D20 - 7.2 The Naive Solution Gradient Ascent The simplest...

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7.2 The Naive Solution: Gradient Ascent The simplest numerical solution of a convex optimisation problem is obtained by gradient ascent, sometimes known as the steepest ascent algorithm. The algorithm starts with an initial estimate for the solution, denoted by α 0 , and then iteratively updates the vector following the steepest ascent path, that is moving in the direction of the gradient of W ( α ) evaluated at the position α t for update t + 1. At each iteration the direction of the update is determined by the steepest ascent strategy but the length of the step still has to be fixed. The length of the update is known as the learning rate . In the sequential or stochastic version, this strategy is approximated by evaluating the gradient for just one pattern at a time, and hence updating a single component α i t by the increment where the parameter η is the learning rate. If η is chosen carefully, the objective function will increase monotonically, and the average direction approximates the local gradient. One can
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