Abstract: The paper characterizes classes of functions for which deep learning can be exponentially better than shallow learning. Deep convolutional networks are a special case of these conditions, though weight sharing is not the main reason for their exponential advantage. This work was supported by the Center for Brains, Minds and Machines (CBMM), funded by NSF STC award CCF -1231216. H.M. is supported in part by ARO Grant W911NF-15-1-0385. Abstract: The paper characterizes classes of functions for which deep learning can be exponentially better than shallow learning. Deep convolutional networks are a special case of these conditions, though weight sharing is not the main reason for their exponential advantage. Keywords: Deep and Shallow Networks, Convolutional Neural Networks, Function Approximation, Deep Learning 1 A theory of deep learning There are at three main sets of theory questions about Deep Neural Networks. The first set of questions is about the power of the architecture -which classes of functions can it approximate and learn well? The second set of questions is about the learning process: why is SGD (Stochastic Gradient Descent) so unreasonably efficient, at least in appearance? The third, more important question is about generalization. Overparametrization may explain why minima are easy to find during training but then why does overfitting seems to be less of a problem than for classical shallow networks? Is this because deep networks are very efficient algorithms for Hierarchical Vector Quantization? In this paper we focus especially on the first set of questions, summarizing several theorems that have appeared online in 2015 [1, 2, 3] , and in 2016 [4, 5] ). We then describe additional results as well as a few conjectures and open questions. The main message is that deep networks have the theoretical guarantee, which shallow networks do not have, that they can avoid the curse of dimensionality for an important class of problems, corresponding to compositional functions, that is functions of functions. An especially interesting subset of such compositional functions are hierarchically local compositional functions where all the constituent functions are local in the sense of bounded small dimensionality. The deep networks that can approximate them without the curse of dimensionality are of the deep convolutional type (though weight sharing is not necessary). Implications of the theorems likely to be relevant in practice are: 1. Certain deep convolutional architectures have a theoretical guarantee that they can be much better than one layer architectures such as kernel machines; 2. the problems for which certain deep networks are guaranteed to avoid the curse of dimensionality (see for a nice review [6] ) correspond to input-output mappings that are compositional. The most interesting set of problems consists of compositional functions composed of a hierarchy of constituent functions that are local: an example is f (x1, · · · , x8) = h3(h21(h11(x1, x2), h12(x3, x4)), h22(h13(x5, x6), h14(x7, x8))). The compositional function f requires only "local" computations (here with just dimension 2) in each of its constituent functions h; 3. the key aspect of convolutional networks that can give them an exponential advantage is not weight sharing but locality at each level of the hierarchy. Deep Learning references start with Hinton's backpropagation and with Lecun's convolutional networks (see for a nice review [7] ). Of course, multilayer convolutional networks have been around at least as far back as the optical processing era of the 70s. The Neocognitron [8] was a convolutional neural network that was trained to recognize characters. The property of compositionality was a main motivation for hierarchical models of visual cortex such as HMAX which can be regarded as a pyramid of AND and OR layers [9] , that is a sequence of conjunctions and disjunctions. Several papers in the '80s focused on the approximation power and learning properties of one-hidden layer networks (called shallow networks here). Very little appeared on multilayer networks, (but see [10, 11, 12] ), mainly because one hidden layer nets performed empirically as well as deeper networks. On the theory side, a review by Pinkus in 1999 [13] concludes that "...there seems to be reason to conjecture that the two hidden layer model may be significantly more promising than the single hidden layer model...". A version of the questions about the importance of hierarchies was asked in [14] as follows: "A comparison with real brains offers another, and probably related, challenge to learning theory. The "learning algorithms" we have described in this paper correspond to one-layer architectures. Are hierarchical architectures with more layers justifiable in terms of learning theory? It seems that the learning theory of the type we have outlined does not offer any general argument in favor of hierarchical learning machines for regression or classification. This is somewhat of a puzzle since the organization of cortex -for instance visual cortex -is strongly hierarchical. At the same time, hierarchical learning systems show superior performance in several engineering applications." Because of the great empirical success of deep learning over the last three years, several papers addressing the question of why hierarchies have appeared. Sum-Product networks, which are equivalent to polynomial networks (see [15, 16] ), are a simple case of a hierarchy that was analyzed [17] but did not provide particularly useful insights. Montufar and Bengio [18] showed that the number of linear regions that can be synthesized by a deep network with ReLU nonlinearities is much larger than by a shallow network. The meaning of this result in terms of approximation theory and of our results is at the moment an open question 1 . Relevant to the present review is the work on hierarchical quadratic networks [16] , together with function approximation results [19, 13] . Also relevant is the conjecture by Shashua (see [20] ) on a connection between deep learning networks and the hierarchical Tucker representations of tensors. In fact, our theorems describe formally the class of functions for which the conjecture holds. This paper describes and extends results pre- 1 We conjecture that the result may be similar to other examples in section 4.2. It says that among the class of functions that are piecewise linear, there exist functions that can be synthesized by deep networks with a certain number of units but require a much large number of units to be synthesized by shallow networks sented in [21, 22, 23] and in [24, 4] which derive