. Spielman, "The Perceptron Strikes Back", in Proceedings of the Sixth Annual Conference on Structure in Complexity Theory, IEEE Computer Society Press (1991) 286-291.

....current address: Computer Science Department, 226 Bell Hall, UB North Campus, Buffalo, NY 14260 2000. Email: regan cs.buffalo.edu, tel. 716) 645 3189, fax: 716) 645 3464. 1 results apprear to require constructions which work under all oracles that have the promise property (see [BT91, BRS91] rather than just the parity oracle. Because of all these applications, there is considerable interest in improving the complexity of the construction, in terms of the running time t(n) of M o , the number r(n) of random bits M uses, and the success probability c(n) that M o accepts when x 2 L. ....

R. Beigel, N. Reingold, and D. Spielman. The perceptron strikes back. In The Proceedings of the 6th Annual IEEE Conference on Structure in Complexity Theory, pages 286--291, 1991.

....on the order of perceptrons which compute arbitrary AC 0 functions. As a step in this direction, we prove Theorem 4 For d 3; any read once Boolean formula of depth d over f; g can be computed by a perceptron of order O(n 1 Gamma 1 3 Delta2 d Gamma3 ) 1 Beigel et al. stated in [5] that Minsky and Papert gave an Omega Gamma p n) lower bound for DNF but this was in error [4] 2 Theorem 4 implies that the class of read once AC 0 formulas can be learned in subexponential time. 2 Preliminaries 2.1 DNF, Decision Lists, Decision Trees, and Perceptrons A disjunctive ....

R. Beigel, N. Reingold and D. Spielman. The perceptron strikes back, in "Proc. 6th Conf. on Structure in Complexity Theory" (1991), 286-291.

....T m t for some m and t. A perceptron is a depth 2 circuit consisting of s of variables or their negations at the bottom, followed by a threshold gate. If the inputs to the gates are all unnegated variables then the perceptron is monotone. Perceptrons form an important class of neural networks [2, 20]. Depth 2 circuits consisting of s of variables or their negations, followed by a threshold gate can be called dual perceptrons. If the inputs to the gates are all unnegated variables then the dual perceptron is monotone. This terminology is justified by the following fact. The dual of a ....

R. Beigel, N. Reingold, and D. Spielman. The perceptron strikes back (preliminary report). In Proceedings of the Sixth Annual Structure in Complexity Theory Conference, pages 286--291, 1991.

....of such a model is the perceptron of Minsky and Papert [MP] which can be viewed as a MAJORITY gate whose inputs are ANDs of the input variables. These original perceptrons are rather limited and their computing power is well understood. But recently, perceptrons have been revived in a new form [BRS]. Along with a probabilistic version, there has emerged what we will call the d perceptron, a constant depth unbounded fan in circuit which has AND and OR gates except for a single MAJORITY gate at the output. It has been shown that such circuits require exponential size (exponentially many ....

....integers or the reals, and outputs the sign of the result. This can be extended to polynomials over the complex numbers, using an ad hoc notion of sign [BS] Furthermore, the d perceptron model is robust, in that other circuit models with a limited use of threshold gates can be mapped into it [BRS, ABFR, Be]. In the study of threshold computations, harmonic analysis has found many interesting applications (for some recent examples, see [Br, KKL, LMN] Of particular interest to us is the work by Linial et al. LMN] 2 where they showed that any AC 0 function (any function computable by a ....

[Article contains additional citation context not shown here]

R. Beigel, N. Reingold and D. Spielman. The perceptron strikes back. Proceedings of the 6th Annual Conference on Structrur in Complexity Theory (1991), 286-291.

.... Perceptron Like Models Zhi Li Zhang Computer Science Department University of Massachusetts Amherst, Massachusetts MA 01003 USA July, 1992 Abstract We examine the size complexity of the symmetric boolean functions in two circuit models containing threshold gates: the d perceptron model [BRS, ABFR] (a single threshold function of constant depth AND OR circuits) and the parity threshold model studied by Bruck [Br] a single threshold function of exclusive ORs) These models are intermediate between the well understood model of constant depth AND OR circuits and the still mysterious model ....

