Gradient-based optimization of absolute errors is tricky, since the gradient is "never" zero. In theory, adaptive methods should be able to damp oscillations so that it converges to the minimum. However, I found none of the'standard' methods were able to do this "out of the box". Learning rate decay could alleviate the problem, but needs manual tuning which I would rather avoid. Does anyone know of a method that can do this?

Loss functions with a large number of saddle points are one of the main obstacles to training many modern machine learning models. Gradient descent (GD) is a fundamental algorithm for machine learning and converges to a saddle point for certain initial data. We call the region formed by these initial values the "attraction region." For quadratic functions, GD converges to a saddle point if the initial data is in a subspace of up to n-1 dimensions. In this paper, we prove that a small modification of the recently proposed Laplacian smoothing gradient descent (LSGD) [Osher, et al., arXiv:1806.06317] contributes to avoiding saddle points without sacrificing the convergence rate of GD. In particular, we show that the dimension of the LSGD's attraction region is at most floor((n-1)/2) for a class of quadratic functions which is significantly smaller than GD's (n-1)-dimensional attraction region.

It is a gradient descent algorithm for classification implemented from scratch using numpy library. It is good practice to shuffle data at first numpy.random.shuffle() Mini Batch Size is size of input data flowing through network at a time for calculating error as a whole Learning Rate Alpha decides the rate at which, weights and biases will update while back propagation Number of Epochs decides number of times, the whole dataset will be used to train the network Set Mini Batch Size to 1/10th of total data available. And update it manually after every train of network to find its optimum value Alpha should be selected such that learning isn't very slow as well as it didn't take long jump or else, network will start diverging from local minima Number of epochs are selected such that network don't overfit itself over noise In ANN, output will depend on every neuron it pass through For output layer, we have label according to which, it is possible to find it's expected value But for all other layers, there is no single solution available So, finding optimum value is little harder for that

In this paper we focus on the problem of finding the optimal weights of the shallowest of neural networks consisting of a single Rectified Linear Unit (ReLU). These functions are of the form $\mathbf{x}\rightarrow \max(0,\langle\mathbf{w},\mathbf{x}\rangle)$ with $\mathbf{w}\in\mathbb{R}^d$ denoting the weight vector. We focus on a planted model where the inputs are chosen i.i.d. from a Gaussian distribution and the labels are generated according to a planted weight vector. We first show that mini-batch stochastic gradient descent when suitably initialized, converges at a geometric rate to the planted model with a number of samples that is optimal up to numerical constants. Next we focus on a parallel implementation where in each iteration the mini-batch gradient is calculated in a distributed manner across multiple processors and then broadcast to a master or all other processors. To reduce the communication cost in this setting we utilize a Quanitzed Stochastic Gradient Scheme (QSGD) where the partial gradients are quantized. Perhaps unexpectedly, we show that QSGD maintains the fast convergence of SGD to a globally optimal model while significantly reducing the communication cost. We further corroborate our numerical findings via various experiments including distributed implementations over Amazon EC2.

The analysis of nonconvex matrix completion has recently attracted much attention in the community of machine learning thanks to its computational convenience. Existing analysis on this problem, however, usually relies on $\ell_{2,\infty}$ projection or regularization that involves unknown model parameters, although they are observed to be unnecessary in numerical simulations, see, e.g. Zheng and Lafferty [2016]. In this paper, we extend the analysis of the vanilla gradient descent for positive semidefinite matrix completion proposed in Ma et al. [2017] to the rectangular case, and more significantly, improve the required sampling complexity from $\widetilde{O}(r^3)$ to $\widetilde{O}(r^2)$. Our technical ideas and contributions are potentially useful in improving the leave-one-out analysis in other related problems.

