(1) Let f be a differentiable and μ-strongly-convex function whose minimum is achieved at x*. Let us assume that the variance on the gradients is controlled: There exists σ > 0 and L≥ 0 such that E; ||V fi(xk) ||² ≤ 0² + L ||xk - x* ||². Prove the following statements: 1. If > 0 and L = 0, SGD with step size n satisfies - x+ j=0 Elf (zk) - f+]≤ 2Σj-onj where Στο Zk In particular, E[ƒ (zk) - f*] converges to 0 if and only if Σ; n; = ∞ and = 0. ล 2. If σ > 0 and L > 0, SGD with a constant step size n satisfies Ek+1−x" ? < (1 -n+r)*Elo - x + (1 - nutrinok(k+2), (3) Is this an informative upper bound? Could you guess which condition has to hold to derive an informative upper bound? 3. Let us observe by definition, SGD with step size nk satisfies: || xk+1− x 2 = || x − x 2 + n Vf(x)||2 – 27k (xk − x*, Vfi(x)). - x+ Derive the optimal step size and comment on it. - (4)
(1) Let f be a differentiable and μ-strongly-convex function whose minimum is achieved at x*. Let us assume that the variance on the gradients is controlled: There exists σ > 0 and L≥ 0 such that E; ||V fi(xk) ||² ≤ 0² + L ||xk - x* ||². Prove the following statements: 1. If > 0 and L = 0, SGD with step size n satisfies - x+ j=0 Elf (zk) - f+]≤ 2Σj-onj where Στο Zk In particular, E[ƒ (zk) - f*] converges to 0 if and only if Σ; n; = ∞ and = 0. ล 2. If σ > 0 and L > 0, SGD with a constant step size n satisfies Ek+1−x" ? < (1 -n+r)*Elo - x + (1 - nutrinok(k+2), (3) Is this an informative upper bound? Could you guess which condition has to hold to derive an informative upper bound? 3. Let us observe by definition, SGD with step size nk satisfies: || xk+1− x 2 = || x − x 2 + n Vf(x)||2 – 27k (xk − x*, Vfi(x)). - x+ Derive the optimal step size and comment on it. - (4)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![(1)
Let f be a differentiable and μ-strongly-convex function whose minimum is achieved at x*. Let us assume that the
variance on the gradients is controlled: There exists σ > 0 and L≥ 0 such that E; ||V fi(xk) ||² ≤ 0² + L ||xk - x* ||². Prove
the following statements:
1. If > 0 and L = 0, SGD with step size n satisfies
-
x+
j=0
Elf (zk) - f+]≤
2Σj-onj
where
Στο
Zk
In particular, E[ƒ (zk) - f*] converges to 0 if and only if Σ; n; = ∞ and
= 0.
ล
2. If σ > 0 and L > 0, SGD with a constant step size n satisfies
Ek+1−x" ? < (1 -n+r)*Elo - x + (1 - nutrinok(k+2),
(3)
Is this an informative upper bound? Could you guess which condition has to hold to derive an informative upper
bound?
3. Let us observe by definition, SGD with step size nk satisfies:
|| xk+1− x 2 = || x − x 2 + n Vf(x)||2 – 27k (xk − x*, Vfi(x)).
-
x+
Derive the optimal step size and comment on it.
-
(4)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F08792f8d-91a6-4520-a2bb-5d9564b28bda%2F227d50b3-d998-4652-8cb5-56597efe9b4f%2Fr3ugotp_processed.png&w=3840&q=75)
Transcribed Image Text:(1)
Let f be a differentiable and μ-strongly-convex function whose minimum is achieved at x*. Let us assume that the
variance on the gradients is controlled: There exists σ > 0 and L≥ 0 such that E; ||V fi(xk) ||² ≤ 0² + L ||xk - x* ||². Prove
the following statements:
1. If > 0 and L = 0, SGD with step size n satisfies
-
x+
j=0
Elf (zk) - f+]≤
2Σj-onj
where
Στο
Zk
In particular, E[ƒ (zk) - f*] converges to 0 if and only if Σ; n; = ∞ and
= 0.
ล
2. If σ > 0 and L > 0, SGD with a constant step size n satisfies
Ek+1−x" ? < (1 -n+r)*Elo - x + (1 - nutrinok(k+2),
(3)
Is this an informative upper bound? Could you guess which condition has to hold to derive an informative upper
bound?
3. Let us observe by definition, SGD with step size nk satisfies:
|| xk+1− x 2 = || x − x 2 + n Vf(x)||2 – 27k (xk − x*, Vfi(x)).
-
x+
Derive the optimal step size and comment on it.
-
(4)
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