....of such a model is the perceptron of Minsky and Papert [MP] which can be viewed as a MAJORITY gate whose inputs are ANDs of the input variables. These original perceptrons are rather limited and their computing power is well understood. But recently, perceptrons have been revived in a new form [BRS]. Along with a probabilistic version, there has emerged what we will call the d perceptron, a constant depth unbounded fan in circuit which has AND and OR gates except for a single MAJORITY gate at the output. It has been shown that such circuits require exponential size (exponentially many ....

[Article contains additional citation context not shown here]

R. Beigel, N. Reingold and D. Spielman. The Perceptron Strikes Back. Proceedings of the 6th Annual Conference on Structrure in Complexity Theory pages 286-291, 1991.

.... trees with linear or low degree test functions [11] 26] 37] Another important tool in examining Boolean function complexity is representing the Boolean functions by polynomials above some field or ring, which facilitates using algebraic or analytical methods (see, e.g. 1] 3] 4] [6], 8] 7] 24] 27] 34] The previous two approaches are unified, i.e. communication complexity tools are applied to the polynomial representations of Boolean functions in [26] 10] 16] or in the full version of [12] In the present work, communication complexity tools will be applied ....

....functions are also logarithmic, relative to the maximum n. 1.4 Circuit Applications: Bounded Depth In the recent literature one can find very interesting lower bounds and techniques for bounded depth circuits with hard to handle gates (e.g. MOD m gates, MAJORITY gates, etc. See for example [6], 8] 34] 10] 12] 15] 4] or see [5] for a survey. Hajnal, Maass, Pudl ak, Szegedy and Tur an [17] proved an exponential lower bound for the size of depth 2 circuits with a MAJORITY gate at the top, and linear threshold gates of small weights on the bottom. Hastad and Goldmann [19] ....

R. Beigel, N. Reingold, and D. A. Spielman, The perceptron strikes back, in Proc. 6th Annual Conference on Structure in Complexity Theory, IEEE Comp.Soc. Press, 1991.

....(log 2 Sn log 2 n) D . First we show the existence of the polynomial H by supposing that we have shown the existence of the polynomial G satisfying the conditions mentioned above. Let s and d be the size and depth of the Boolean circuit corresponding to g. Let 0 # 1 = # 2. Beigel et al. [BRS91, Lemma 6] showed that for any Boolean circuit of depth d and size s there exists a polynomial G # (x) of degree O log(1 # 1 ) log s log 2 n d = O(z # ) that agrees with g # (x) on all except # 1 2 n inputs 2 . Define H to be H(x) G(x) G # (x) If g(x) 0 then g # (x) 0 and hence ....

R. Beigel, N. Reingold, D. Spielman, The perceptron strikes back. In Proceedings of the 6th Annual IEEE Structure in Complexity Theory Conference, 1991.

....not known if depth reduction theorems such as Corollaries 10, 18, and 19 can be generalized to composite moduli. Tarui (1991) has shown that AC 0 can be simulated by probabilistic depth two threshold circuits of size 2 log O(1) n with one sided error. Related results may also be found in Beigel et al. 1990). These developments in circuit complexity go hand in hand with significant progress being made in understanding the relationships that exist among various subclasses of PSPACE, such as the polynomial hierarchy, PP, and the counting hierarchy, starting with the seminal result of Toda (1991) These ....

Beigel, R., Reingold, N., and Spielman, D. (1990), The perceptron strikes back, in "Proceedings, 6th IEEE Structure in Complexity Theory Conference," pp. 286--291.

....described in this survey have direct applications to circuit complexity, particularly in showing how to use threshold gates to simulate certain AND OR circuits and also to use some number of threshold gates to simulate many of them. For example, Allender [All89] Beigel, Reingold, and Spielman [BRS91] and Tarui [Tar93] show how to use Theorem 3.14 to simulate boundeddepth AND OR circuits by small depth 2 circuits with a single threshold gate on top. 24 Lance Fortnow One can similarly draw an analogy between Mod k P computations and the class ACC of bounded depth circuits with Mod k gates ....