new upper bounds for the approximation by deep networks of certain important classes of functions which avoid the curse of dimensionality. The upper bound for the approximation by shallow networks of general functions was well known to be exponential. It seems natural to assume that, since there is no general way for shallow networks to exploit a compositional prior, lower bounds for the approximation by shallow networks of compositional functions should also be exponential. In fact, examples of specific functions that cannot be represented efficiently by shallow networks have been given very recently by Telgarsky [25] and by Shamir [26] . We provide in theorem 5 another example of a class of compositional functions for which there is a gap between shallow and deep networks. In this section, we state theorems about the approximation properties of shallow and deep networks. The general paradigm is as follows. We are interested in determining how complex a network ought to be to theoretically guarantee approximation of an unknown target function f up to a given accuracy > 0. To measure the accuracy, we need a norm · on some normed linear space X. As we will see the norm used in the results of this paper is the sup norm in keeping with the standard choice in approximation theory. Notice, however, that from the point of view of machine learning, the relevant norm is the L2 norm. In this sense, several of our results are stronger than needed. On the other hand, our main results on compositionality require the sup norm in order to be independent from the unknown distribution of the inputa data. This is important for machine learning. Let VN be the be set of all networks of a given kind with complexity N which we take here to be the total number of units in the network (e.g., all shallow networks with N units in the hidden layer). It is assumed that the class of networks with a higher complexity include those with a lower complexity; i.e., VN ⊆ VN+1. The degree of approximation is defined by dist(f, VN ) = inf (1) γ ) will be sufficient to guarantee an approximation with accuracy at least . Since f is unknown, in order to obtain theoretically proved upper bounds, we need to make some assumptions on the class of functions from which the unknown target function is chosen. This a priori information is codified by the statement that f ∈ W for some subspace W ⊆ X. This subspace is usually a smoothness class characterized by a smoothness parameter m. Here it will be generalized to a smoothness and compositional class, characterized by the parameters m and d (d = 2 in the example of Figure 1 ; in general is the size of the kernel in a convolutional network). This section characterizes conditions under which deep networks are "better" than shallow network in approximating functions. Thus we compare shallow (one-hidden layer) networks with deep networks as shown in Figure 1 . Both types of networks use the same small set of operations -dot products, linear combinations, a fixed nonlinear function of one variable, possibly convolution and pooling. Each node in the networks we consider usually corresponds to a node in the graph of the function to be approximated, as shown in the Figure. In particular each node in the network contains a certain number of units. A unit is a neuron which computes where w is the vector of weights on the vector input x. Both t and the real number b are parameters tuned by learning. We assume here that each node in the networks computes the linear combination of r such units Notice that for our main example of a deep network corresponding to a binary tree graph, the resulting architecture is an idealized version of the plethora of deep convolutional neural networks described in the literature. In particular, it has only one output at the top unlike most of the deep architectures with many channels and many top-level outputs. Correspondingly, each node computes a single value instead of multiple channels, using the combination of several units (see Equation 3 ). Our approach and basic results apply rather directly to more complex networks (see third note in section 6). A careful analysis and comparison with simulations will be described in future work. The logic of our theorems is as follows. • Both shallow (a) and deep (b) networks are universal, that is they can approximate arbitrarily well any continuous function of n variables on a compact domain. The result for shallow networks is classical. Since shallow networks can be viewed as a special case of deep networks, it clear that for any continuous function of n variables, there exists also a deep network that approximates the function arbitrarily well on a compact domain. • We consider a special class of functions of n variables on a compact domain that are a hierarchical compositions of local functions such as The structure of the function in equation 4 is represented by a graph of the binary tree type. This is the simplest example of compositional functions, reflecting dimensionality d = 2 for the constituent functions h. In general, d is arbitrary but fixed and independent of the dimensionality n of the compositional function f . In our results we will often think of n increasing while d is fixed. In section 4 we will consider the more general compositional case. • The approximation of functions with a compositional structure -can be achieved with the same degree of accuracy by deep and shallow networks but that the number of parameters are much smaller for the deep networks than for the shallow network with equivalent approximation accuracy. It is intuitive that a hierarchical network matching the structure of a compositional function should be "better" at approximating it than a generic shallow network but universality of shallow networks asks for non-obvious characterization of "better". Our result makes clear that the intuition is indeed correct. Figure 1 The top graphs are associated to functions; each of the bottom diagrams depicts the ideal network approximating the function above. In a) a shallow universal network in 8 variables and N units approximates a generic function of 8 variables f (x1, · · · , x8). Inset b) shows a binary tree hierarchical network at the bottom in n = 8 variables, which approximates well functions of the form f (x1, · · · , x8) = h3(h21(h11(x1, x2), h12(x3, x4)), h22(h13(x5, x6), h14(x7, x8))) as represented by the binary graph above. In the approximating network each of the n − 1 nodes in the graph of the function corresponds to a set of Q = N n−1 ReLU units computing the ridge function Each term in the ridge function corresponds to a unit in the node (this is somewhat