Recently mean field theory has been successfully used to analyze properties of wide, random neural networks. It gave rise to a prescriptive theory for initializing feedforward neural networks with orthogonal weights, which ensures that both the forward propagated activations and the backpropagated gradients are near isometries and as a consequence training is orders of magnitude faster. Despite strong empirical performance, the mechanisms by which critical initializations confer an advantage in the optimization of deep neural networks are poorly understood. Here we show a novel connection between the maximum curvature of the optimization landscape (gradient smoothness) as measured by the Fisher information matrix (FIM) and the spectral radius of the inputoutput Jacobian, which partially explains why more isometric networks can train much faster. Furthermore, given that orthogonal weights are necessary to ensure that gradient norms are approximately preserved at initialization, we experimentally investigate the benefits of maintaining orthogonality throughout training, and we conclude that manifold optimization of weights performs well regardless of the smoothness of the gradients. Moreover, we observe a surprising yet robust behavior of highly isometric initializations  even though such networks have a lower FIM condition number \emph{at initialization}, and therefore by analogy to convex functions should bemore »
Robust Regression via Model Based Methods
The mean squared error loss is widely used in many applications, including autoencoders, multitarget regression, and matrix factorization, to name a few. Despite computational advantages due to its differentiability, it is not robust to outliers. In contrast, ℓ𝑝 norms are known to be robust, but cannot be optimized via, e.g., stochastic gradient descent, as they are nondifferentiable. We propose an algorithm inspired by socalled modelbased optimization (MBO), which replaces a nonconvex objective with a convex model function and alternates between optimizing the model function and updating the solution. We apply this to robust regression, proposing SADM, a stochastic variant of the Online Alternating Direction Method of Multipliers (OADM) to solve the inner optimization in MBO. We show that SADM converges with the rate 𝑂(log𝑇/𝑇) . Finally, we demonstrate experimentally (a) the robustness of ℓ𝑝 norms to outliers and (b) the efficiency of our proposed modelbased algorithms in comparison with gradient methods on autoencoders and multitarget regression.
 Publication Date:
 NSFPAR ID:
 10313470
 Journal Name:
 Joint European Conference on Machine Learning and Knowledge Discovery in Databases
 Sponsoring Org:
 National Science Foundation
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