Exercise · ZW-CONVERGENCE-WITH-LIPSCHITZ-GRADIENT-004 · ★☆☆ Basic
Which inequality follows from the Lipschitz condition on the gradient?
Choose one option.
Show answer
- B. \( f(y)\le f(x)+\nabla f(x)^T(y-x)+\frac{L}{2}\|y-x\|^2 \)
Explanation
This is the form of the descent lemma derived from the Lipschitz condition. The option with the reversed inequality and no quadratic term does not follow from the hypothesis.
Review the concept: Convergence with Lipschitz gradient
Result guaranteeing that gradient descent with a suitable step size converges for differentiable functions whose gradient \( \nabla f \) is Lipschitz continuous with constant \( L \).
\[f(x_k)-f^* \le \frac{L\|x_0-x^*\|^2}{2k}\]
More practice problems: convergence with lipschitz gradient
- ZW-CONVERGENCE-WITH-LIPSCHITZ-GRADIENT-001 What does it mean for ∇ f to be Lipschitz with constant L ?
- ZW-CONVERGENCE-WITH-LIPSCHITZ-GRADIENT-002 If ∇ f is Lipschitz with constant L , which gradient step is the standard choice?
- ZW-CONVERGENCE-WITH-LIPSCHITZ-GRADIENT-003 What is the function-value convergence rate for convex gradient descent with step 1/L ?
- ZW-CONVERGENCE-WITH-LIPSCHITZ-GRADIENT-005 If ∇ f is Lipschitz, what does convexity of f add for the descent sequence?
- ZW-CONVERGENCE-WITH-LIPSCHITZ-GRADIENT-006 A function has L=4 . What step t is safe under the rule t=1/L ?
- ZW-CONVERGENCE-WITH-LIPSCHITZ-GRADIENT-007 For f(x)=x^2 on ℝ , what is the Lipschitz constant L of the gradient?
- ZW-CONVERGENCE-WITH-LIPSCHITZ-GRADIENT-008 If ∇ f is Lipschitz with L=3 and \|x_0-x^*\|=2 , what bound does the O(1/k) rate give for k=6 ?
- ZW-CONVERGENCE-WITH-LIPSCHITZ-GRADIENT-009 For f(x)=(1)/(2)x^2 with t=1/L , what is x_1 if x_0=4 ?
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