different from todays deep networks, but equivalent to them, see text and note in 6). In a binary tree with n inputs, there are log2n levels and a total of n − 1 nodes. Similar to the shallow network, a hierarchical network is universal, that is, it can approximate any continuous function; the text proves that it can approximate a compositional functions exponentially better than a shallow network. No invariance -that is weight sharing -is assumed here. Notice that the key property that makes convolutional deep nets exponentially better than shallow for compositional functions is the locality of the constituent functions -that is their low dimensionality. Weight sharing corresponds to all constituent functions at one level to be the same (h11 = h12 etc.). Inset c) shows a different mechanism that can be exploited by the deep network at the bottom to reduce the curse of dimensionality in the compositional function at the top: leveraging different degrees of smoothness of the constituent functions, see Theorem 6 in the text. Notice that in c) the input dimensionality must be ≥ 2 in order for deep nets to have an advantage over shallow nets. The simplest examples of functions to be considered for a), b) and c) are polynomials with a structure corresponding to the graph at the top. In the perspetive of machine learning, we assume that the shallow networks do not have any structural information on the function to be learned (here its compositional structure), because they cannot represent it directly and cannot exploit the advantage of a smaller number of parameters. In any case, in the context of approximation theory, we will exhibit and cite lower bounds of approximation by shallow networks for the class of compositional functions. Deep networks with standard architectures on the other hand do represent compositionality in their architecture and can be adapted to the details of such prior information. We approximate functions of n variables of the form of Equation (4) with networks in which the activation nonlinearity is a smoothed version of the so called ReLU, originally called ramp by Breiman and given by σ(x) = x+ = max(0, x) . The architecture of the deep networks reflects Equation (4) with each node hi being a ridge function, comprising one or more neurons. be the space of all continuous functions on I n , with f = maxx∈In |f (x)|. Let SN,n denote the class of all shallow networks with N units of the form The number of trainable parameters here is (n + 2)N ∼ n. Let m ≥ 1 be an integer, and W n m be the set of all functions of n variables with continuous partial derivatives of orders up to m < ∞ such that f where D k denotes the partial derivative indicated by the multiinteger k ≥ 1, and |k|1 is the sum of the components of k. For the hierarchical binary tree network, the analogous spaces are defined by considering the compact set W n,2 m to be the class of all compositional functions f of n variables with a binary tree architecture and constituent functions h in W 2 m . We define the corresponding class of deep networks DN,2 to be the set of all deep networks with a binary tree architecture, where each of the constituent nodes is in SM,2, where N = |V |M , V being the set of non-leaf vertices of the tree. We note that in the case when n is an integer power of 2, the total number of parameters involved in a deep network in DN,2 -that is, weights and biases, is 4N . Two observations are critical to understand the meaning of our results: • compositional functions of n variables are a subset of func- m . Deep networks can exploit in their architecture the special structure of compositional functions, whereas shallow networks are blind to it. Thus from the point of view of shallow networks, functions in W n,2 m are just functions in W n m ; this is not the case for deep networks. • the deep network does not need to have exactly the same compositional architecture as the compositional function to be approximated. It is sufficient that the acyclic graph representing the structure of the function is a subgraph of the graph representing the structure of the deep network. The degree of approximation estimates depend on the graph associated with the network and are thus an upper bound on what could be achieved by a network exactly matched to the function architecture. The following two theorems estimate the degree of approximation for shallow and deep networks. The first theorem is about shallow networks. Theorem 1. Let σ : R → R be infinitely differentiable, and not a polynomial. For f ∈ W n m the complexity of shallow networks that provide accuracy at least is −n/m ) and is the best possible. Notes In [27, Theorem 2.1] , the theorem is stated under the condition that σ is infinitely differentiable, and there exists b ∈ R such that σ (k) (b) = 0 for any integer k ≥ 0. It is proved in [28] that the second condition is equivalent to σ not being a polynomial. The proof in [27] relies on the fact that under these conditions on σ, the algebraic polynomials in n variables of (total or coordinatewise) degree < q are in the uniform closure of the span of O(q n ) functions of the form x → σ( w, x + b) (see Appendix 4.1). The estimate itself is an upper bound on the degree of approximation by such polynomials. Since it is based on the approximation of the polynomial space contained in the ridge functions implemented by shallow networks, one may ask whether it could be improved by using a different approach. The answer relies on the concept of nonlinear n-width of the compact set W n m (cf. [29, 4] ). The n-width results imply that the estimate in Theorem (1) is the best possible among all reasonable [29] methods of approximating arbitrary functions in W n m . The estimate of Theorem 1 is the best possible if the only a priori information we are allowed to assume is that the target function belongs to f ∈ W n m . The exponential dependence on the dimension n of the number −n/m of parameters needed to obtain an accuracy O( ) is known as the curse of dimensionality. Note that the constants involved in O in the theorems will depend upon the norms of the derivatives of f as well as σ. A simple but useful corollary follows from the proof of Theorem 1 about polynomials (which are a smaller space than spaces of Sobolev functions). Let us denote with P n k the linear space of polynomials of degree at most k in n variables. Then Corollary 1. Let σ : R → R be infinitely differentiable, and not a polynomial. Every f ∈ P n k can be realized with an arbitrary accuracy by shallow network with r units, r = n+k k ≈ k n . Our second and main theorem is about deep networks with smooth activations