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Register a new userOn the Locality of the Natural Gradient for Deep Learning
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Authors
Nihat Ay
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We study the natural gradient method for learning in deep Bayesian networks, including neural networks. There are two natural geometries associated with such learning systems consisting of visible and hidden units. One geometry is related to the full system, the other one to the visible sub-system. These two geometries imply different natural gradients. In a first step, we demonstrate a great simplification of the natural gradient with respect to the first geometry, due to locality properties of the Fisher information matrix. This simplification does not directly translate to a corresponding simplification with respect to the second geometry. We develop the theory for studying the relation between the t
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In this paper, we develop an efficient sketchy empirical natural gradient method (SENG) for large-scale deep learning problems. The empirical Fisher information matrix is usually low-rank since the sampling is only practical on a small amount of data at each iteration. Although the corresponding natural gradient direction lies in a small subspace, both the computational cost and memory requirement are still not tractable due to the high dimensionality. We design randomized techniques for different neural network structures to resolve these challenges. For layers with a reasonable dimension, sketching can be performed on a regularized least squares subproblem. Otherwise, since the gradient is a vectorization of the product between two matrices, we apply sketching on the low-rank approximations of these matrices to compute the most expensive parts. A distributed version of SENG is also developed for extremely large-scale applications. Global convergence to stationary points is established under some mild assumptions and a fast linear convergence is analyzed under the neural tangent kernel (NTK) case. Extensive experiments on convolutional neural networks show the competitiveness of SENG compared with the state-of-the-art methods. On the task ResNet50 with ImageNet-1k, SENG achieves 75.9% Top-1 testing accuracy within 41 epochs. Experiments on the distributed large-batch training show that the scaling efficiency is quite reasonable.
The Fisher information matrix (FIM) has been applied to the realm of deep learning. It is closely related to the loss landscape, the variance of the parameters, second order optimization, and deep learning theory. The exact FIM is either unavailable in closed form or too expensive to compute. In practice, it is almost always estimated based on empirical samples. We investigate two such estimators based on two equivalent representations of the FIM. They are both unbiased and consistent with respect to the underlying true FIM. Their estimation quality is characterized by their variance given in closed form. We bound their variances and analyze how the parametric structure of a deep neural network can impact the variance. We discuss the meaning of this variance measure and our bounds in the context of deep learning.
Variational Bayesian neural nets combine the flexibility of deep learning with Bayesian uncertainty estimation. Unfortunately, there is a tradeoff between cheap but simple variational families (e.g.~fully factorized) or expensive and complicated inference procedures. We show that natural gradient ascent with adaptive weight noise implicitly fits a variational posterior to maximize the evidence lower bound (ELBO). This insight allows us to train full-covariance, fully factorized, or matrix-variate Gaussian variational posteriors using noi
Black-box optimization is primarily important for many compute-intensive applications, including reinforcement learning (RL), robot control, etc. This paper presents a novel theoretical framework for black-box optimization, in which our method performs stochastic update with the implicit natural gradient of an exponential-family distribution. Theoretically, we prove the convergence rate of our framework with full matrix update for convex functions. Our theoretical results also hold for continuous non-differentiable black-box functions. Our methods are very simple and contain less hyper-parameters than CMA-ES cite{hansen2006cma}. Empirically, our method with full matrix update achieves competitive performance compared with one of the state-of-the-art method CMA-ES on benchmark test problems. Moreover, our methods can achieve high optimization precision on some challenging test functions (e.g., $l_1$-norm ellipsoid test problem and Levy test problem), while methods with explicit natural gradient, i.e., IGO cite{ollivier2017information} with full matrix update can not. This shows the efficiency of our methods.
Why and how that deep learning works well on different tasks remains a mystery from a theoretical perspective. In this paper we draw a geometric picture of the deep learning system by finding its analogies with two existing geometric structures, the geometry of quantum computations and the geometry of the diffeomorphic template matching. In this framework, we give the geometric structures of different deep learning systems including convolutional neural networks, residual networks, recursive neural networks, recurrent neural networks and the equilibrium prapagation framework. We can also analysis the relationship between the geometrical structures and their performance of different networks in an algorithmic level so that the geometric framework may guide the design of the structures and algorithms of deep learning systems.
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