We interpret the variational inference of the Stochastic Gradient Descent (SGD) as minimizing a new potential function named the \textit{quasi-potential}. We analytically construct the quasi-potential function in the case when the loss function is convex and admits only one global minimum point. We show in this case that the quasi-potential function is related to the noise covariance structure of SGD via a partial differential equation of Hamilton-Jacobi type. This relation helps us to show that anisotropic noise leads to faster escape than isotropic noise. We then consider the dynamics of SGD in the case when the loss function is non-convex and admits several different local minima. In this case, we demonstrate an example that shows how the noise covariance structure plays a role in "implicit regularization", a phenomenon in which SGD favors some particular local minimum points. This is done through the relation between the noise covariance structure and the quasi-potential function. Our analysis is based on Large Deviations Theory (LDT), and they are validated by numerical experiments.

The gradient noise (GN) in the stochastic gradient descent (SGD) algorithm is often considered to be Gaussian in the large data regime by assuming that the classical central limit theorem (CLT) kicks in. This assumption is often made for mathematical convenience, since it enables SGD to be analyzed as a stochastic differential equation (SDE) driven by a Brownian motion. We argue that the Gaussianity assumption might fail to hold in deep learning settings and hence render the Brownian motion-based analyses inappropriate. Inspired by non-Gaussian natural phenomena, we consider the GN in a more general context and invoke the generalized CLT (GCLT), which suggests that the GN converges to a heavy-tailed $\alpha$-stable random variable. Accordingly, we propose to analyze SGD as an SDE driven by a L\'{e}vy motion. Such SDEs can incur `jumps', which force the SDE transition from narrow minima to wider minima, as proven by existing metastability theory. To validate the $\alpha$-stable assumption, we conduct extensive experiments on common deep learning architectures and show that in all settings, the GN is highly non-Gaussian and admits heavy-tails. We further investigate the tail behavior in varying network architectures and sizes, loss functions, and datasets. Our results open up a different perspective and shed more light on the belief that SGD prefers wide minima.

Much of recent machine learning has focused on deep learning, in which neural network weights are trained through variants of stochastic gradient descent. An alternative approach comes from the field of neuroevolution, which harnesses evolutionary algorithms to optimize neural networks, inspired by the fact that natural brains themselves are the products of an evolutionary process. Neuroevolution enables important capabilities that are typically unavailable to gradient-based approaches, including learning neural network building blocks (for example activation functions), hyperparameters, architectures and even the algorithms for learning themselves. Neuroevolution also differs from deep learning (and deep reinforcement learning) by maintaining a population of solutions during search, enabling extreme exploration and massive parallelization. Finally, because neuroevolution research has (until recently) developed largely in isolation from gradient-based neural network research, it has developed many unique and effective techniques that should be effective in other machine learning areas too.

Why does training deep neural networks using stochastic gradient descent (SGD) result in a generalization error that does not worsen with the number of parameters in the network? To answer this question, we advocate a notion of effective model capacity that is dependent on {\em a given random initialization of the network} and not just the training algorithm and the data distribution. We provide empirical evidences that demonstrate that the model capacity of SGD-trained deep networks is in fact restricted through implicit regularization of {\em the $\ell_2$ distance from the initialization}. We also provide theoretical arguments that further highlight the need for initialization-dependent notions of model capacity. We leave as open questions how and why distance from initialization is regularized, and whether it is sufficient to explain generalization.

The development of the perceptron was a big step towards the goal of creating useful connectionist networks capable of learning complex relations between inputs and outputs. In the late 1950's, the connectionist community understood that what was needed for further development of connectionist models was a mathematically-derived (and thus potentially more flexible and powerful) rule for learning. By early 1960's, the Delta Rule [also known as the Widrow & Hoff Learning rule or the Least Mean Square (LMS) rule] was invented by Widrow and Hoff. This rule is similar to the perceptron learning rule by McClelland & Rumelhart, 1988, but is also characterized by a mathematical utility and elegance missing in the perceptron and other early learning rules. The Delta Rule uses the difference between target activation (i.e., target output values) and obtained activation to drive learning.