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(i) Gradient Descent (GD) (also referred to as batch gradient descent): here we use the full gradi- ent, as in we take the average over all n terms, so our update rule is: B(k+1) = B(k) EVL:(B(k), k = 0,1,2, .... n i=1 Page 5 (ii) Stochastic Gradient Descent (SGD): instead of considering all n terms, at the k-th step we choose an index i̟ randomly from {1,.., n}, and update B(k+1) = B(k) – ak V Lip (B(k)), k = 0, 1, 2, .... Here, we are approximating the full gradient VL(B) using VL, (B). iii) Mini-Batch Gradient Descent: GD (using all terms) and SGD (using a single term) represents the two possible extremes. In mini-batch GD we choose batches of size 1 < B < n randomly at each step, call their indices {ik,, ik2 ..., ikg}, and then we update B Blk+1) = B(k) _ ak VLi,(8(*), k = 0, 1, 2, . .., В j=1 WI

(i) Gradient Descent (GD) (also referred to as batch gradient descent): here we use the full gradi- ent, as in we take the average over all n terms, so our update rule is: B(k+1) = B(k) EVL:(B(k), k = 0,1,2, .... n i=1 Page 5 (ii) Stochastic Gradient Descent (SGD): instead of considering all n terms, at the k-th step we choose an index i̟ randomly from {1,.., n}, and update B(k+1) = B(k) – ak V Lip (B(k)), k = 0, 1, 2, .... Here, we are approximating the full gradient VL(B) using VL, (B). iii) Mini-Batch Gradient Descent: GD (using all terms) and SGD (using a single term) represents the two possible extremes. In mini-batch GD we choose batches of size 1 < B < n randomly at each step, call their indices {ik,, ik2 ..., ikg}, and then we update B Blk+1) = B(k) _ ak VLi,(8(*), k = 0, 1, 2, . .., В j=1 WI

Advanced Engineering Mathematics
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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(g) It should be obvious that a closed form expression for Bridge exists. Write down the closed form expression, and compute the exact numerical value on the training dataset with ø = 0.5. What to submit: Your working, and a print out of the value of the ridge solution based on (X_train, Y_train). Include a screen shot of any code used for this section and a copy of your python code in solutions.py. We will now solve the ridge problem but using numerical techniques. As noted in the lectures, there are a few variants of gradient descent that we will briefly outline here. Recall that in gradient descent our update rule is Blk+1) B(k) – akVL(B(k), k = 0, 1, 2, ... , = - where L(B) is the loss function that we are trying to minimize. In machine learning, it is often the case that the loss function takes the form L(B) 1 L:(B), i=1 i.e. the loss is an average of n functions that we have lablled L;. It then follows that the gradient is also an average of the form VL(B) = EVL:(3). i=1 We can now define some popular variants of gradient descent . (i) Gradient Descent (GD) (also referred to as batch gradient descent): here we use the full gradi- ent, as in we take the average over all n terms, so our update rule is: Blk+1) = B(k) . EVL:(B(k), k = 0,1, 2, .... i=: Page 5 (ii) Stochastic Gradient Descent (SGD): instead of considering all n terms, at the k-th step we choose an index ir randomly from {1,..., n}, and update B(k+1) = B(k) – akVLi, (B(k), k = 0, 1,2, .... Here, we are approximating the full gradient VL(B) using VLi, (B). (iii) Mini-Batch Gradient Descent: GD (using all terms) and SGD (using a single term) represents the two possible extremes. In mini-batch GD we choose batches of size 1 < B < n randomly at each step, call their indices {ik, ik2... , ikp}, and then we update B Blk+1) = B(k) В j=1 VLi, (B(k), k = 0, 1, 2, ..., so we are still approximating the full gradient but using more than a single element as is done in SGD.
Transcribed Image Text:(g) It should be obvious that a closed form expression for Bridge exists. Write down the closed form expression, and compute the exact numerical value on the training dataset with ø = 0.5. What to submit: Your working, and a print out of the value of the ridge solution based on (X_train, Y_train). Include a screen shot of any code used for this section and a copy of your python code in solutions.py. We will now solve the ridge problem but using numerical techniques. As noted in the lectures, there are a few variants of gradient descent that we will briefly outline here. Recall that in gradient descent our update rule is Blk+1) B(k) – akVL(B(k), k = 0, 1, 2, ... , = - where L(B) is the loss function that we are trying to minimize. In machine learning, it is often the case that the loss function takes the form L(B) 1 L:(B), i=1 i.e. the loss is an average of n functions that we have lablled L;. It then follows that the gradient is also an average of the form VL(B) = EVL:(3). i=1 We can now define some popular variants of gradient descent . (i) Gradient Descent (GD) (also referred to as batch gradient descent): here we use the full gradi- ent, as in we take the average over all n terms, so our update rule is: Blk+1) = B(k) . EVL:(B(k), k = 0,1, 2, .... i=: Page 5 (ii) Stochastic Gradient Descent (SGD): instead of considering all n terms, at the k-th step we choose an index ir randomly from {1,..., n}, and update B(k+1) = B(k) – akVLi, (B(k), k = 0, 1,2, .... Here, we are approximating the full gradient VL(B) using VLi, (B). (iii) Mini-Batch Gradient Descent: GD (using all terms) and SGD (using a single term) represents the two possible extremes. In mini-batch GD we choose batches of size 1 < B < n randomly at each step, call their indices {ik, ik2... , ikp}, and then we update B Blk+1) = B(k) В j=1 VLi, (B(k), k = 0, 1, 2, ..., so we are still approximating the full gradient but using more than a single element as is done in SGD.
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