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Optimization Part II. G.Anuradha. Review of previous lecture- Steepest Descent. Choose the next step so that the function decreases:. For small changes in x we can approximate F ( x ):. where. If we want the function to decrease:. We can maximize the decrease by choosing:. Example.

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## Optimization Part II

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**Optimization Part II**G.Anuradha**Review of previous lecture-Steepest Descent**Choose the next step so that the function decreases: For small changes in x we can approximate F(x): where If we want the function to decrease: We can maximize the decrease by choosing:**Necessary and sufficient conditions for a function with**single variable**Functions with two variables**Sufficient conditions Necessary conditions**Effect of learning rate**More the learning rate the trajectory becomes oscillatory. This will make the algorithm unstable The upper limit for learning rates can be set for quadratic functions**Stable Learning Rates (Quadratic)**Stability is determined by the eigenvalues of this matrix. Eigenvalues of [I - aA]. (li - eigenvalue of A) Stability Requirement:**Newton’s Method**Take the gradient of this second-order approximation and set it equal to zero to find the stationary point:**This is used for finding line minimization methods and their**stopping criteria • Initial bracketing • Line searches • Newton’s method • Secant method • Sectioning method**Initial Bracketing**• Helps in finding the range which contains the relative minimum • Bracketing some assumed minimum in the starting interval is required • Two schemes are used for this purpose

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