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.
Finally, unlike in the lazy limit where preactivations vary very weakly during training, in the feature limit feature learning occurs. It is equivalent to saying that the tangent kernel (which can always be defined) evolves in time @cite_21 @cite_11 @cite_26 . What are the characteristic time scales and magnitude of this evolution?
@cite_21 @cite_26 @cite_11
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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 @math (which maps input vectors to output vectors) follows the kernel gradient of the functional cost (which is convex, in contrast to the parameter cost) w.r.t. a new kernel: 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 it 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 prove the positive-definiteness of the limiting NTK when the data is supported on the sphere and the non-linearity is non-polynomial. We then focus on the setting of least-squares regression and show that in the infinite-width limit, the network function @math 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. 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. Due to the limited amount of training samples, finetuning pre-trained deep models online is prone to overfitting. In this paper, we propose a sequential training method for convolutional neural networks (CNNs) to effectively transfer pre-trained deep features for online applications. We regard a CNN as an ensemble with each channel of the output feature map as an individual base learner. Each base learner is trained using different loss criterions to reduce correlation and avoid over-training. To achieve the best ensemble online, all the base learners are sequentially sampled into the ensemble via important sampling. To further improve the robustness of each base learner, we propose to train the convolutional layers with random binary masks, which serves as a regularization to enforce each base learner to focus on different input features. The proposed online training method is applied to visual tracking problem by transferring deep features trained on massive annotated visual data and is shown to significantly improve tracking performance. Extensive experiments are conducted on two challenging benchmark data set and demonstrate that our tracking algorithm can outperform state-of-the-art methods with a considerable margin. In this paper, we propose a novel approach for efficient training of deep neural networks in a bottom-up fashion using a layered structure. Our algorithm, which we refer to as deep cascade learning, is motivated by the cascade correlation approach of Fahlman and Lebiere, who introduced it in the context of perceptrons. We demonstrate our algorithm on networks of convolutional layers, though its applicability is more general. Such training of deep networks in a cascade directly circumvents the well-known vanishing gradient problem by ensuring that the output is always adjacent to the layer being trained. We present empirical evaluations comparing our deep cascade training with standard end–end training using back propagation of two convolutional neural network architectures on benchmark image classification tasks (CIFAR-10 and CIFAR-100). We then investigate the features learned by the approach and find that better, domain-specific, representations are learned in early layers when compared to what is learned in end–end training. This is partially attributable to the vanishing gradient problem that inhibits early layer filters to change significantly from their initial settings. While both networks perform similarly overall, recognition accuracy increases progressively with each added layer, with discriminative features learned in every stage of the network, whereas in end–end training, no such systematic feature representation was observed. We also show that such cascade training has significant computational and memory advantages over end–end training, and can be used as a pretraining algorithm to obtain a better performance.