EE425X程序设计代做、program编程代做、代写Java，Python程序

日期：2021-02-20 11:03

Homework 1b: Linear Regression part 2.

EE425X - Machine Learning: A Signal Processing Perspective

Homework 1 focused on learning the parameter θ for linear regression. In this homework we will first understand

how to use the learnt parameter to predict the output for a given query input. We will also understand

bias-variance tradeoff and how to decide the model dimension when limited training data is available. This

HW will rely heavily on the code from the previous homework.

Generate Data Code: Generate m + mtest data points satisfying

y = θ

Tx + e

with θ being ONE fixed n length vector for all of them. Use n = 100, θ = [100, ?99, 98, ?97...1]0, σ2e = 0.01||θ||22,e ～ N (0, σ2e), x ～ N (0, I), and assume mutual independence of the different inputs and noise values (e).

1. Use code from Homework 1 (using any one approach is okay) to learn θ. Vary m and show a plot of both

estimation error in θ,||θ ? ?θ||22/||θ||2

and a second plot of the “Monte Carlo estimate” of the prediction error on the test data (test data MSE).

Normalized-Test-MSE := E[(ytest ? y?)2]/E[y2test], with ?y := ?θ

Txtest

Monte Carlo estimate means: compute (ytest ?y?)

2

for mtest different input-output pairs and then average

the result.

(a) Vary m: use m = 80, m = 100, m = 120, m = 400. If your code is unable to return an estimate of θ,

you can report the errors to be ∞ (and for the plot just use a large value say 100000 to replace ∞.

(b) Repeat this experiment with σ2e = 0.1||θ||22.

Thus this part will produce four plots.

2. In this second part, suppose you have only m = 80 training data points satisfying y = θTx + e, with

n = 100. Notice n is the same as in the first part. I had a typo earlier which has now been fixed.

What you will have concluded from part 1 is that you cannot learn θ correctly in this case because m is

even smaller than n.

Let us assume you do not have the option to increase m. What can you do? All you can do is reduce n

to a value nsmall ≤ m. Experiment with different values of nsmall to come up with the best one. Do this

experiment for two values of σ2e: σ2e = 0.01||θ||22and σ2e = 0.1||θ||22.

How to decide which entries of x to throw away? For now, just throw away the last n?nsmall + 1 entries.

So for nsmall = 1, let xsmall be just the first entry, and so on. So for nsmall = 30 for example, xsmall

will be the first 30 entries of x. There are many other better ways which we will learn about later in the

course.

Start with nsmall = 1 and keep increasing its value and each time compute Normalized-Test-MSE by

learning a value of θ first (using m = 80 of course). Obtain a plot. Use the plot and what you learn in

class to decide what value of nsmall is best.

3. Interpret your results based on the Bias-Variance tradeoff discussion. See Section 11 of Summary-Notes

and what will be taught in the next few classes.

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