and is recent (preliminary versions appeared in [3, 2, 4] ). We formulate it in the binary tree case for simplicity but it extends immediately to functions that are compositions of constituent functions of a fixed number of variables d instead than of d = 2 variables as in the statement of the theorem (in convolutional networks d corresponds to the size of the kernel). m consider a deep network with the same compositonal architecture and with an activation function σ : R → R which is infinitely differentiable, and not a polynomial. The complexity of the network to provide approximation with accuracy at least is Proof To prove Theorem 2, we observe that each of the constituent functions being in W 2 m , (1) applied with n = 2 implies that each of these functions can be approximated from SN,2 up to accuracy = cN −m/2 . Our assumption that f ∈ W N,2 m implies that each of these constituent functions is Lipschitz continuous. Hence, it is easy to deduce that, for example, if P , P1, P2 are approximations to the constituent functions h, h1, h2, respectively within an accuracy of , then since for some constant c > 0 independent of the functions involved. This, together with the fact that there are (n − 1) nodes, leads to (6). Also in this case the proof provides the following corollary about the subset T n k of the space P n k which consists of compositional polynomials with a binary tree graph and constituent polynomial functions of degree k (in 2 variables) Corollary 2. Let σ : R → R be infinitely differentiable, and not a polynomial. Let n = 2 l . Then f ∈ T n k can be realized by a deep network with a binary tree graph and a total of r units with r = (n − 1) It is important to emphasize that the assumptions on σ in the theorems are not satisfied by the ReLU function x → x+, but they are satisfied by smoothing the function in an arbitrarily small interval around the origin. This suggests that the result of the theorem should be valid also for the non-smooth ReLU. Section 4.1 provides formal results. Stronger results than the theorems of this section (see [5] ) hold for networks where each unit evaluates a Gaussian non-linearity; i.e., Gaussian networks of the form where the approximation is on the entire Euclidean space. In summary, when the only a priori assumption on the target function is about the number of derivatives, then to guarantee an accuracy of , we need a shallow network with O( −n/m ) trainable parameters. If we assume a hierarchical structure on the target function as in Theorem 2, then the corresponding deep network yields a guaranteed accuracy of with O( −2/m ) trainable parameters. Note that Theorem 2 applies to all f with a compositional architecture given by a graph which correspond to, or is a subgraph of, the graph associated with the deep network -in this case the graph corresponding to W n,d m . Theorem 2 leads naturally to the notion of effective dimensionality that we formalize in the next section Definition 1. The effective dimension of a class W of functions (for a given norm) is said to be d if for every > 0, any function in W can be recovered within an accuracy of (as measured by the norm) using an appropriate network (either shallow or deep) with −d parameters. The main class of functions we considered in previous papers consists of functions as in Figure 1 b that we called compositional functions. The term "compositionality" was used with the meaning it has in language and vision, where higher level concepts are composed of a small number of lower level ones, objects are composed of parts, sentences are composed of words and words are composed of syllables. Notice that this meaning of compositionality is narrower than the mathematical meaning of composition of functions. The compositional functions we have described in previous papers may be more precisely called functions composed of hierarchically local functions. Here we generalize formally our previous results to the broader class of compositional functions (beyond the hierarchical locality of Figure 1b to Figure 1c and Figure 2 ) by restating formally a few comments of previous papers. Let us begin with one of the previous examples. Consider +F y 2 + 2Gx + 2Hy + I) Since Q is nominally a polynomial of coordinatewise degree 2 11 , [27, Lemma 3.2] shows that a shallow network with 2 11 + 1 units is able to approximate Q arbitrarily well on I 2 . However, because of the hierarchical structure of Q, [27, Lemma 3.2] shows also that a hierarchical network with 9 units can approximate the quadratic expression, and 10 further layers, each with 3 units can approximate the successive powers. Thus, a hierarchical network with 11 layers and 39 units can approximate Q arbitrarily well. We note that even if Q is nominally of degree 2 11 , each of the monomial coefficients in Q is a function of only 9 variables, A, · · · , I. A different example is This is obviously a Lipschitz continuous function of 2 variables. The effective dimension of this class is 2, and hence, a shallow network would require at least c −2 parameters to approximate it within . However, the effective dimension of the class of univariate Lipschitz continuous functions is 1. Hence, if we take into account the fact that Q is a composition of a polynomial of degree 2 in 2 variables and the univariate Lipschitz continuous function t → |t|, then it is easy to see that the same approximation can be achieved by using a two layered network with O( −1 ) parameters. To formulate our most general result that includes the examples above as well as the constraint of hierarchical locality, we first define formally a compositional function in terms of a directed acyclic graph. Let G be a directed acyclic graph (DAG), with the set of nodes V . A G-function is defined as follows. Each of the source node obtains an input from R. Each in-edge of every other node represents an input real variable, and the node itself represents a function of these input real variables, called a constituent function. The out-edges fan out the result of this evaluation. We assume that there is only one sink node, whose output is the G-function. Thus, ignoring the compositionality of this function, it is a function of n variables, where n is the number of source nodes in G. Theorem 3. Let G be a DAG,n be the number of source nodes, and for each v ∈ V , let dv be the number of in-edges of v. Let f : R n → R be a compositional G-function, where each of the constitutent function is in W dv mv . Consider shallow and deep networks with infinitely smooth activation function as in Theorem 1. Then deep networks -with an associated graph that corresponds to the graph of f -avoid