R. Beigel, N. Reingold, and D. Spielman. The perceptron strikes back. In Proceedings of the 6th IEEE Structure in Complexity Theory Conference, pages 286--291. IEEE, New York, 1991.

....not known if depth reduction theorems such as Corollaries 10, 18, and 19 can be generalized to composite moduli. Tarui (1991) has shown that AC 0 can be simulated by probabilistic depth two threshold circuits of size 2 log O(1) n with one sided error. Related results may also be found in Beigel et al. 1990). These developments in circuit complexity go hand in hand with significant progress being made in understanding the relationships that exist among various subclasses of PSPACE, such as the polynomial hierarchy, PP, and the counting hierarchy, starting with the seminal result of Toda (1991) These ....

Beigel, R., Reingold, N., and Spielman, D. (1990), The perceptron strikes back, in "Proceedings, 6th IEEE Structure in Complexity Theory Conference," pp. 286--291.

....of such a model is the perceptron of Minsky and Papert [MP] which can be viewed as a MAJORITY gate whose inputs are ANDs of the input variables. These original perceptrons are rather limited and their computing power is well understood. But recently, perceptrons have been revived in a new form [BRS]. Along with a probabilistic version, there has emerged what we will call the d perceptron, a constant depth unbounded fan in circuit which has AND and OR gates except for a single MAJORITY gate at the output. It has been shown that such circuits require exponential size (exponentially many ....

....integers or the reals, and outputs the sign of the result. This can be extended to polynomials over the complex numbers, using an ad hoc notion of sign [BS] Furthermore, the d perceptron model is robust, in that other circuit models with a limited use of threshold gates can be mapped into it [BRS, ABFR, Be]. In the study of threshold computations, harmonic analysis has found many interesting applications (for some recent examples, see [Br, KKL, LMN] Of particular interest to us is the work by Linial et al. LMN] where they showed that any AC 0 function (any function computable by a poly size ....

[Article contains additional citation context not shown here]

R. Beigel, N. Reingold and D. Spielman. The perceptron strikes back. Proceedings: Structure in Complexity Theory, 6th Annual Conference (1991), 286-291.

.... fixed k 0, each OE x produced by M is a depth k formula of such polynomial fan in conjuncts and disjuncts, with negations allowed, then L 2 AC 0 (C) Some results, the first and last easy, which also illustrate the notation are: a) P PP = P #P [Sim75, Val79] b) P PP[O(logn) PP [BRS91] c) P PP tt = PP [FR91] d) P MP tt = P # P[1] GKR 92] It is well known that the class P #P[1] is unchanged if one fixes the oracle f to be the #P complete function f SAT described in section 1, or if one fixes y x = x (but then the oracle function f depends on M ) The last result ....

.... be closed under union or intersection at all; this problem is described in detail at the end of [GKR 92] In all previous cases in the literature where a complexity class C has been shown to be closed under intersection or other Boolean operations, even the very difficult theorem for PP from [BRS91] cited above, the theorem provides a construction that only expands the lengths of the representations linearly. That is, let RA and RB be representations for two languages A; B in C; for instance, they can be the codes of machines that compute the P predicates and auxiliary functions for the ....

R. Beigel, N. Reingold, and D. Spielman. The perceptron strikes back. In Proc. 6th Annual IEEE Conference on Structure in Complexity Theory, pages 286--291, 1991.

....Saks [44] discovered that rational functions could be used for lower bounds on the size of threshold circuits. Toda [53] used polynomials to prove upper bounds on the power of the polynomial hierarchy. This led to a series of upper bounds on the power of the polynomial hierarchy [54, 52] AC [2, 3, 52, 18], and ACC [58, 20, 30, 37] and related classes [21, 42] Beigel and Gill [15] pushed the use of polynomials in order to prove closure properties of counting classes. This was extended by Hertrampf [34] see also [16] Beigel, Reingold, and Spielman [19] discovered that rational functions could ....

