Constantin Octavian Puiu

Constantin Octavian Puiu

In optimization for Machine learning (ML), it is typical that curvature-matrix (CM) estimates rely on an exponential average (EA) of local estimates (giving EA-CM algorithms). This approach has little principled justification, but is very often used in practice. In this paper, we draw a connection between EA-CM algorithms and what we call a "Wake of Quadratic regularized models". The outlined connection allows us to understand what EA-CM algorithms are doing from an optimization perspective. Generalizing from the established connection, we propose a new family of algorithms, "KL-Divergence Wake-Regularized Models" (KLD-WRM). We give three different practical instantiations of KLD-WRM, and show numerically that these outperform K-FAC on MNIST.

Constantin Octavian Puiu

K-FAC (arXiv:1503.05671, arXiv:1602.01407) is a tractable implementation of Natural Gradient (NG) for Deep Learning (DL), whose bottleneck is computing the inverses of the so-called ``Kronecker-Factors'' (K-factors). RS-KFAC (arXiv:2206.15397) is a K-FAC improvement which provides a cheap way of estimating the K-factors inverses. In this paper, we exploit the exponential-average construction paradigm of the K-factors, and use online numerical linear algebra techniques to propose an even cheaper (but less accurate) way of estimating the K-factors inverses. In particular, we propose a K-factor inverse update which scales linearly in layer size. We also propose an inverse application procedure which scales linearly as well (the one of K-FAC scales cubically and the one of RS-KFAC scales quadratically). Overall, our proposed algorithm gives an approximate K-FAC implementation whose preconditioning part scales linearly in layer size (compare to cubic for K-FAC and quadratic for RS-KFAC). Importantly however, this update is only applicable in some circumstances (typically for all FC layers), unlike the RS-KFAC approach (arXiv:2206.15397). Numerical results show RS-KFAC's inversion error can be reduced with minimal CPU overhead by adding our proposed update to it. Based on the proposed procedure, a correction to it, and RS-KFAC, we propose three practical algorithms for optimizing generic Deep Neural Nets. Numerical results show that two of these outperform RS-KFAC for any target test accuracy on CIFAR10 classification with a slightly modified version of VGG16_bn. Our proposed algorithms achieve 91$\%$ test accuracy faster than SENG (the state of art implementation of empirical NG for DL; arXiv:2006.05924) but underperform it for higher test-accuracy.

Constantin Octavian Puiu

K-FAC is a successful tractable implementation of Natural Gradient for Deep Learning, which nevertheless suffers from the requirement to compute the inverse of the Kronecker factors (through an eigen-decomposition). This can be very time-consuming (or even prohibitive) when these factors are large. In this paper, we theoretically show that, owing to the exponential-average construction paradigm of the Kronecker factors that is typically used, their eigen-spectrum must decay. We show numerically that in practice this decay is very rapid, leading to the idea that we could save substantial computation by only focusing on the first few eigen-modes when inverting the Kronecker-factors. Importantly, the spectrum decay happens over a constant number of modes irrespectively of the layer width. This allows us to reduce the time complexity of K-FAC from cubic to quadratic in layer width, partially closing the gap w.r.t. SENG (another practical Natural Gradient implementation for Deep learning which scales linearly in width). Randomized Numerical Linear Algebra provides us with the necessary tools to do so. Numerical results show we obtain $\approx2.5\times$ reduction in per-epoch time and $\approx3.3\times$ reduction in time to target accuracy. We compare our proposed K-FAC sped-up versions SENG, and observe that for CIFAR10 classification with VGG16_bn we perform on par with it.