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Title: Directional Analysis of Stochastic Gradient Descent via von Mises-Fisher Distributions in Deep Learning

Paper content:
Although stochastic gradient descent (SGD) is a driving force behind the recent success of deep learning, our understanding of its dynamics in a high-dimensional parameter space is limited.
In recent years, some researchers have used the stochasticity of minibatch gradients, or the signal-to-noise ratio, to better characterize the learning dynamics of SGD.
Inspired from these work, we here analyze SGD from a geometrical perspective by inspecting the stochasticity of the norms and directions of minibatch gradients.
We propose a model of the directional concentration for minibatch gradients through von Mises-Fisher (VMF) distribution, and show that the directional uniformity of minibatch gradients increases over the course of SGD.
We empirically verify our result using deep convolutional networks and observe a higher correlation between the gradient stochasticity and the proposed directional uniformity than that against the gradient norm stochasticity, suggesting that the directional statistics of minibatch gradients is a major factor behind SGD.
Stochastic gradient descent (SGD) has been a driving force behind the recent success of deep learning.
Despite a series of work on improving SGD by incorporating the second-order information of the objective function BID26 BID21 BID6 BID22 BID7 , SGD is still the most widely used optimization algorithm for training a deep neural network.
The learning dynamics of SGD, however, has not been well characterized beyond that it converges to an extremal point BID1 due to the non-convexity and highdimensionality of a usual objective function used in deep learning.Gradient stochasticity, or the signal-to-noise ratio (SNR) of the stochastic gradient, has been proposed as a tool for analyzing the learning dynamics of SGD.
BID28 identified two phases in SGD based on this.
In the first phase, "drift phase", the gradient mean is much higher than its standard deviation, during which optimization progresses rapidly.
This drift phase is followed by the "diffusion phase", where SGD behaves similarly to Gaussian noise with very small means.
Similar observations were made by BID18 and BID4 who have also divided the learning dynamics of SGD into two phases.
BID28 have proposed that such phase transition is related to information compression.
Unlike them, we notice that there are two aspects to the gradient stochasticity.
One is the L 2 norm of the minibatch gradient (the norm stochasticity), and the other is the directional balance of minibatch gradients (the directional stochasticity).
SGD converges or terminates when either the norm of the minibatch gradient vanishes to zeros, or when the angles of the minibatch gradients are uniformly distributed and their non-zero norms are close to each other.
That is, the gradient stochasticity, or the SNR of the stochastic gradient, is driven by both of these aspects, and it is necessary for us to investigate not only the holistic SNR but also the SNR of the minibatch gradient norm and that of the minibatch gradient angles.In this paper, we use a von Mises-Fisher (vMF hereafter) distribution, which is often used in directional statistics BID20 , and its concentration parameter κ to characterize the directional balance of minibatch gradients and understand the learning dynamics of SGD from the perspective of directional statistics of minibatch gradients.
We prove that SGD increases the direc-tional balance of minibatch gradients.
We empirically verify this with deep convolutional networks with various techniques, including batch normalization BID12 and residual connections BID9 , on MNIST and CIFAR-10 ( BID15 ).
Our empirical investigation further reveals that the proposed directional stochasticity is a major drive behind the gradient stochasticity compared to the norm stochasticity, suggesting the importance of understanding the directional statistics of the stochastic gradient.Contribution We analyze directional stochasticity of the minibatch gradients via angles as well as the concentration parameter of the vMF distribution.
Especially, we theoretically show that the directional uniformity of the minibatch gradients modeled by the vMF distribution increases as training progresses, and verify this by experiments.
In doing so, we introduce gradient norm stochasticity as the ratio of the standard deviation of the minibatch gradients to their expectation and theoretically and empirically show that this gradient norm stochasticity decreases as the batch size increases.Related work Most studies about SGD dynamics have been based on two-phase behavior BID28 BID18 BID4 .
BID18 investigated this behavior by considering a shallow neural network with residual connections and assuming the standard normal input distribution.
They showed that SGD-based learning under these setups has two phases; search and convergence phases.
BID28 on the other hand investigated a deep neural network with tanh activation functions, and showed that SGD-based learning has drift and diffusion phases.
They have also proposed that such SNR transition (drift + diffusion) is related to the information transition divided into empirical error minimization and representation compression phases.
However, Saxe et al. (2018) have reported that the information transition is not generally associated with the SNR transition with ReLU BID23 ) activation functions.
BID4 instead looked at the inner product between successive minibatch gradients and presented transient and stationary phases.Unlike our work here, the experimental verification of the previous work conducted under limited settings -the shallow network BID18 , the specific activation function BID28 , and only MNIST dataset BID28 BID4 -that conform well with their theoretical assumptions.
Moreover, their work does not offer empirical result about the effect of the latest techniques including both batch normalization BID12 layers and residual connections BID9 .
Stochasticity of gradients is a key to understanding the learning dynamics of SGD BID28 and has been pointed out as a factor behind the success of SGD (see, e.g., BID17 BID14 .
In this paper, we provide a theoretical framework using von Mises-Fisher distribution, under which the directional stochasticity of minibatch gradients can be estimated and analyzed, and show that the directional uniformity increases over the course of SGD.
Through the extensive empirical evaluation, we have observed that the directional uniformity indeed improves over the course of training a deep neural network, and that its trend is monotonic when batch normalization and skip connections were used.
Furthermore, we demonstrated that the stochasticity of minibatch gradients is largely determined by the directional stochasticity rather than the gradient norm stochasticity.Our work in this paper suggests two major research directions for the future.
First, our analysis has focused on the aspect of optimization, and it is an open question how the directional uniformity relates to the generalization error although handling the stochasticity of gradients has improved SGD BID24 BID11 BID29 BID13 .
Second, we have focused on passive analysis of SGD using the directional statistics of minibatch gradients, but it is not unreasonable to suspect that SGD could be improved by explicitly taking into account the directional statistics of minibatch gradients during optimization.

assistant: text: One of theoretical issues in deep learning