Stochastic gradient descent (SGD) is a key ingredient in the training of deep neural networks and yet its geometrical significance appears elusive. We study a deterministic model in which the trajectories of our dynamical systems are described via geodesics of a family of metrics arising from the diffusion matrix. These metrics encode information about the highly non-isotropic gradient noise in SGD. We establish a parallel with General Relativity models, where the role of the electromagnetic field is played by the gradient of the loss function. We compute an example of a two layer network.

Decentralized optimization has been demonstrated to be very useful in machine learning. This work studies the communication-efficiency issue in decentralized optimization. We analyze the Periodic Decentralized Stochastic Gradient Descent (PD-SGD) algorithm, a straightforward combination of federated averaging and decentralized SGD. For the setting of for non-convex objective and non-identically distributed data, we prove that PD-SGD converges to a critical point. In particular, the number of local SGDs trades off communication and local computation. From an algorithmic perspective, we analyze a novel version of PD-SGD, which alternates between multiple local updates and multiple decentralized SGDs. We also show that when we periodically shrink the length of local updates, this generalized PD-SGD can better balance the communication-convergence trade-off both theoretically and empirically.

Figure 1: A two-dimensional convex function represented via contour lines. The function value is constant on the boundary of each such ellipse, and decreases as the ellipse becomes smaller and smaller. Let us assume we want to minimize this function starting from a point \(A\). The red line shows the path followed by a gradient descent optimizer converging to the minimum point \(B\), while the green dashed line represents the direct line joining \(A\) and \(B\). In today's post, we will discuss an interesting property concerning the trajectory of gradient descent iterates, namely the length of the Gradient Descent curve.

Two types of zeroth-order stochastic algorithms have recently been designed for nonconvex optimization respectively based on the first-order techniques SVRG and SARAH/SPIDER. This paper addresses several important issues that are still open in these methods. First, all existing SVRG-type zeroth-order algorithms suffer from worse function query complexities than either zeroth-order gradient descent (ZO-GD) or stochastic gradient descent (ZO-SGD). In this paper, we propose a new algorithm ZO-SVRG-Coord-Rand and develop a new analysis for an existing ZO-SVRG-Coord algorithm proposed in Liu et al. 2018b, and show that both ZO-SVRG-Coord-Rand and ZO-SVRG-Coord (under our new analysis) outperform other exiting SVRG-type zeroth-order methods as well as ZO-GD and ZO-SGD. Second, the existing SPIDER-type algorithm SPIDER-SZO (Fang et al. 2018) has superior theoretical performance, but suffers from the generation of a large number of Gaussian random variables as well as a $\sqrt{\epsilon}$-level stepsize in practice. In this paper, we develop a new algorithm ZO-SPIDER-Coord, which is free from Gaussian variable generation and allows a large constant stepsize while maintaining the same convergence rate and query complexity, and we further show that ZO-SPIDER-Coord automatically achieves a linear convergence rate as the iterate enters into a local PL region without restart and algorithmic modification.

Expressiveness of deep models was recently addressed via the connection between neural networks (NNs) and kernel learning, where first-order dynamics of NN during a gradient-descent (GD) optimization were related to gradient similarity kernel, also known as Neural Tangent Kernel (NTK). In the majority of works this kernel is considered to be time-invariant, with its properties being defined entirely by NN architecture and independent of the learning task at hand. In contrast, in this paper we empirically explore these properties along the optimization and show that in practical applications the NN kernel changes in a very dramatic and meaningful way, with its top eigenfunctions aligning toward the target function learned by NN. Moreover, these top eigenfunctions serve sort of basis functions for NN output - a function represented by NN is spanned almost completely by them for the entire optimization process. Further, since the learning along top eigenfunctions is typically fast, their alignment with the target function improves the overall optimization performance. In addition, we study how the neural spectrum is affected by learning rate decay, typically done by practitioners, showing various trends in the kernel behavior. We argue that the presented phenomena may lead to a more complete theoretical understanding behind NN learning.

Training neural-networks is computationally hard. However, in practice they are trained efficiently using gradient-based algorithms, achieving remarkable performance on natural data. To bridge this gap, we observe the property of local correlation: correlation between small patterns of the input and the target label. We focus on learning deep neural-networks with a variant of gradient-descent, when the target function is a tree-structured Boolean circuit. We show that in this case, the existence of correlation between the gates of the circuit and the target label determines whether the optimization succeeds or fails. Using this result, we show that neural-networks can learn the (log n)-parity problem for most product distributions. These results hint that local correlation may play an important role in differentiating between distributions that are hard or easy to learn.

This paper provides a framework to analyze stochastic gradient algorithms in a mean squared error (MSE) sense using the asymptotic normality result of the stochastic gradient descent (SGD) iterates. We perform this analysis by taking the asymptotic normality result and applying it to the finite iteration case. Specifically, we look at problems where the gradient estimators are biased and have reduced variance and compare the iterates generated by these gradient estimators to the iterates generated by the SGD algorithm. We use the work of Fabian to characterize the mean and the variance of the distribution of the iterates in terms of the bias and the covariance matrix of the gradient estimators. We introduce the sliding window SGD (SW-SGD) algorithm, with its proof of convergence, which incurs a lower MSE than the SGD algorithm on quadratic and convex problems. Lastly, we present some numerical results to show the effectiveness of this framework and the superiority of SW-SGD algorithm over the SGD algorithm.

What enables Stochastic Gradient Descent (SGD) to achieve better generalization than Gradient Descent (GD) in Neural Network training? This question has attracted much attention. In this paper, we study the distribution of the Stochastic Gradient Noise (SGN) vectors during the training. We observe that for batch sizes 256 and above, the distribution is best described as Gaussian at-least in the early phases of training. This holds across data-sets, architectures, and other choices.

Autoencoders enable data dimensionality reduction and are a key component of many (deep) learning systems. This article explores the use of the XCSF online evolutionary reinforcement learning system to perform autoencoding. Initial results using a neural network representation and combining artificial evolution with stochastic gradient descent, suggest it is an effective approach to data reduction. The approach adaptively subdivides the input domain into local approximations that are simpler than a global neural network solution. By allowing the number of neurons in the autoencoders to evolve, this further enables the emergence of an ensemble of structurally heterogeneous solutions to cover the problem space. In this case, networks of differing complexity are typically seen to cover different areas of the problem space. Furthermore, the rate of gradient descent applied to each layer is tuned via self-adaptive mutation, thereby reducing the parameter optimisation task.