Can multilayer neural networks -- typically constructed as highly complex structures with many nonlinearly activated neurons across layers -- behave in a non-trivial way that yet simplifies away a major part of their complexities? In this work, we uncover a phenomenon in which the behavior of these complex networks -- under suitable scalings and stochastic gradient descent dynamics -- becomes independent of the number of neurons as this number grows sufficiently large. We develop a formalism in which this many-neurons limiting behavior is captured by a set of equations, thereby exposing a previously unknown operating regime of these networks. While the current pursuit is mathematically non-rigorous, it is complemented with several experiments that validate the existence of this behavior.
As mentioned, several recent works have studied the MF limit in the two-layers network case. The works @cite_44 @cite_18 @cite_45 @cite_35 establish the MF limit, and in particular, @cite_44 proves that this holds as soon as the number of neurons exceeds the data dimension. @cite_44 @cite_18 utilize this limit to prove that (noisy) SGD can converge to (near) global optimum under different assumptions. For a specific class of activations and data distribution, @cite_1 proves that this convergence is exponentially fast using the displacement convexity property of the MF limit. Taking the same viewpoint, @cite_48 proves a convergence result for a specifically chosen many-neurons limit. @cite_35 @cite_4 study the fluctuations around the MF limit. Our analysis of the multilayer case requires substantial extension and new ideas, uncovering certain properties that are not obvious from the two-layers analysis (see also Section ).
@cite_35 @cite_18 @cite_4 @cite_48 @cite_1 @cite_44 @cite_45
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Multilayer neural networks are among the most powerful models in machine learning, yet the fundamental reasons for this success defy mathematical understanding. Learning a neural network requires optimizing a nonconvex high-dimensional objective (risk function), a problem that is usually attacked using stochastic gradient descent (SGD). Does SGD converge to a global optimum of the risk or only to a local optimum? In the former case, does this happen because local minima are absent or because SGD somehow avoids them? In the latter, why do local minima reached by SGD have good generalization properties? In this paper, we consider a simple case, namely two-layer neural networks, and prove that—in a suitable scaling limit—SGD dynamics is captured by a certain nonlinear partial differential equation (PDE) that we call distributional dynamics (DD). We then consider several specific examples and show how DD can be used to prove convergence of SGD to networks with nearly ideal generalization error. This description allows for “averaging out” some of the complexities of the landscape of neural networks and can be used to prove a general convergence result for noisy SGD. Multi-layer neural networks are among the most powerful models in machine learning, yet the fundamental reasons for this success defy mathematical understanding. Learning a neural network requires to optimize a non-convex high-dimensional objective (risk function), a problem which is usually attacked using stochastic gradient descent (SGD). Does SGD converge to a global optimum of the risk or only to a local optimum? In the first case, does this happen because local minima are absent, or because SGD somehow avoids them? In the second, why do local minima reached by SGD have good generalization properties? In this paper we consider a simple case, namely two-layers neural networks, and prove that -in a suitable scaling limit- SGD dynamics is captured by a certain non-linear partial differential equation (PDE) that we call distributional dynamics (DD). We then consider several specific examples, and show how DD can be used to prove convergence of SGD to networks with nearly ideal generalization error. This description allows to 'average-out' some of the complexities of the landscape of neural networks, and can be used to prove a general convergence result for noisy SGD. Neural networks exhibit good generalization behavior in the over-parameterized regime, where the number of network parameters exceeds the number of observations. Nonetheless, current generalization bounds for neural networks fail to explain this phenomenon. In an attempt to bridge this gap, we study the problem of learning a two-layer over-parameterized neural network, when the data is generated by a linearly separable function. In the case where the network has Leaky ReLU activations, we provide both optimization and generalization guarantees for over-parameterized networks. Specifically, we prove convergence rates of SGD to a global minimum and provide generalization guarantees for this global minimum that are independent of the network size. Therefore, our result clearly shows that the use of SGD for optimization both finds a global minimum, and avoids overfitting despite the high capacity of the model. This is the first theoretical demonstration that SGD can avoid overfitting, when learning over-specified neural network classifiers. Deep neural networks (DNNs) have demonstrated dominating performance in many fields; since AlexNet, networks used in practice are going wider and deeper. On the theoretical side, a long line of works has been focusing on training neural networks with one hidden layer. The theory of multi-layer networks remains largely unsettled. In this work, we prove why stochastic gradient descent (SGD) can find @math on the training objective of DNNs in @math . We only make two assumptions: the inputs are non-degenerate and the network is over-parameterized. The latter means the network width is sufficiently large: @math in @math , the number of layers and in @math , the number of samples. Our key technique is to derive that, in a sufficiently large neighborhood of the random initialization, the optimization landscape is almost-convex and semi-smooth even with ReLU activations. This implies an equivalence between over-parameterized neural networks and neural tangent kernel (NTK) in the finite (and polynomial) width setting. As concrete examples, starting from randomly initialized weights, we prove that SGD can attain 100 training accuracy in classification tasks, or minimize regression loss in linear convergence speed, with running time polynomial in @math . Our theory applies to the widely-used but non-smooth ReLU activation, and to any smooth and possibly non-convex loss functions. In terms of network architectures, our theory at least applies to fully-connected neural networks, convolutional neural networks (CNN), and residual neural networks (ResNet).