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Title: Stochastic Geodesic Optimization for Neural Networks

Paper content:
We develop a novel and efficient algorithm for optimizing neural networks inspired by a recently proposed geodesic optimization algorithm.
Our algorithm, which we call Stochastic  Geodesic Optimization (SGeO), utilizes an adaptive coefficient on top of Polyak's Heavy Ball method effectively controlling the amount of weight put on the previous update to the parameters based on the change of direction in the optimization path.
Experimental results on strongly convex functions with Lipschitz gradients and deep Autoencoder benchmarks show that SGeO reaches lower errors than established first-order methods and competes well with lower or similar errors to a recent second-order method called K-FAC (Kronecker-Factored Approximate Curvature).
We also incorporate Nesterov style lookahead gradient into our algorithm (SGeO-N) and observe notable improvements.
First order methods such as Stochastic Gradient Descent (SGD) with Momentum (Sutskever et al., 2013) and their variants are the methods of choice for optimizing neural networks.
While there has been extensive work on developing second-order methods such as Hessian-Free optimization (Martens, 2010) and Natural Gradients (Amari, 1998; Martens & Grosse, 2015) , they have not been successful in replacing them due to their large per-iteration costs, in particular, time and memory.
Although Nesterov's accelerated gradient and its modifications have been very effective in deep neural network optimization (Sutskever et al., 2013) , some research have shown that Nesterov's method might perform suboptimal for strongly convex functions (Aujol et al., 2018) without looking at local geometry of the objective function.
Further, in order to get the best of both worlds, search for optimization methods which combine the the efficiency of first-order methods and the effectiveness of second-order updates is still underway.
In this work, we introduce an adaptive coefficient for the momentum term in the Heavy Ball method as an effort to combine first-order and second-order methods.
We call our algorithm Geodesic Optimization (GeO) and Stochastic Geodesic Optimization (SGeO) (for the stochastic case) since it is inspired by a geodesic optimization algorithm proposed recently (Fok et al., 2017) .
The adaptive coefficient effectively weights the momentum term based on the change in direction on the loss surface in the optimization process.
The change in direction can contribute as implicit local curvature information without resorting to the expensive second-order information such as the Hessian or the Fisher Information Matrix.
Our experiments show the effectiveness of the adaptive coefficient on both strongly-convex functions with Lipschitz gradients and general non-convex problems, in our case, deep Autoencoders.
GeO can speed up the convergence process significantly in convex problems and SGeO can deal with illconditioned curvature such as local minima effectively as shown in our deep autoencoder benchmark experiments.
SGeO has similar time-efficiency as first-order methods (e.g. Heavy Ball, Nesterov) while reaching lower reconstruction error.
Compared to second-order methods (e.g., K-FAC), SGeO has better or similar reconstruction errors while consuming less memory.
The structure of the paper is as follows: In section 2, we give a brief background on the original geodesic and contour optimization introduced in Fok et al. (2017) , neural network optimization methods and the conjugate gradient method.
In section 3, we introduce our adaptive coefficient specifically designed for strongly-convex problems and then modify it for general non-convex cases.
In section 4, we discuss some of the related work in the literature.
Section 5 illustrates the algorithm's performance on convex and non-convex benchmarks.
More details and insights regarding the algorithm and the experiments can be found in the Appendix.
We proposed a novel and efficient algorithm based on adaptive coefficients for the Heavy Ball method inspired by a geodesic optimization algorithm.
We compared SGeO against SGD with Nesterov's Momentum and regular momentum (Heavy Ball) and a recently proposed second-order method, K-FAC, on three deep autoencoder optimization benchmarks and three strongly convex functions with Lipschitz gradients.
We saw that SGeO is able to outperform all first-order methods that we compared to, by a notable margin.
SGeO is easy to implement and the computational overhead it has over the first-order methods, which is calculating the dot product, is marginal.
It can also perform as effectively as or better than second-order methods (here, K-FAC) without the need for expensive higher-order operations in terms of time and memory.
We believe that SGeO opens new and promising directions in high dimensional optimization research and in particular, neural network optimization.
We are working on applying SGeO to other machine learning paradigms such as CNNs, RNNs and Reinforcement Learning.
It remains to analyse the theoretical properties of SGeO such as its convergence rate in convex and non-convex cases which we leave for future work.

assistant: text: We utilize an adaptive coefficient on top of regular momentum inspired by geodesic optimization which significantly speeds up training in both convex and non-convex functions.