.... that p represents a function f with error ffl if for all inputs x 1 ; x n for at least a 1 Gamma ffl fraction of the random inputs r 1 ; r m p(x 1 ; x n ; r 1 ; r m ) f(x 1 ; xm ) Toda and Ogiwara [54] Tarui [52] and Beigel, Reingold, and Spielman [18] extended techniques of Valiant and Vazirani [56] to construct probablistic polynomials that represent the OR function. Lemma 12. Toda, Ogiwara Tarui Beigel, Reingold, Spielman) There exists a probabilistic polynomial of degree O with O random inputs that represents OR(x 1 ; ....

[Article contains additional citation context not shown here]

R. Beigel, N. Reingold, and D. Spielman. The perceptron strikes back. In Proc. 6th Structures, pp. 286--291, 1991.

....bounds in a new way. This allows us to prove the new result that an AC circuit with one majority gate cannot approximate parity. Our proof depends only on basic properties of integer polynomials. 1. Introduction Linial, Mansour, and Nisan [9] Tarui [16] and Beigel, Reingold, and Spielman [2] have shown that polynomial size, bounded depth circuits can be closely approximated as the sign of a low degree polynomial over the rationals. This result closely ties the class AC [7, 18] to the class of lowdegree polynomials over the rationals, much as previous work by Razborov [12] and ....

....PP with probability 1. 17 5. Relation to Bounded Depth Circuits In this section we describe a method for constructing from an AC circuit with a majority gate at its root a voting polynomial which closely approximates it. The construction is based on that of Beigel, Reingold, and Spielman [2]. The following key lemma is a simplification of their Lemma 5. Note that in this construction bits will be represented by the real values 0 and 1. Lemma 5.1 For any ffl 0 and any distribution of the inputs there exists a degree O(log(1=ffl) log(n) polynomial on x 1 : x n which computes ....

Richard Beigel, Nick Reingold, and Daniel Spielman. The perceptron strikes back. In Proceedings of the 6th Annual Conference on Structure in Complexity Theory, pages 286--291, 1991.

No context found.

R. Beigel, N. Reingold, and D. Spielman, The perceptron strikes back. In Proceedings of the 6th Ann. Conf. Structure in Complexity Theory, 1991b, 286--291.

....Saks [44] discovered that rational functions could be used for lower bounds on the size of threshold circuits. Toda [53] used polynomials to prove upper bounds on the power of the polynomial hierarchy. This led to a series of upper bounds on the power of the polynomial hierarchy [54, 52] AC 0 [2, 3, 52, 18], and ACC [58, 20, 30, 37] and related classes [21, 42] Beigel and Gill [15] pushed the use of polynomials in order to prove closure properties of counting classes. This was extended by Hertrampf [34] see also [16] Beigel, Reingold, and Spielman [19] discovered that rational functions could ....

.... say that p represents a function f with error ffl if for all inputs x 1 ; x n for at least a 1 Gamma ffl fraction of the random inputs r 1 ; r m p(x 1 ; x n ; r 1 ; r m ) f(x 1 ; xm ) Toda and Ogiwara [54] Tarui [52] and Beigel, Reingold, and Spielman [18] extended techniques of Valiant and Vazirani [56] to construct probablistic polynomials that represent the OR function. Lemma 12. Toda, Ogiwara Tarui Beigel, Reingold, Spielman) There exists a probabilistic polynomial of degree O i log(1=ffl) log 2 n j with O i log(1=ffl) log 2 n ....

[Article contains additional citation context not shown here]

R. Beigel, N. Reingold, and D. Spielman. The perceptron strikes back. In Proc. 6th Structures, pp. 286--291, 1991.

....find ways to reduce the use of one resource. Typically this entails a modest increase in some other resources. Recently, quasipolynomial size circuits have been the setting for unexpected resource tradeoffs (Allender 1989, Allender Hertrampf 1994, Yao 1990, Beigel Tarui 1994, Tarui 1993, and Beigel et al. 1991). In this paper we show how to reduce the number of majority gates in many kinds of quasipolynomial size circuits from polylog to 1. 2 Beigel For example, consider constant depth quasipolynomial size circuits that consist of AND , OR , NOT , and majority gates. We show how to reduce the number ....

