Two distinct limits for deep learning as the net width @math have been proposed, depending on how the weights of the last layer scale with @math . In the "lazy-learning" regime, the dynamics becomes linear in the weights and is described by a Neural Tangent Kernel @math . By contrast, in the "feature-learning" regime, the dynamics can be expressed in terms of the density distribution of the weights. Understanding which regime describes accurately practical architectures and which one leads to better performance remains a challenge. We answer these questions and produce new characterizations of these regimes for the MNIST data set, by considering deep nets @math whose last layer of weights scales as @math at initialization, where @math is a parameter we vary. We performed systematic experiments on two setups (A) fully-connected Softplus momentum full batch and (B) convolutional ReLU momentum stochastic. We find that (1) @math separates the two regimes. (2) for (A) and (B) feature learning outperforms lazy learning, a difference in performance that decreases with @math and becomes hardly detectable asymptotically for (A) but is very significant for (B). (3) In both regimes, the fluctuations @math induced by initial conditions on the learned function follow @math , leading to a performance that increases with @math . This improvement can be instead obtained at intermediate @math values by ensemble averaging different networks. (4) In the feature regime there exists a time scale @math , such that for @math the dynamics is linear. At @math , the output has grown by a magnitude @math and the changes of the tangent kernel @math become significant. Ultimately, it follows @math for ReLU and Softplus activation, with @math & @math when depth grows.
More recently, it has been shown that beyond initialization, the learning dynamics in this limit also simplifies @cite_31 @cite_5 @cite_6 @cite_8 @cite_29 @cite_15 and is entirely described by the neural tangent kernel (NTK) @cite_31 defined as: where @math are two inputs and @math is the gradient with respect to the parameters @math . For @math , @math does not vary at initialization (and thus it does not depend on the specific choice of @math ) and does not evolve in time. The dynamics is guaranteed to converge on a time independent of @math to a global minimum of the loss, and as for usual kernel learning the function only evolves in the space spanned by the functions @math , where @math is the training set.
@cite_8 @cite_15 @cite_29 @cite_6 @cite_5 @cite_31
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A longstanding goal in deep learning research has been to precisely characterize training and generalization. However, the often complex loss landscapes of neural networks have made a theory of learning dynamics elusive. In this work, we show that for wide neural networks the learning dynamics simplify considerably and that, in the infinite width limit, they are governed by a linear model obtained from the first-order Taylor expansion of the network around its initial parameters. Furthermore, mirroring the correspondence between wide Bayesian neural networks and Gaussian processes, gradient-based training of wide neural networks with a squared loss produces test set predictions drawn from a Gaussian process with a particular compositional kernel. While these theoretical results are only exact in the infinite width limit, we nevertheless find excellent empirical agreement between the predictions of the original network and those of the linearized version even for finite practically-sized networks. This agreement is robust across different architectures, optimization methods, and loss functions. We investigate how the final parameters found by stochastic gradient descent are influenced by over-parameterization. We generate families of models by increasing the number of channels in a base network, and then perform a large hyper-parameter search to study how the test error depends on learning rate, batch size, and network width. We find that the optimal SGD hyper-parameters are determined by a "normalized noise scale," which is a function of the batch size, learning rate, and initialization conditions. In the absence of batch normalization, the optimal normalized noise scale is directly proportional to width. Wider networks, with their higher optimal noise scale, also achieve higher test accuracy. These observations hold for MLPs, ConvNets, and ResNets, and for two different parameterization schemes ("Standard" and "NTK"). We observe a similar trend with batch normalization for ResNets. Surprisingly, since the largest stable learning rate is bounded, the largest batch size consistent with the optimal normalized noise scale decreases as the width increases. How well does a classic deep net architecture like AlexNet or VGG19 classify on a standard dataset such as CIFAR-10 when its "width" --- namely, number of channels in convolutional layers, and number of nodes in fully-connected internal layers --- is allowed to increase to infinity? Such questions have come to the forefront in the quest to theoretically understand deep learning and its mysteries about optimization and generalization. They also connect deep learning to notions such as Gaussian processes and kernels. A recent paper [, 2018] introduced the Neural Tangent Kernel (NTK) which captures the behavior of fully-connected deep nets in the infinite width limit trained by gradient descent; this object was implicit in some other recent papers. A subsequent paper [, 2019] gave heuristic Monte Carlo methods to estimate the NTK and its extension, Convolutional Neural Tangent Kernel (CNTK) and used this to try to understand the limiting behavior on datasets like CIFAR-10. The current paper gives the first efficient exact algorithm (based upon dynamic programming) for computing CNTK as well as an efficient GPU implementation of this algorithm. This results in a significant new benchmark for performance of a pure kernel-based method on CIFAR-10, being 10 higher than the methods reported in [, 2019], and only 5 lower than the performance of the corresponding finite deep net architecture (once batch normalization etc. are turned off). We give the first non-asymptotic proof showing that a fully-trained sufficiently wide net is indeed equivalent to the kernel regression predictor using NTK. Our experiments also demonstrate that earlier Monte Carlo approximation can degrade the performance significantly, thus highlighting the power of our exact kernel computation, which we have applied even to the full CIFAR-10 dataset and 20-layer nets. 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). At initialization, artificial neural networks (ANNs) are equivalent to Gaussian processes in the infinite-width limit, thus connecting them to kernel methods. We prove that the evolution of an ANN during training can also be described by a kernel: during gradient descent on the parameters of an ANN, the network function (which maps input vectors to output vectors) follows the so-called kernel gradient associated with a new object, which we call the Neural Tangent Kernel (NTK). This kernel is central to describe the generalization features of ANNs. While the NTK is random at initialization and varies during training, in the infinite-width limit it converges to an explicit limiting kernel and stays constant during training. This makes it possible to study the training of ANNs in function space instead of parameter space. Convergence of the training can then be related to the positive-definiteness of the limiting NTK. We then focus on the setting of least-squares regression and show that in the infinite-width limit, the network function follows a linear differential equation during training. The convergence is fastest along the largest kernel principal components of the input data with respect to the NTK, hence suggesting a theoretical motivation for early stopping. Finally we study the NTK numerically, observe its behavior for wide networks, and compare it to the infinite-width limit.