Exercise · ZW-CONVERGENCE-WITH-LIPSCHITZ-GRADIENT-010 · ★★★ Advanced
Let \( f(x)=2x^2 \) on \( \mathbb{R} \), with \( x_0=1 \) and step \( t=1/L \). What is \( x_1 \)?
Choose one option.
Show answer
- D. \( 0 \)
Explanation
Here \( \nabla f(x)=4x \), so \( L=4 \) and \( t=1/4 \). Then \( x_1=1-\frac{1}{4}\cdot 4=0 \). Using \( L=2 \) would give \( x_1=-1 \), a typical mistake.
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-004 Which inequality follows from the Lipschitz condition on the gradient?
- 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 ?