About Nonlinear maximizer tanh
Mso-para-margin-right:0in; mso-para-margin-bottom:10.0pt; mso-para-margin-left:0in; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin;}table.MsoTableGrid {mso-style-name:"Table Grid"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-priority:59; mso-style-unhide:no; border:solid black 1.0pt; mso-border-themecolor:text1; mso-border-alt:solid black .5pt; mso-border-themecolor:text1; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-border-insideh:.5pt solid black; mso-border-insideh-themecolor:text1; mso-border-insidev:.5pt solid black; mso-border-insidev-themecolor:text1; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin;}1. In neuralnetworks, what is the role of nonlinear activation functions such as sigmoid, tanh,and ReLU?a) They speed upthe gradient calculation in backpropagation, as compared to linear unitsb) They help tolearn nonlinear decision boundariesc) They alwaysoutput values between 0 and 1d) All of the above Answer: (b) They help to learn nonlinear decision boundaries Non-linear activation functions such as ReLU, Tanh, and Sigmoid can help the network learn complex data, compute and learn almost any function representing a question, and provide accurate predictions. A neural network must be able to take any input from -infinity to +infinity, but it should be able to map it to an output that ranges between {0,1} or between {-1,1} in some cases - thus the need for activation function. Non-linearity is needed in activation functions because its aim in a neural network is to produce a nonlinear decision boundary via non-linear combinations of the weight and inputs. What is an activation function? An activation function in a neural network defines how the weighted sum of the input is transformed into an output from a node or nodes in a layer of the network. Activation functions in general are used to convert linear outputs of a neuron into nonlinear outputs, ensuring that a neural network can learn nonlinear behavior. Why do we need an activation function in neural network? Without activation function, weight and bias would only have a linear transformation, or neural network is just a linear regression model, a linear equation is polynomial of one degree only which is simple to solve but limited in terms of ability to solve complex problems or higher degree polynomials. On the other hand, with activation function added to neural network executes the non-linear transformation to input and make it capable to solve complex problems such as language translations and image classifications. Why non-linear activation functions used in the hidden layer of neural network? Differentiable non-linear activation functions are used in the hidden layers of a neural network. This allows the model to learn more complex functions than a network trained using a linear activation function. This helps
Key features of the Nonlinear maximizer tanh that are enabled through these drivers include:
- The extended tanh method for solving systems of nonlinear
- How tanh has to do with nonlinearity - Cross Validated
- The tanh method: a tool for solving certain classes of nonlinear
- The tanh method: I. Exact solutions of nonlinear evolution and
- The tanh method for traveling wave solutions of nonlinear equations
- Programmable Tanh-, ELU-, Sigmoid-, and Sin-Based Nonlinear
- Expanding the Tanh-Function Method for Solving Nonlinear Equations
Objective functionW(Z % plot only where the constraints are satisfiedsurf(X,Y,W,'LineStyle','none');view(68,20)hold onplot3(.4396, .0373, .8152,'o','MarkerEdgeColor','r', ... 'MarkerSize',8); % best pointxlabel('x')ylabel('y')zlabel('z')hold offThe nonlinear constraints must be written in the form c(x) . We compute all the symbolic constraints and their derivatives, and place them in a function handle using matlabFunction.The gradients of the constraints should be column vectors; they must be placed in the objective function as a matrix, with each column of the matrix representing the gradient of one constraint function. This is the transpose of the form generated by jacobian, so we take the transpose below.We place the nonlinear constraints into a function handle. fmincon expects the nonlinear constraints and gradients to be output in the order [c ceq gradc gradceq]. Since there are no nonlinear equality constraints, we output [] for ceq and gradceq.c1 = x1^4 - 5*sinh(x2/5);c2 = x2^2 - 5*tanh(x1/5) - 1;c = [c1 c2];gradc = jacobian(c,x).'; % transpose to put in correct formconstraint = matlabFunction(c,[],gradc,[],'vars',{x});The interior-point algorithm requires its Hessian function to be written as a separate function, instead of being part of the objective function. This is because a nonlinearly constrained function needs to include those constraints in its Hessian. Its Hessian is the Hessian of the Lagrangian; see the User's Guide for more information.The Hessian function takes two input arguments: the position vector x, and the Lagrange multiplier structure lambda. The parts of the lambda structure that you use for nonlinear constraints are lambda.ineqnonlin and lambda.eqnonlin. For the current constraint, there are no linear equalities, so we use the two multipliers lambda.ineqnonlin(1) and lambda.ineqnonlin(2).We calculated the Hessian of the objective function in the first example. Now we calculate the Hessians of the two constraint functions, and make function handle versions with matlabFunction.hessc1 = jacobian(gradc(:,1),x); % constraint = first c columnhessc2 = jacobian(gradc(:,2),x);hessfh = matlabFunction(hessf,'vars',{x});hessc1h = matlabFunction(hessc1,'vars',{x});hessc2h = matlabFunction(hessc2,'vars',{x});To make the final Hessian, we put the three Hessians together, adding the appropriate Lagrange multipliers to the constraint functions.myhess = @(x,lambda)(hessfh(x) + ... lambda.ineqnonlin(1)*hessc1h(x) + ... lambda.ineqnonlin(2)*hessc2h(x));Set the options to use the interior-point algorithm, the gradient, and the Hessian, have the objective function return both the objective and the gradient, and run the solver:options = optimoptions('fmincon', ... 'Algorithm','interior-point', ... 'SpecifyObjectiveGradient',true, ... 'SpecifyConstraintGradient',true, ... 'HessianFcn',myhess, ... 'Display','final');% fh2 = objective without Hessianfh2 = matlabFunction(f,gradf,'vars',{x});[xfinal,fval,exitflag,output] = fmincon(fh2,[-1;2],... [],[],[],[],[],[],constraint,options)Local minimum found that satisfies the constraints.Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of
