[30 points] In this problem, we will compare the performance of three different types of algorithms on a synthetic training set. All three algorithms will attempt to learn the a true vector w that has generated a noisy dataset. First, though, you will need to generate this dataset. Follow the steps below for the data generation: (a) Pick a weight vector wR10 whose Euclidean norm equals 1 in the following manner: i. Generate w such that each element of w is distributed standard normal, N(0,1). ii. Normalize w by w so that it's Euclidean norm is equal to 1 . Note: For reference, this is the equation for the Euclidian norm of a vector : =ii2 (b) Next, generate a training set of size m of the form {(x1,y1),,(xm,ym)} where each xiR10 is a random vector and each yi{0,1} is a label derived from a function of xi : i. To generate the xi s use the same procedure as in 1(a)i, but do not normalize. Do this m times to produce {x1,xm} ii. For the yi s, you will generate their binary values at random using the following function: GenLABEL(xi)={10X(wxi),XUNIFORM(0,1)otherwise Where is the sigmoid function and UNIFORM (0,1) is the uniform distribution between 0 and 1 . Informally, GenLabeL simply returns a 1 with probability (wxi) and a 0 otherwise. With the data generation complete, the goal is to learn w using three different algorthms: - Algorithm 1 is logistic regression (you may use built-in methods for this). - Algorithm 2 is gradient descent where you train a model of the form (wx) (with parameter w) with respect to square loss, i.e. the loss function is 21((wx)y)2, averaged over the points in the training set (code this up yourself, including calculating the gradient). - Algorithm 3 is stochastic gradient descent again with respect to square loss, where during each iteration we use the gradient at one random point from the training set. You should measure success as follows: compute ww where w is the weight vector output by your algorithm after training. For each value of m, do the following several times You should measure success as follows: compute ww where w is the weight vector output by your algorithm after training. For each value of m, do the following several times (say 10 times) and take the average: (a) Generate a fresh w and fresh training data using the procedures outlined above (b) Train your algorithm and obtain w 1 (c) Calculate ww Plot the results for all three algorithms for m=50,100,150,200,250. For each algorithm, also record the time taken for the entire experiment