the curse of dimensionality in approximating f for increasing n, whereas shallow networks cannot directly avoid the curse. In particular, the complexity of the best approximating shallow network is exponential in n Figure 2 The figure shows the graphs of functions that may have small effective dimensionality, depending on the number of units per node required for good approximation. where m = minv∈V mv, while the complexity of the deep network is Following definition 1 we call dv/mv the effective dimension of function v. Then, deep networks can avoid the curse of dimensionality if the constituent functions of a compositional function have a small effective dimension; i.e., have fixed, "small" dimensionality or fixed, "small" "roughness. A different interpretation of Theorem 3 is the following. Proposition 1. If a family of functions f : R n → R of smoothness m has an effective dimension < n/m, then the functions are compositional in a manner consistent with the estimates in Theorem 3. Notice that the functions included in this theorem are functions that are either local or the composition of simpler functions or both. Figure 2 shows some examples in addition to the examples at the top of Figure Figure 1 . As before, there is a simple corollary for polynomial functions: Corollary 3. Let σ : R → R be infinitely differentiable, and not a polynomial. With the set up as in Theorem 3, let f be DAG polynomial; i.e., a DAG function, each of whose constituent functions is a polynomial of degree k. Then f can be represented by a deep network with O(|VN |k d ) units, where |VN | is the number of non-leaf vertices, and d is the maximal indegree of the nodes. For example, if G is a full binary tree with 2 n leaves, then the nominal degree of the G polynomial as in Corollary 3 is k Notice that polynomials in S n k are sparse with a number of terms which is not exponential in n, that is it is not O(k n ) but linear in n (that is O(nk)) or at most polynomial in n. The results we described so far use smooth activation functions. We already mentioned why relaxing the smoothness assumption should not change our results in a fundamental way. While studies on the properties of neural networks with smooth activation abound, the results on non-smooth activation functions are much more sparse. Here we briefly recall some of them. In the case of shallow networks, the condition of a smooth activation function can be relaxed to prove density (see [13] , Proposition 3.7): Proposition 2. Let σ =: R → R be in C 0 , and not a polynomial. Then shallow networks are dense in C 0 . In particular, ridge functions using ReLUs of the form r i=1 ci( wi, x + bi)+, with wi, x ∈ R n , ci, bi ∈ R are dense in C. Networks with non-smooth activation functions are expected to do relatively poorly in approximating smooth functions such as polynomials in the sup norm. "Good" degree of approximation rates (modulo a constant) have been proved in the L2 norm. Define B the unit ball in R n . Call C m (B n ) the set of all continuous functions with continuous derivative up to degree m defined on the unit ball. We define the Sobolev space W with respect to the Sobolev norm p (see for details [13] page 168). We define the space B  It is shown in [13, Corollary 6.10] that These approximation results with respect to the L 2 norm cannot be applied to derive bounds for compositional networks. Indeed, in the latter case, as we remarked already, estimates in the uniform norm are needed to control the propagation of the errors from one layer to the next, see Theorem 2. Results in this direction are given in [30] , and more recently in [31] and [5] (see Theorem 3.1). In particular, using a result in [31] and following the proof strategy of Theorem 2 it is possible to derive the following results on the approximation of Lipshitz continuous functions with deep and shallow ReLU networks that mimics our Theorem 2: Theorem 4. Let f be a L-Lipshitz continuous function of n variables. Then, the complexity of a network which is a linear combination of ReLU providing an approximation with accuracy at least is wheres that of a deep compositional architecture is Our general Theorem 3 can be extended in a similar way. Theorem 4 is an example of how the analysis of smooth activation functions can be adapted to ReLU. Indeed, it shows how deep compositional networks with standard ReLUs can avoid the curse of dimensionality. In the above results, the regularity of the function class is quantified by the magnitude of Lipshitz constant. Whether the latter is the best notion of smoothness for ReLU based networks, and if the above estimates can be improved, are interesting questions that we defer to a future work. A result that is more intuitive and may reflect what networks actually do is described in Appendix 4.3. Though the construction described there provides approximation in the L2 norm but not in the sup norm, this is not a problem under any discretization of real number required for computer simulations (see Appendix). Figures 3, 4, 5, 6 provide a sanity check and empirical support for our main results and for the claims in the introduction. There are several units per node of the tree. In our setup here the network with an associated binary tree graph was set up so that each layer had the same number of units and shared parameters. The number of units for the shallow and binary tree neural networks had the same number of parameters. On the left the function is composed of a single ReLU per node and is approximated by a network using ReLU activations. On the right the compositional function is f (x1, x2, x3, x4) = h2(h11(x1, x2), h12(x3, x4)) and is approximated by a network with a smooth ReLU activation (also called softplus). The functions h1, h2, h3 are as described in Figure 3 . In order to be close to the function approximation case, a large data set of 60K training examples was used for both training sets. We used for SGD the Adam [32] optimizer. In order to get the best possible solution we ran 200 independent hyper parameter searches using random search [33] and reported the one with the lowest training error. The hyper parameters search was over the step size, the decay rate, frequency of decay and the mini-batch size. The exponential decay hyper parameters for Adam were fixed to the recommended values according to the original paper [32] . The implementations were based on TensorFlow [34] . The figure shows on the top the graph of the function to be approximated, while the bottom part of the figure shows a deep neural network with the same graph structure. The left and right node inf the first layer has each n units giving a total