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This method is only feasible when the projection is efficiently computable on a computer.

## What I Shared at the CONSPIRE Conference (1/2): Certainty, Uncertainty, and the Path of Descent

Under suitable assumptions, this method converges. This method is a specific case of the forward-backward algorithm for monotone inclusions which includes convex programming and variational inequalities. Another extension of gradient descent is due to Yurii Nesterov from , [12] and has been subsequently generalized. He provides a simple modification of the algorithm that enables faster convergence for convex problems.

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Nevertheless, there is the opportunity to improve the algorithm by reducing the constant factor. The optimized gradient method OGM [14] reduces that constant by a factor of two and is an optimal first-order method for large-scale problems. For constrained or non-smooth problems, Nesterov's FGM is called the fast proximal gradient method FPGM , an acceleration of the proximal gradient method. Yet another extension, that reduces the risk of getting stuck in a local minimum, as well as speeds up the convergence considerably in cases where the process would otherwise zig-zag heavily, is the momentum method , which uses a momentum term in analogy to "the mass of Newtonian particles that move through a viscous medium in a conservative force field".

The basic intuition behind gradient descent can be illustrated by a hypothetical scenario. A person is stuck in the mountains and is trying to get down i. There is heavy fog such that visibility is extremely low.

Therefore, the path down the mountain is not visible, so they must use local information to find the minima. They can use the method of gradient descent, which involves looking at the steepness of the hill at their current position, then proceeding in the direction with the steepest descent i.

If they were trying to find the top of the mountain i. Using this method, they would eventually find their way down the mountain. However, assume also that the steepness of the hill is not immediately obvious with simple observation, but rather it requires a sophisticated instrument to measure, which the person happens to have at the moment. It takes quite some time to measure the steepness of the hill with the instrument, thus they should minimize their use of the instrument if they wanted to get down the mountain before sunset. The difficulty then is choosing the frequency at which they should measure the steepness of the hill so not to go off track.

In this analogy, the person represents the algorithm, and the path taken down the mountain represents the sequence of parameter settings that the algorithm will explore. The steepness of the hill represents the slope of the error surface at that point. The instrument used to measure steepness is differentiation the slope of the error surface can be calculated by taking the derivative of the squared error function at that point.

The direction they choose to travel in aligns with the gradient of the error surface at that point. The amount of time they travel before taking another measurement is the learning rate of the algorithm. From Wikipedia, the free encyclopedia. For the analytical method called "steepest descent", see Method of steepest descent.

Function ; import static java. Bertsekas, Nonlinear Programming , Athena Scientific , 2nd edition, pp. Paris Mathematical Programming, Series A. Berlin, Heidelberg: Springer. Adaptive filter theory.

Pearson Education India, Functional Analysis 2nd ed. Pergamon Press. New York: Cambridge University Press. Springer Vieweg.

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Combettes and J. Bauschke, R. Burachik , P. Combettes, V. Elser, D. Luke, and H. Wolkowicz, Editors , pp. Springer, New York, Journal of Complexity. Neural Networks.

Archived from the original PDF on 8 May Retrieved 17 October Willamette University. Retrieved 2 October Optimization : Algorithms , methods , and heuristics.

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Unconstrained nonlinear. Golden-section search Interpolation methods Line search Nelder—Mead method Successive parabolic interpolation. Trust region Wolfe conditions. Newton's method. Constrained nonlinear. Barrier methods Penalty methods. Augmented Lagrangian methods Sequential quadratic programming Successive linear programming.

Convex optimization. Cutting-plane method Reduced gradient Frank—Wolfe Subgradient method.

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Affine scaling Ellipsoid algorithm of Khachiyan Projective algorithm of Karmarkar. Can you spell these 10 commonly misspelled words? Build a city of skyscrapers—one synonym at a time. Login or Register. Save Word. Log In. Definition of descent. Keep scrolling for more. Examples of descent in a Sentence The book describes his descent into a deep depression after the death of his wife.

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