What are the three prerequisites for a network to function optimally? Let's break them down and examine them separately.

Two layers neural networks consist of 3 input neurons and one output neuron. The input vector is (x1, x2, x3) where x3 is always =1.0. x1, x2 and the desired output d are given by the table below. When using the initial weight vector w = (1.0, 1.0, -3.0) there was an error as shown in the figure. The equation for the decision boundary was: x1*1 + x2*1 - 1*3=0. , (i.e. net = x1*w1+x2*w2+x3*w3), f(net) is the sign of net   What are the new values of the vector w after the first input (1,1,1), note that input x3 is always 1.0? What are the new values of the vector w after the second input (9.4, 6.4, 1.0)? Suggest a vector w such that the total error is ZERO for the 10 points.

The cost function of a general neural network is defined as J(ŷ,y) = m // [4 (9(0), y(1) The loss function L(ŷ), y() is defined by the logistic loss function Ly, y) = [ylogy) + (1-y)log (1 - ¹)] Please list the stochastic gradient descent update rule, batch gradient descent update rule, and mini-batch gradient descent update rule. Explain the main difference of these three update rules.

1. Consider the network depicted in Figure 1; suppose that each node starts with the behavior B, and each node has a threshold of q = 1/2 for switching to behavior A. a. Now, let e and f form a two-node set S of initial adopters of behavior A. If other nodes follow the threshold rule for choosing behaviors, which nodes will eventually switch to A? b. Find a cluster of density greater than 1 – q = 1/2 in the part of the graph outside S that blocks behavior A from spreading to all nodes, starting from S, at threshold q. Figure 1: Starting from nodes e and f, the new behavior A fails to spread to the entire graph.