of 2n units in the first layer. The second layer has a total of 2n units. The first layer has a convolution of size n to mirror the structure of the function to be learned. The compositional function we approximate has the form f (x1, x2, x3, x4) = h2(h11(x1, x2), h12(x3, x4)) with h11, h12 and h2 as indicated in the figure. So far we have shown that there are deep networks -for instance of the convolutional type -that can avoid the curse of dimensionality if the functions they are learning are blessed with compositionality. There are no similar guarantee for shallow networks: for shallow networks approximating generic continuous functions the lower and the upper bound are both exponential [13] . From the point of view of machine learning, it is obvious that shallow networks, unlike deep ones, cannot exploit in their architecture the reduced number of parameters associated with priors corresponding to compositional functions. In past papers we listed a few examples, some of which are also valid lower bounds from the point of view of approximation theory: • The polynomial considered earlier can be approximated by deep networks with a smaller number of parameters than shallow networks is based on polynomial approximation of functions of the type g(g(g())). Here, however, a formal proof of the impossibility of good approximation by shallow networks is not available. For a lower bound we need at least one example of a compositional function which cannot be approximated by shallow networks with a non-exponential degree of approximation. • Such an example, for which a proof of the lower bound exists since a few decades, consider a function which is a linear combination of n tensor product Chui-Wang spline wavelets, where each wavelet is a tensor product cubic spline. It is shown in [11, 12] that is impossible to implement such a function using a shallow neural network with a sigmoidal activation function using O(n) neurons, but a deep network with the activation function (x+) 2 can do so. In this case, as we mentioned, there is a formal proof of a gap between deep and shallow networks. Similarly, Eldan and Shamir [35] show other cases with separations that are exponential in the input dimension. • As we mentioned earlier, Telgarsky proves an exponential gap between certain functions produced by deep networks and their approximation by shallow networks. The theorem [25] can be summarized as saying that a certain family of classification problems with real-valued inputs cannot be approximated well by shallow networks with fewer than exponentially many nodes whereas a deep network achieves zero error. This corresponds to high-frequency, sparse trigonometric polynomials in our case. His upper bound can be proved directly from our main theorem by considering the real-valued polynomials x1x2...x d defined on the cube (−1, 1) d which is obviously a compositional function with a binary tree graph. • We exhibit here another example to illustrate a limitation of shallow networks in approximating a compositional function. Let n ≥ 2 be an integer, B ⊂ R n be the unit ball of R n . We consider the class W of all compositional functions f = f2 • f1, where f1 : R n → R, and |k|≤4 where AN is either the class SN of all shallow networks with N units or DN of deep networks with two layers, the first with n inputs, and the next with one input. The both cases, the activation function is a C ∞ function σ : R → R that is not a polynomial. In contrast, there exists c3 > 0 such that The constants c1, c2, c3 may depend upon n. PROOF. The estimate (12) follows from the estimates already given for deep networks. To prove (11), we use Lemma 3.2 in [12] . Let φ be a C ∞ function supported on [0, 1], and we consider fN (x) = φ(|4 N x| 2 ). We may clearly choose φ so that fN ∞ = 1. Then it is clear that each fN ∈ W . Clearly, We choose (14) Since fN is supported on {x ∈ R n : |x| ≤ 4 −N }, we may imitate the proof of Lemma 3.2 in [12] with g * k (t) = σ(t+b * k ). Let x0 ∈ B be such that (without loss of generality) fN (x0) = maxx∈B |fN (x)|, and µ0 be the Dirac measure supported at x0. We group {w * k } in m = N/(n − 1) disjoint groups of n − 1 vectors each. For each group, we take vectors {v } such that v is orthogonal to the w * k 's in group . The argument in the proof of Lemma 3.2 in [12] can be modified to get a measure µ with total variation 2 m such that It is easy to deduce from here as in [12] using the duality principle that Together with (13) and (14), this implies (11). So by now plenty of examples of lower bounds exist showing a gap between shallow and deep networks. A particularly interesting case is the product function, that is the monomial f (x1, · · · , xn) = x1x2 · · · xn which is, from our point of view, the prototypical compositional functions. Keeping in mind the issue of lower bounds, the question here has to do with the minimum integer r(n) such that the function f is in the closure of the span of σ( w k , x + b k ), with k = 1, · · · , r(n), and w k , b k ranging over their whole domains. Such a result has been claimed for the case of smooth ReLUs, using unusual group techniques and is sketched in the Appendix of [36] . Notice, in support of the conjecture, that assuming that a shallow network with (non-smooth) ReLUs has a lower bound of r(q) = O(q) will lead to an apparent contradiction with Hastad theorem (which is about representation not approximation of Boolean functions) by restricting xi from xi ∈ (−1, 1) to xi ∈ {−1, +1}. Hastad theorem [37] establishes the inapproximability of the parity function by shallow circuits of non-exponential size. As mentioned already, the approximating deep network does not need to exactly match the architecture of the compositional function as long as the graph or tree associated with the function is contained in the graph associated with the network. This is of course good news: the compositionality prior embedded in the architecture of the network does not to reflect exactly the graph of a new function to be learned. We have shown that for a given class of compositional functions characterized by an associated graph there exist a deep network that approximates such a function better than a shallow network. The same network approximates well functions characterized by subgraphs of the original class. The proofs of our theorems show that linear combinations of compositional functions are universal in the sense that they can approximate any function and that deep networks with a number of units that increases exponentially with layers can approximate any function. Notice that deep compositional networks can interpolate if they are