Gradient descent

Summary

In mathematics, gradient descent (also often called steepest descent) is a iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, because this is the direction of steepest descent. Conversely, stepping in the direction of the gradient will lead to a local maximum of that function; the procedure is then known as gradient ascent. It is particularly useful in machine learning for minimizing the cost or loss function. Gradient descent should not be confused with local search algorithms, although both are iterative methods for optimization. Gradient descent is generally attributed to Augustin-Louis Cauchy, who first suggested it in 1847. Jacques Hadamard independently proposed a similar method in 1907. Its convergence properties for non-linear optimization problems were first studied by Haskell Curry in 1944, with

Official source

https://en.wikipedia.org/wiki/Gradient_descent

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Stochastic Zeroth-Order Optimisation Algorithms with Variance Reduction

Ahmad Ajalloeian

Introduction of optimisation problems in which the objective function is black box or obtaining the gradient is infeasible, has recently raised interest in zeroth-order optimisation methods. As an example finding adversarial examples for Deep Learning models (Chen et al. (2017); Moosavi-Dezfooli et al. (2016)) is one of the most common applications in which zeroth-order methods could be used. These optimisation methods use only function values at certain points to estimate the gradient. Most current approaches iteratively sample a random search direction along which they compute an estimation of the gradient (Nesterov and Spokoiny (2017); Conn et al. (2009); Wibisono et al. (2012)). However, due to the high variance in the search direction, these methods usually need d times more iterations than the standard gradient methods, where d is the dimensionality of the problem. So it seems that the main effort for improving the zeroth-order methods should be in reducing the variance of the gradient estimate. In this work we will analyse the gradient-free oracle which uses random directions sampled form a Gaussian distribution. Our analysis shows that in smooth and strongly convex setting, we have a convergence rate of O( d/T) which clearly shows the dependency to the dimension of the problem. Furthermore we propose some variance reduction methods to make the zeroth-order optimisation faster. We experiment our proposed methods in Python to compare their convergence in stochastic and non-stochastic setting. Our empirical results show that in a setting that number of allowed function evaluation is fixed, using a variance reduction method (e.g. momentum) can make the convergence of zeroth-order methods happen faster.

2019

We describe the first gradient methods on Riemannian manifolds to achieve accelerated rates in the non-convex case. Under Lipschitz assumptions on the Riemannian gradient and Hessian of the cost function, these methods find approximate first-order critical points faster than regular gradient descent. A randomized version also finds approximate second-order critical points. Both the algorithms and their analyses build extensively on existing work in the Euclidean case. The basic operation consists in running the Euclidean accelerated gradient descent method (appropriately safe-guarded against non-convexity) in the current tangent space, then moving back to the manifold and repeating. This requires lifting the cost function from the manifold to the tangent space, which can be done for example through the Riemannian exponential map. For this approach to succeed, the lifted cost function (called the pullback) must retain certain Lipschitz properties. As a contribution of independent interest, we prove precise claims to that effect, with explicit constants. Those claims are affected by the Riemannian curvature of the manifold, which in turn affects the worst-case complexity bounds for our optimization algorithms.

SPRINGER2022

Gradient flow dynamics of shallow ReLU networks for square loss and orthogonal inputs

Nicolas Henri Bernard Flammarion, Loucas Pillaud-Vivien

The training of neural networks by gradient descent methods is a cornerstone of the deep learning revolution. Yet, despite some recent progress, a complete theory explaining its success is still missing. This article presents, for orthogonal input vectors, a precise description of the gradient flow dynamics of training one-hidden layer ReLU neural networks for the mean squared error at small initialisation. In this setting, despite non-convexity, we show that the gradient flow converges to zero loss and characterise its implicit bias towards minimum variation norm. Furthermore, some interesting phenomena are highlighted: a quantitative description of the initial alignment phenomenon and a proof that the process follows a specific saddle to saddle dynamics.

2022