I need help to solve problem Manually fit a linear function h_(\theta )(vec(x))=vec(\theta )^(T)*vec(x) based on the following training instances using the stochastic gradient descent algorithm. The initial values of parameters are \theta _(0)=0.1,\theta _(1)=0.1,\theta _(2)=0.1. The learning rate \alpha is 0.1. Please update each parameter at least five times.X_1,X_2,Y 0 0 20 1 31 0 31 1 4
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I need help to solve problem
Manually fit a linear function h_(\theta )(vec(x))=vec(\theta )^(T)*vec(x) based on the following training instances using the stochastic gradient descent
X_1,X_2,Y
0 0 2
0 1 3
1 0 3
1 1 4
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- Appendix A 10-Fold Cross Validation for Parameter Selection Cross Validation is the standard method for evaluation in empirical machine learning. It can also be used for parameter selection if we make sure to use the training set only. To select parameter A of algorithm A(X) over an enumerated range d E [A1,..., A] using dataset D, we do the following: 1. Split the data D into 10 disjoint folds. 2. For each value of A e (A1,..., Ar]: (a) For i = 1 to 10 Train A(A) on all folds but ith fold Test on ith fold and record the error on fold i (b) Compute the average performance of A on the 10 folds. 3. Pick the value of A with the best average performance Now, in the above, D only includes the training data and the parameter A is chosen without the knowledge of the test data. We then re-train on the entire train set D using the chosen A and evaluate the result on the test set.Machine Learning Problem Perform the optimization problem of finding the minimum of J(x) = (2x-3)2 by: (i) defining theta, J(theta), h(theta) as defined in the Stanford Machine Learning videos in Coursera; (ii) plotting J(theta) vs theta by hand then use a program (iii) determining its minimum using gradient descent approach starting from a random initial value of theta = 5. Perform the search for the minimum using the gradient descent approach by hand calculations, i.e., step 1, step 2, etc. showing your work completelyMachine Learning You are given the scatter of points (x,y) = (1, 1.5), (4, 3.5), (7, 9), (10, 8). Set up an optimization problem to perform linear regression by hand along the lines of the previous problem by defining theta, etc. Then perform hand calculations to implement linear regression using gradient descent, and report the optimal value of theta that you obtain and what that means. Again, list all the steps - 1, 2,...etc. without omitting any details.
- Gradient descent is a widely used optimization algorithm in machine learning and deep learning. It is used to find the minimum value of a differentiable function by iteratively adjusting the parameters of the function in the direction of the steepest decrease of the function's value. In gradient descent, the function is first differentiated to find its gradient, which is a vector of the partial derivatives with respect to each of the parameters. The gradient points in the direction of the steepest increase of the function, so to find the minimum, we need to move in the opposite direction, i.e., in the direction of the negative gradient. The algorithm starts from an initial guess for the parameters and iteratively adjusts the parameters by subtracting a small fraction of the gradient from the current parameters. The fraction is determined by the learning rate, which is a hyperparameter that controls the step size of the update. The algorithm updates the parameters…Consider the same house rent prediction problem where you are supposed to predict price of a house based on just its area. Suppose you have n samples with their respective areas, x(1), x(2), ... , x(n), their true house rents y(1), y(2),..., y(n). Let's say, you train a linear regres- sor that predicts f(x()) = 00 + 01x(e). The parameters 6o and 0, are scalars and are learned by minimizing mean-squared-error loss with L2-regularization through gradient descent with a learning rate a and the regularization strength constant A. Answer the following questions. 1. Express the loss function(L) in terms of x), y@), n, 0, 01, A. 2. Compute L 3. Compute 4. Write update rules for 6, and O11. Let f(x₁, x₂) = x₁ cos(x₂) e¯*₁. What is the gradient Vf(x) of f?
- implement Bresenham line drawing algorithm:For Slope |m|<1:Either value of x is increasedOR both x and y is increased using decision parameter.Consider the same house rent prediction problem where you are supposed to predict price of a house based on just its area. Suppose you have n samples with their respective areas, x(¹), x(²),...,x(n), their true house rents y(¹), y(2),..., y(n). Let's say, you train a linear regres- sor that predicts f(x)) = 0 + 0₁x). The parameters, and 0₁ are scalars and are learned by minimizing mean-squared-error loss with L1-regularization through gradient descent with a learning rate a and the regularization strength constant A. Answer the following questions. 1. Express the loss function(L) in terms of x(i),y(i), n, 00, 01, X. 2. Compute L 200 ƏL 3. Compute 20₁ 4. Write update rules for 0o and 0₁ Hint: d|w| dw undefined -1 w>0 w=0 w <0Let us analyze the linearity and convexity of deep neural networks. Recall that a function g: R" → Rm is linear if for all a, b € R, and x, y € R", g(ax +by) = ag(x) + bg(y). Say that a function f: R" → R is convex if and only if f((1 t)x+ty) ≤ (1 – t)f(x) + tf(y) for t = [0, 1] and all x, y € R". Select all that are true. The following fully connected network without activation functions is linear: g3 (92(91(x))), where gi(x) = Wix and W; are matrices Leaky ReLU = max{0.01x, x} is convex A combination of ReLUs such as ReLU(x) – ReLU(x - 1) is convex ResNet-50, which has ReLU activations, is nonlinear and convex (assume only 1 output activation).
- It is known that a natural law obeys the quadratic relationship y = ax“. What is the best line of the form y = px + q that can be used to model data and minimize Mean-Squared-Error if all of the data points are drawn uniformly at random from the domain [0,1]? r* ur, a,I need to optimize the Rosenbrock function using the steepest descent method and the modified newton method. It also uses the line search method (golden section). The function is given by: f(x) = 100*(x(2)-x(1)^2)^2+(1-x(1))^2 The starting point is x=[-1 0]T.Prove valid. (x) ( ~ Ax > Dx) / (Ex) (Ax v Bx) (Ey) ( ~ (By v ~ Dy) > ~ ( ~ Dy v ~ Ay)) (x) ((Ax v Dx) > ( ~ Ax v ~ Dx))