overparametrized with respect to the data, even if the data reflect a non-compositional function. As an aside, note that the simplest compositional function -addition -is trivial in the sense that it offers no approximation advantage to deep networks. The key function is multiplication which is for us the prototypical compositional functions. As a consequence, polynomial functions are compositional -they are linear combinations of monomials which are compositional. However, their compositional structure does not confer any advantage in terms of approximation, because of the exponential number of compositional terms. As we mentioned earlier, networks corresponding to graphs that include the graph of the function to be learned can exploit compositionality. The relevant number of parameters to be optimized, however, is the number of parameters r in the network and not the number of parameters r * (r * < r) of the optimal deep network with a graph exactly matched to the graph of the function to be learned. As an aside, the price to be paid in using a non-optimal prior depend on the learning algorithm. For instance, under sparsity constraints it may be possible to pay a smaller price than r (but higher than r * ). In this sense, some of the densely connected deep networks used in practice -which contain sparse graphs possibly relevant for the function to be learned and which are still "smaller" than the exponential number of units required to represent a generic function of n variables -may be capable in some cases of exploiting an underlying compositionality structure without paying an exhorbitant price in terms of required complexity. The approach followed in our main theorems suggest the following considerations (see Appendix 1 for a brief introduction). The structure of a deep network is reflected in polynomials that are best approximated by it -for instance generic polynomials or sparse polynomials (in the coefficients) in d variables of order k. The tree structure of the nodes of a deep network reflects the structure of a specific sparse polynomial. Generic polynomial of degree k in d variables are difficult to learn because the number of terms, trainable parameters and associated VC-dimension are all exponential in d. On the other hand, functions approximated well by sparse polynomials can be learned efficiently by deep networks with a tree structure that matches the polynomial. We recall that in a similar way several properties of certain Boolean functions can be "read out" from the terms of their Fourier expansion corresponding to "large" coefficients, that is from a polynomial that approximates well the function. Classical results [37] about the depth-breadth tradeoff in circuits design show that deep circuits are more efficient in representing certain Boolean functions than shallow circuits. Hastad proved that highly-variable functions (in the sense of having high frequencies in their Fourier spectrum), in particular the parity function cannot even be decently approximated by small constant depth circuits (see also [38] ). A closely related result follow immediately from our main theorem since functions of real variables of the form x1x2... Notice that Hastad's results on Boolean functions have been often quoted in support of the claim that deep neural networks can represent functions that shallow networks cannot. For instance Bengio and LeCun [39] write "We claim that most functions that can be represented compactly by deep architectures cannot be represented by a compact shallow architecture".". Finally, we want to mention a few other observations on Boolean functions that shows an interesting connection with our approach. It is known that within Boolean functions the AC 0 class of polynomial size constant depth circuits is characterized by Fourier transforms where most of the power spectrum is in the low order coefficients. Such functions can be approximated well by a polynomial of low degree and can be learned well by considering only such coefficients. There are two algorithms [40] that allow learning of certain Boolean function classes: 1. the low order algorithm that approximates functions by considering their low order Fourier coefficients and 2. the sparse algorithm which learns a function by approximating its significant coefficients. Decision lists and decision trees can be learned by the first algorithm. Functions with small L1 norm can be approximated well by the second algorithm. Boolean circuits expressing DNFs can be approximated by the first one but even better by the second. In fact, in many cases a function can be approximated by a small set of coefficients but these coefficients do not correspond to low-order terms. All these cases are consistent with the notes about sparse functions in section 6. The key property of the theory of compositional functions sketched here is that certain deep networks can learn them avoiding the curse of dimensionality because of the blessing of compositionality via a small effective dimension. We state here several comments and conjectures. • Assumptions of the compositionality type may have more direct practical implications and be at least as effective as assumptions about function smoothness in countering the curse of dimensionality in learning and approximation. • The estimates on the n-width imply that there is some function in either W n m (theorem 1) or W n,2 m (theorem 2) for which the approximation cannot be better than that suggested by the theorems. • The main question that may be asked about the relevance of the theoretical results of this paper and networks used in practice has to do with the many "channels" used in the latter and with our assumption that each node in the networks computes a scalar function -the linear combination of r units (Equation 3). The following obvious but interesting extension of Theorem 1 to vector-valued functions says that the number of hidden units required for a given accuracy in each component of the function is the same as in the scalar case considered in our theorems (of course the number of weigths is larger): Corollary 4. Let σ : R → R be infinitely differentiable, and not a polynomial. For a vector-valued function f : R n → R q with components fi ∈ W n m , i = 1, · · · , q the number of hidden units in shallow networks with n inputs, q outputs that provide accuracy at least in each of the components of f is The demonstration follows the proof of theorem 1, see also Appendix 4.1. It amounts to realizing that the hidden units (or linear combinations of them) can be equivalent to the monomials of a generic polynomial of degree k in n variables that can be used by a different