II with constant l >> Suppose the function f: R R is strongly convex and twice differentiable,
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II with constant l >">
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Suppose the function f: R"→ R is strongly convex and twice differentiable, i.e V2 f(x) > II with constant l > 0. Also, its gradient is Lipschitz continuous with constant L> 0, i.e. we have that Vf(x) - Vf(y)| ≤L||x-y|| for any z, y. • (5 pt) Prove f(y) ≥ f(x) + (Vf(x), y-x)+||-||² • (10 pt) Prove f(y) ≥ f(x) - Vf(x)||2 (hint: f(y)2 min, f(y)) (15 pt) Then if we run gradient descent for t iterations with step size a = prove it will give a linear convergence rate, i.e. f(x¹+¹) - f* ≤ (1-1)(f(x) - ƒ*) by using exact line search, Suppose the function f: R"→ R is strongly convex and twice differentiable, i.e V2 f(x) > II with constant l > 0. Also, its gradient is Lipschitz continuous with constant L> 0, i.e. we have that Vf(x) - Vf(y)| ≤L||x-y|| for any z, y. • (5 pt) Prove f(y) ≥ f(x) + (Vf(x), y-x)+||-||² • (10 pt) Prove f(y) ≥ f(x) - Vf(x)||2 (hint: f(y)2 min, f(y)) (15 pt) Then if we run gradient descent for t iterations with step size a = prove it will give a linear convergence rate, i.e. f(x¹+¹) - f* ≤ (1-1)(f(x) - ƒ*) by using exact line search,
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To prove fy geq fx 12 abla fx2 we can use the hint and consider the minimum value of fy over all y L... View the full answer
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Income Tax Fundamentals 2013
ISBN: 9781285586618
31st Edition
Authors: Gerald E. Whittenburg, Martha Altus Buller, Steven L Gill
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