Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userNormalization effects on shallow neural networks and related asymptotic expansions
222
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We consider shallow (single hidden layer) neural networks and characterize their performance when trained with stochastic gradient descent as the number of hidden units $N$ and gradient descent steps grow to infinity. In particular, we investigate the effect of different scaling schemes, which lead to different normalizations of the neural network, on the networks statistical output, closing the gap between the $1/sqrt{N}$ and the mean-field $1/N$ normalization. We develop an asymptotic expansion for the neural networks statistical output pointwise with respect to the scaling parameter as the number of hidden units grows to infinity. Based on this expansion, we demonstrate mathematically that to leading order in $N$, there is no bias-variance trade off, in that both bias and variance (both explicitly characterized) decrease as the number of hidden units increases and time grows. In addition, we show that to leading order in $N$, the variance of the neural networks statistical output decays as the implied normalization by the scaling parameter approaches the mean field normalization. Numerical studies on the MNIST and CIFAR10 datasets show that test and train accuracy monotonically improve as the neural networks normalization gets closer to the mean field normalization.
rate research
Read More
We consider the approximation rates of shallow neural networks with respect to the variation norm. Upper bounds on these rates have been established for sigmoidal and ReLU activation functions, but it has remained an important open problem whether these rates are sharp. In this article, we provide a solution to this problem by proving sharp lower bounds on the approximation rates for shallow neural networks, which are obtained by lower bounding the $L^2$-metric entropy of the convex hull of the neural network basis functions. In addition, our methods also give sharp lower bounds on the Kolmogorov $n$-widths of this convex hull, which show that the variation spaces corresponding to shallow neural networks cannot be efficiently approximated by linear methods. These lower bounds apply to both sigmoidal activation functions with bounded variation and to activation functions which are a power of the ReLU. Our results also quantify how much stronger the Barron spectral norm is than the variation norm and, combined with previous results, give the asymptotics of the $L^infty$-metric entropy up to logarithmic factors in the case of the ReLU activation function.
Dropout is a regularisation technique in neural network training where unit activations are randomly set to zero with a given probability emph{independently}. In this work, we propose a generalisation of dropout and other multiplicative noise injection schemes for shallow and deep neural networks, where the random noise applied to different units is not independent but follows a joint distribution that is either fixed or estimated during training. We provide theoretical insights on why such adaptive structured noise injection (ASNI) may be relevant, and empirically confirm that it helps boost the accuracy of simple feedforward and convolutional neural networks, disentangles the hidden layer representations, and leads to sparser representations. Our proposed method is a straightforward modification of the classical dropout and does not require additional computational overhead.
We consider the teacher-student setting of learning shallow neural networks with quadratic activations and planted weight matrix $W^*inmathbb{R}^{mtimes d}$, where $m$ is the width of the hidden layer and $dle m$ is the data dimension. We study the optimization landscape associated with the empirical and the population squared risk of the problem. Under the assumption the planted weights are full-rank we obtain the following results. First, we establish that the landscape of the empirical risk admits an energy barrier separating rank-deficient $W$ from $W^*$: if $W$ is rank deficient, then its risk is bounded away from zero by an amount we quantify. We then couple this result by showing that, assuming number $N$ of samples grows at least like a polynomial function of $d$, all full-rank approximate stationary points of the empirical risk are nearly global optimum. These two results allow us to prove that gradient descent, when initialized below the energy barrier, approximately minimizes the empirical risk and recovers the planted weights in polynomial-time. Next, we show that initializing below this barrier is in fact easily achieved when the weights are randomly generated under relatively weak assumptions. We show that provided the network is sufficiently overparametrized, initializing with an appropriate multiple of the identity suffices to obtain a risk below the energy barrier. At a technical level, the last result is a consequence of the semicircle law for the Wishart ensemble and could be of independent interest. Finally, we study the minimizers of the empirical risk and identify a simple necessary and sufficient geometric condition on the training data under which any minimizer has necessarily zero generalization error. We show that as soon as $Nge N^*=d(d+1)/2$, randomly generated data enjoys this geometric condition almost surely, while that ceases to be true if $N<N^*$.
We consider the variation space corresponding to a dictionary of functions in $L^2(Omega)$ and present the basic theory of approximation in these spaces. Specifically, we compare the definition based on integral representations with the definition in terms of convex hulls. We show that in many cases, including the dictionaries corresponding to shallow ReLU$^k$ networks and a dictionary of decaying Fourier modes, that the two definitions coincide. We also give a partial characterization of the variation space for shallow ReLU$^k$ networks and show that the variation space with respect to the dictionary of decaying Fourier modes corresponds to the Barron spectral space.
Generative adversarial networks (GANs) are highly effective unsupervised learning frameworks that can generate very sharp data, even for data such as images with complex, highly multimodal distributions. However GANs are known to be very hard to train, suffering from problems such as mode collapse and disturbing visual artifacts. Batch normalization (BN) techniques have been introduced to address the training. Though BN accelerates the training in the beginning, our experiments show that the use of BN can be unstable and negatively impact the quality of the trained model. The evaluation of BN and numerous other recent schemes for improving GAN training is hindered by the lack of an effective objective quality measure for GAN models. To address these issues, we first introduce a weight normalization (WN) approach for GAN training that significantly improves the stability, efficiency and the quality of the generated samples. To allow a methodical evaluation, we introduce squared Euclidean reconstruction error on a test set as a new objective measure, to assess training performance in terms of speed, stability, and quality of generated samples. Our experiments with a standard DCGAN architecture on commonly used datasets (CelebA, LSUN bedroom, and CIFAR-10) indicate that training using WN is generally superior to BN for GANs, achieving 10% lower mean squared loss for reconstruction and significantly better qualitative results than BN. We further demonstrate the stability of WN on a 21-layer ResNet trained with the CelebA data set. The code for this paper is available at https://github.com/stormraiser/gan-weightnorm-resnet
suggested questions
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
