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; [||Vf; (x)||² | xk] ≤0² + L ||xk − x* ||². Prove the following statements: 1. If σ > 0 and L = 0, SGD with step size ŋk satisfies | E || xo - Ef (zk) - f*] ≤ =0 (1) j=0 where Σj=0jxj Zk= (2) ΣΕ In particular, E[f (zk) - f*] converges to 0 if and only if Σ, n; = ∞ and 2. If σ > 0 and L > 0, SGD with a constant step size n satisfies = 0. E||xk+1 - x* ||² ≤ (1 - 2nμ+n²L)*E ||xo-x* ||² + (1 − 2nµ + n²L); - ησε 2μ-nL (3) What is the restriction on the stepsize? 3. Let us observe by definition, SGD with step size n satisfies: ||K+1 – x = ||xk - xu t ng Vi(x)|| − 20k (k – x*, Vf(x)). (4) Derive the optimal step size and comment on it.
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; [||Vf; (x)||² | xk] ≤0² + L ||xk − x* ||². Prove the following statements: 1. If σ > 0 and L = 0, SGD with step size ŋk satisfies | E || xo - Ef (zk) - f*] ≤ =0 (1) j=0 where Σj=0jxj Zk= (2) ΣΕ In particular, E[f (zk) - f*] converges to 0 if and only if Σ, n; = ∞ and 2. If σ > 0 and L > 0, SGD with a constant step size n satisfies = 0. E||xk+1 - x* ||² ≤ (1 - 2nμ+n²L)*E ||xo-x* ||² + (1 − 2nµ + n²L); - ησε 2μ-nL (3) What is the restriction on the stepsize? 3. Let us observe by definition, SGD with step size n satisfies: ||K+1 – x = ||xk - xu t ng Vi(x)|| − 20k (k – x*, Vf(x)). (4) Derive the optimal step size and comment on it.
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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![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; [||Vf; (x)||² | xk] ≤0² + L ||xk − x* ||².
Prove the following statements:
1. If σ > 0 and L = 0, SGD with step size ŋk satisfies
| E || xo -
Ef (zk) - f*] ≤
=0
(1)
j=0
where
Σj=0jxj
Zk=
(2)
ΣΕ
In particular, E[f (zk) - f*] converges to 0 if and only if Σ, n; = ∞ and
2. If σ > 0 and L > 0, SGD with a constant step size n satisfies
= 0.
E||xk+1 - x* ||² ≤ (1 - 2nμ+n²L)*E ||xo-x* ||² + (1 − 2nµ + n²L);
-
ησε
2μ-nL
(3)
What is the restriction on the stepsize?
3. Let us observe by definition, SGD with step size n satisfies:
||K+1 – x = ||xk - xu t ng Vi(x)|| − 20k (k – x*, Vf(x)).
(4)
Derive the optimal step size and comment on it.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F08792f8d-91a6-4520-a2bb-5d9564b28bda%2F8c4ec1af-62f8-447c-8a1b-99f03fbd9528%2F4yzfbg_processed.png&w=3840&q=75)
Transcribed Image Text: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; [||Vf; (x)||² | xk] ≤0² + L ||xk − x* ||².
Prove the following statements:
1. If σ > 0 and L = 0, SGD with step size ŋk satisfies
| E || xo -
Ef (zk) - f*] ≤
=0
(1)
j=0
where
Σj=0jxj
Zk=
(2)
ΣΕ
In particular, E[f (zk) - f*] converges to 0 if and only if Σ, n; = ∞ and
2. If σ > 0 and L > 0, SGD with a constant step size n satisfies
= 0.
E||xk+1 - x* ||² ≤ (1 - 2nμ+n²L)*E ||xo-x* ||² + (1 − 2nµ + n²L);
-
ησε
2μ-nL
(3)
What is the restriction on the stepsize?
3. Let us observe by definition, SGD with step size n satisfies:
||K+1 – x = ||xk - xu t ng Vi(x)|| − 20k (k – x*, Vf(x)).
(4)
Derive the optimal step size and comment on it.
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