We study deep neural networks with polynomial activations, particularly their expressive power. For a fixed architecture and activation degree, a polynomial neural network defines an algebraic map from weights to polynomials. The image of this map is the functional space associated to the network, and it is an irreducible algebraic variety upon taking closure. This paper proposes the dimension of this variety as a precise measure of the expressive power of polynomial neural networks. We obtain several theoretical results regarding this dimension as a function of architecture, including an exact formula for high activation degrees, as well as upper and lower bounds on layer widths in order for deep polynomials networks to fill the ambient functional space. We also present computational evidence that it is profitable in terms of expressiveness for layer widths to increase monotonically and then decrease monotonically. Finally, we link our study to favorable optimization properties when training weights, and we draw intriguing connections with tensor and polynomial decompositions.
The polynomial networks studied in this paper are not arithmetic circuits, but feedforward deep networks with polynomial @math -th power activations. This is a vast generalization of a setting considered in several recent papers @cite_7 @cite_23 @cite_12 , that study shallow (two layer) networks with quadratic activations ( @math ). These papers show that if the width of the intermediate layer is at least twice the input dimension, then the quadratic loss has no bad'' local minima. This result in line with our Proposition , which explains in this case the functional space is convex and the ambient space. We also point out that polynomial activations are required for the functional space of the network to span a finite dimensional vector space @cite_26 @cite_7 .
@cite_26 @cite_23 @cite_12 @cite_7
2016576580 2295840581 2115881141 2080210666
In their paper on the ''chasm at depth four'', Agrawal and Vinay have shown that polynomials in m variables of degree O(m) which admit arithmetic circuits of size 2^o^(^m^) also admit arithmetic circuits of depth four and size 2^o^(^m^). This theorem shows that for problems such as arithmetic circuit lower bounds or black-box derandomization of identity testing, the case of depth four circuits is in a certain sense the general case. In this paper we show that smaller depth four circuits can be obtained if we start from polynomial size arithmetic circuits. For instance, we show that if the permanent of nxn matrices has circuits of size polynomial in n, then it also has depth 4 circuits of size n^O^(^n^l^o^g^n^). If the original circuit uses only integer constants of polynomial size, then the same is true for the resulting depth four circuit. These results have potential applications to lower bounds and deterministic identity testing, in particular for sums of products of sparse univariate polynomials. We also use our techniques to reprove two results on: -the existence of nontrivial boolean circuits of constant depth for languages in LOGCFL; -reduction to polylogarithmic depth for arithmetic circuits of polynomial size and polynomially bounded degree. We show an exponential separation between two well-studied models of algebraic computation, namely read-once oblivious algebraic branching programs (ROABPs) and multilinear depth three circuits. In particular we show the following: 1. There exists an explicit n-variate polynomial computable by linear sized multilinear depth three circuits (with only two product gates) such that every ROABP computing it requires 2^ Omega(n) size. 2. Any multilinear depth three circuit computing IMM_ n,d (the iterated matrix multiplication polynomial formed by multiplying d, n * n symbolic matrices) has n^ Omega(d) size. IMM_ n,d can be easily computed by a poly(n,d) sized ROABP. 3. Further, the proof of 2 yields an exponential separation between multilinear depth four and multilinear depth three circuits: There is an explicit n-variate, degree d polynomial computable by a poly(n,d) sized multilinear depth four circuit such that any multilinear depth three circuit computing it has size n^ Omega(d) . This improves upon the quasi-polynomial separation result by Raz and Yehudayoff [2009] between these two models. The hard polynomial in 1 is constructed using a novel application of expander graphs in conjunction with the evaluation dimension measure used previously in Nisan [1991], Raz [2006,2009], Raz and Yehudayoff [2009], and Forbes and Shpilka [2013], while 2 is proved via a new adaptation of the dimension of the partial derivatives measure used by Nisan and Wigderson [1997]. Our lower bounds hold over any field. We compute upper and lower bounds on the VC dimension and pseudodimension of feedforward neural networks composed of piecewise polynomial activation functions. We show that if the number of layers is fixed, then the VC dimension and pseudo-dimension grow as W log W, where W is the number of parameters in the network. This result stands in opposition to the case where the number of layers is unbounded, in which case the VC dimension and pseudo-dimension grow as W2 . We combine our results with recently established approximation error rates and determine error bounds for the problem of regression estimation by piecewise polynomial networks with unbounded weights. Abstract We examine a powerful model of parallel computation: polynomial size threshold circuits of bounded depth (the gates compute threshold functions with polynomial weights). Lower bounds are given to separate polynomial size threshold circuits of depth 2 from polynomial size threshold circuits of depth 3 and from probabilistic polynomial size circuits of depth 2. With regard to the unreliability of bounded depth circuits, it is shown that the class of functions computed reliably with bounded depth circuits of unreliable ∨, ∧, ¬ gates is narrow. On the other hand, functions computable by bounded depth, polynomial-size threshold circuits can also be computed by such circuits of unreliable threshold gates. Furthermore we examine to what extent imprecise threshold gates (which behave unpredictably near the threshold value) can compute nontrivial functions in bounded depth and a bound is given for the permissible amount of imprecision. We also discuss threshold quantifiers and prove an undefinability result for graph connectivity.