## Description

1. Age regression: Train an age regressor that analyzes a (48 × 48 = 2304)-pixel grayscale face image

and outputs a real number ˆy that estimates how old the person is (in years). Your regressor should be

implemented using linear regression. The training and testing data are available here:

• https://s3.amazonaws.com/jrwprojects/age_regression_Xtr.npy

• https://s3.amazonaws.com/jrwprojects/age_regression_ytr.npy

• https://s3.amazonaws.com/jrwprojects/age_regression_Xte.npy

• https://s3.amazonaws.com/jrwprojects/age_regression_yte.npy

Note: you must complete this problem using only linear algebraic operations in numpy – you may not

use any off-the-shelf linear regression software, as that would defeat the purpose.

(a) Analytical solution [20 points]: Compute the optimal weights w = (w1, . . . , w2304) and bias

term b for a linear regression model by deriving the expression for the gradient of the cost function

w.r.t. w and b, setting it to 0, and then solving. The cost function is

fMSE(w, b) = 1

2n

Xn

i=1

(ˆy

(i) − y

(i)

)

2

where ˆy = g(x; w, b) = x

>w + b and n is the number of examples in the training set

Dtr = {(x

(1), y(1)), . . . ,(x

(n)

, y(n)

)}, each x

(i) ∈ R

2304 and each y

(i) ∈ {0, 1}. After optimizing w

and b only on the training set, compute and report the cost fMSE on the training set Dtr and

(separately) on the testing set Dte. Suggestion: to solve for w and b simultaneously, use the

trick shown in class whereby each image (represented as a vector x) is appended with a constant

1 term (to yield an appended representation x˜). Then compute the optimal w˜ (comprising the

original w and an appended b term) using the closed formula:

w˜ =

X˜ X˜ >

−1

Xy˜

For appending, you might find the functions np.hstack, np.vstack, np.atleast 2d useful. After

optimizing w˜ and b (using ˜fMSE), compute and report the cost fMSE on the training set Dtr and

(separately) the testing set Dte.

(b) Gradient descent [25 points]: Pick a random starting value for w ∈ R

2304 and b ∈ R and

a small learning rate (e.g., = .001). (In my code, I sampled each component of w and b

from a Normal distribution with standard deviation 0.01; use np.random.randn). Then, using

the expression for the gradient of the cost function, iteratively update w, b to reduce the cost

fMSE(w, b). Stop after conducting T gradient descent iterations (I suggest T = 5000 with a step

size (aka learning rate) of = 0.003). After optimizing w and b only on the training set, compute

and report the cost fMSE on the training set Dtr and (separately) on the testing set Dte. After

optimizing w and b (using ˜fMSE), compute and report the cost fMSE on the training set Dtr and

(separately) the testing set Dte.

Note: as mentioned during class, on this particular dataset it would take a very long time using

gradient descent to reach weights as the w found by the analytical solution. For T = 5000, your

training cost on part (b) will be higher than for part (a). However, the testing cost should actually

be lower since the relatively small number of gradient descent steps prevents w from growing too

large and hence acts as an implicit regularizer.

1

(c) Regularization [15 points]: Same as (b) above, but change the cost function to include a

penalty for |w|

2 growing too large:

˜fMSE(w) = 1

2n

Xn

i=1

(ˆy

(i) − y

(i)

)

2 +

α

2n

w>w

where α ∈ R

+. Set α = 0.1 (this worked well for me) and then optimize ˜fMSE w.r.t. w and

b. After optimizing w and b (using ˜fMSE), compute and report the cost fMSE (without the L2

term) on the training set Dtr and (separately) the testing set Dte. Important: the regularization

should be applied only to the w, not the b. I suggest a regularization strength of α = 0.1.

Note: as mentioned during class, since part (b) already provides implicit regularization by limiting

the number of gradient descent steps (to T = 5000), you should not expect to see much (or any)

difference between parts (c) and (b) on this dataset. In general, however, the L2 regularization

term can make a big difference.

(d) Visualizing the machine’s behavior [10 points]: After training the regressors in parts (a),

(b), and (c), create a 48 × 48 image representing the learned weights w (without the b term)

from each of the different training methods. Use plt.imshow(). How are the weight vectors from

the different methods different? Next, using the regressor in part (c), predict the ages of all the

images in the test set and report the RMSE (in years). Then, show the top 5 most egregious

errors, i.e., the test images whose ground-truth label y is farthest from your machine’s estimate

yˆ. Include the images, along with associated y and ˆy values, in a PDF. 4

Submission: Put your solution in a Python file called homework2 WPIUSERNAME.py

(or homework2 WPIUSERNAME1 WPIUSERNAME2.py for teams), and show the most egregious errors for part (c)

in homework2 errors WPIUSERNAME.pdf.

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