....; x n and random Boolean inputs r 1 ; r m . A probabilistic perceptron computes a Boolean function f if, for all x 1 ; x n , for at least three fourths of all r 1 ; r m , the circuit s output is equal to f(x 1 ; x n ) Let G = fAND;OR;NOTg; Tarui (1993) and Beigel et al. 1991) proved that every Boolean function computed by a ThreshCG (depth d, size s, majorities 1) circuit where the majority gate is at the root is, in fact, computed by a probabilistic perceptron having order (log s) O(d) and weight 2 (log s) O(d) The following extensions of their result are ....

R. Beigel and J. Tarui, On ACC. This Journal. R. Beigel, N. Reingold, and D. Spielman, The perceptron strikes back. In Proc. 6th Ann. Conf. Structure in Complexity Theory, 1991, 286--291.

.... to PH is the class qAC 0 obtained by taking the size bound to be quasipolynomial, i.e. 2 log O(1) n ) Indeed, corresponding results in terms of shallow circuits and their improvements have been shown in a series of subsequent work by Allender (1989) Allender Hertrampf (1994) Beigel et al. 1991), Kannan et al. 1993) Tarui (1993) and Toda Ogiwara (1992) Many other results obtained by considering polynomial representations are explained by Beigel (1993) 1.3. Results. Yao (1985) obtained the first nontrivial upper bound on the computing power of ACC circuits. In this paper we ....

....computing power of ACC circuits. In this paper we simplify Yao s proof and improve his result (thus both contributions are of type (1) above) For a polynomial p(x 1 ; x n ) over Z, the ring of integers, define the norm of p to be the sum of the absolute values of the coefficients of p. (Beigel Tarui (1991) and respectively Yao (1985) use the word size to denote what we call norm and the logarithm of what we call norm. Define SYM to be the class of languages L for which there exist a family fr n (x 1 ; x n )g of degree log O(1) n On ACC 3 norm 2 log O(1) n polynomials over Z ....

[Article contains additional citation context not shown here]

R. Beigel, N. Reingold, and D. Spielman, The perceptron strikes back. In Proceedings of the 6th Ann. Conf. Structure in Complexity Theory, 1991, 286--291.

....bounds in a new way. This allows us to prove the new result that an AC 0 circuit with one majority gate cannot approximate parity. Our proof depends only on basic properties of integer polynomials. 1. Introduction Linial, Mansour, and Nisan [9] Tarui [16] and Beigel, Reingold, and Spielman [2] have shown that polynomial size, bounded depth circuits can be closely approximated as the sign of a low degree polynomial over the rationals. This result closely ties the class AC 0 [7, 18] to the class of lowdegree polynomials over the rationals, much as previous work by Razborov [12] and ....

....2 PP A with probability 1. 5. Relation to Bounded Depth Circuits In this section we describe a method for constructing from an AC 0 circuit with a majority gate at its root a voting polynomial which closely approximates it. The construction is based on that of Beigel, Reingold, and Spielman [2]. The following key lemma is a simplification of their Lemma 5. Note that in this construction bits will be represented by the real values 0 and 1. Lemma 5.1 For any ffl 0 and any distribution of the inputs there exists a degree O(log(1=ffl) log(n) polynomial on x 1 : x n which computes ....

Richard Beigel, Nick Reingold, and Daniel Spielman. The perceptron strikes back. In Proceedings of the 6th Annual Conference on Structure in Complexity Theory, pages 286--291, 1991.

No context found.

R. Beigel, N. Reingold, and D. A. Spielman, The perceptron strikes back, in Proc. 6th Annual Conference on Structure in Complexity Theory, IEEE Comp.Soc. Press, 1991.

No context found.

R. Beigel, N. Reingold, D. Spielman, The perceptron strikes back. In Proceedings of the 6th Annual IEEE Structure in Complexity Theory Conference, 1991.

No context found.

R. Beigel, N. Reingold, D. Spielman, The perceptron strikes back. In Proceedings of the 6th Annual IEEE Structure in Complexity Theory Conference, 1991.

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