set of coefficients for each of the fi. This argument of course does not mean that during learning this is what happens; it provides one way to perform the approximation and an associated upper bound. The corollary above leads to a simple argument that generalizes our binary tree results to standard, multi-channel deep convolutional networks by introducing a set of virtual linear units as ouputs of one layer and inputs of the next one. This in turn leads to the following prediction: for consistent approximation accuracy across the layers, the rank of the weights matrices between units in successive layers should have a rank in the order of the number of the dimensionality in the first layer (inputs and outputs have to be defined wrt support of the convolution kernel). This suggests rank-deficient weight matrices in present networks. • We have used polynomials (but see Appendix 4.3) to prove results about complexity of approximation in the case of neural networks. Neural network learning with SGD may or may not synthesize polynomial, depending on the smoothness of the activation function and on the target. This is not a problem for theoretically establishing upper bounds on the degree of convergence because results using the framework on nonlinear width guarantee the "polynomial" bounds are optimal. • Both shallow and deep representations may or may not reflect invariance to group transformations of the inputs of the function ( [41, 22] ). Invariance -also called weight sharing -decreases the complexity of the network. Since we are interested in the comparison of shallow vs deep architectures, we have considered the generic case of networks (and functions) for which invariance is not assumed. In fact, the key advantage of deep vs. shallow network -as shown by the proof of the theorem -is the associated hierarchical locality (the constituent functions in each node are local that is have a small dimensionality) and not invariance (which designates shared weights that is nodes at the same level sharing the same function). One may then ask about the relation of these results with i-theory [42] . The original core of i-theory describes how pooling can provide either shallow or deep networks with invariance and selectivity properties. Invariance of course helps but not exponentially as hierarchical locality does. • There are several properties that follow from the theory here which are attractive from the point of view of neuroscience. A main one is the robustness of the results with respect to the choice of nonlinearities (linear rectifiers, sigmoids, Gaussians etc.) and pooling. • In a machine learning context, minimization over a training set of a loss function such as the square loss yields an empirical approximation of the regression function p(y/x). Our hypothesis of compositionality becomes an hypothesis about the structure of the conditional probability function. • Consider again the case of section 4 of a multivariate function f : [0, 1] d → R. Suppose to discretize it by a set of piecewise constant splines and their tensor products. Each coordinate is effectively replaced by n boolean variables.This results in a d-dimensional table with N = n d entries. This in turn corresponds to a boolean function f : {0, 1} N → R. Here, the assumption of compositionality corresponds to compressibility of a d-dimensional table in terms of a hierarchy of d − 1 2-dimensional tables. Instead of n d entries there are (d − 1)n 2 entries. This has in turn obvious connections with HVQ (Hierarchical Vector Quantization), discussed in Appendix 5. • As Appendix 4.3 shows, every function f can be approximated by an epsilon-close binary function fB. Binarization of f : R n → R is done by using k partitions for each variable xi and indicator functions. Thus f → fB : {0, 1} kn → R and sup|f − fB| ≤ , with depending on k and bounded Df . • fB can be written as a polynomial (a Walsh decomposition) fB ≈ pB. It is always possible to associate a p b to any f , given . • The binarization argument suggests a direct way to connect results on function approximation by neural nets with older results on Boolean functions. The latter are special cases of the former results. • One can think about tensors in terms of d-dimensional tables. The framework of hierarchical decompositions of tensors -in particular the Hierarchical Tucker format -is closely connected to our notion of compositionality. Interestingly, the hierarchical Tucker decomposition has been the subject of recent papers on Deep Learning (for instance see [20] ). This work, as well more classical papers [43] , does not characterize directly the class of functions for which these decompositions are effective. Notice that tensor decompositions assume that the sum of polynomial functions of order d is sparse (see eq. at top of page 2030 of [43] ). Our results provide a rigorous grounding for the tensor work related to deep learning. There is obviously a wealth of interesting connections with approximation theory that should be explored. Notice that the notion of separation rank of a tensor is very closely related to the effective r in Equation 26. • We suggest to define binary sparsity of f , in terms of the sparsity of the boolean function pB; binary sparsity implies that an approximation to f can be learned by non-exponential deep networks via binarization. Notice that if the function f is compositional the associated Boolean functions fB is sparse; the converse is not true. • In may situations, Tikhonov regularization corresponds to cutting high order Fourier coefficients. Sparsity of the coefficients subsumes Tikhonov regularization in the case of a Fourier representation. Notice that as an effect the number of Fourier coefficients is reduced, that is trainable parameters, in the approximating trigonometric polynomial. Sparsity of Fourier coefficients is a general constraint for learning Boolean functions. • Sparsity in a specific basis. A set of functions may be defined to be sparse in a specific basis when the number of parameters necessary for its -approximation increases less than exponentially with the dimensionality. An open question is the appropriate definition of sparsity. The notion of sparsity we suggest here is the effective r in Equation 26. For a general function r ≈ k n ; we may define sparse functions those for which r << k n in
Another relevant and insightful work is REF , which considers width vs. depth and provide general results on expressibility of functions with a compositional nature.
Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of Dimensionality: a Review
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