Gradient Descent Converges to Minimizers: Optimal and Adaptive Step-Size RulesAs mentioned in Chap. 3 , gradient descent (GD) and its variants provide the core optimization methodology in machine learning problems. Given a C 1 or C 2 function \\(f: \\mathbb {R}^{n} ightarrow \\mathbb ...
Gradient descent only converges to minimizers. Conference on learning theory, 1246-1257, 2016.[11] Simon S. Du, Chi Jin, Jason D. Lee, Michael I. Jordan, Barnabas Poczos, Aarti Singh. Gradient descent can take exponential time to escape saddle points. Advances in neural information ...
From its origin, it has been an important theoretical task for GD methods (as well as other iterative methods such as Newton’s) to ensure that the sequence \(\{x_n\}\) in (1) converges to a (global) minimum point. We now argue that from a practical viewpoint, theoretical guarantee...
This is an idealization of the usual way to train neural networks with a large hidden layer. We show that, when initialized correctly and in the many-particle limit, this gradient flow, although non-convex, converges to global minimizers. The proof involves Wasserstein gradient flows, a by-...
We can now describe the main result from our recent work with Lénaïc [13]: under assumptions described below, for the function FF defined above, if the Wasserstein gradient flow converges to a measure, this measure has to be a global minimum of FF (note that we cannot prove it is ...
classical regime, n > D, where D is the fixed number of weights; consistency (informally the expected error of the empirical minimizer converges to the best in the class) and generalization (the empirical error of the minimizer converges to the expected error of the minimizer) are ...
update. For convex error surface, batch gradient descent method converges to global minimum. For nonconvex surface, this method converges tolocal minimum. For the N number of samples and M number of features, the computational complexity per iteration of batch gradient descent method will be O(...
Newton's method, often called the tangent method, is based on the tangent of the current position to determine the next position. Compared with the gradient descent method with first-order convergence, Newton's method has second-order convergence with a fast convergence speed. However, the invers...
The main convergence result is obtained by defining a projected gradient, and proving that the gradient projection method forces the sequence of projected gradients to zero. A consequence of this result is that if the gradient projection method converges to a nondegenerate point of a linearly ...
2.1Stochastic gradient descent (SGD) Given a dataset withmtraining examples{ti}i∈[m], i.e.,mtuples if these examples are stored as a table in a database, typical ML tasks essentially solve an optimization problem of minimizing a finite sum overmtraining examples with respect to modelx. ...