Tape the Gradient Offset of a Quadratic CppAD Tape
tape_gradoffset(pfun)An Rcpp_ADFun object. The independent argument to the function are the dynamic parameters of pfun.
An Rcpp_ADFun object.
A quadratic function can be written as $$f(x;\theta) = \frac{1}{2} x^T W(\theta) x + b(\theta)^Tx + c.$$ The gradient of \(f(x; \theta)\) with respect to \(x\) is $$\Delta f(x; \theta) = \frac{1}{2}(W(\theta) + W(\theta)^T)x + b(\theta).$$ The Hessian is $$H f(x; \theta) = \frac{1}{2}(W(\theta) + W(\theta)^T),$$ which does not depend on \(x\), so the gradient of the function can be rewritten as $$\Delta f(x;\theta) = H f(x; \theta) x + b(\theta)^T.$$ The tape calculates \(b(\theta)\) as $$b(\theta) = \Delta f(x;\theta) - H f(x; \theta) x,$$ which does not depend on \(x\).
For creating this tape, the values of pfun$xtape and pfun$dyntape are used.
Other tape builders:
tape_Hessian(),
tape_Jacobian(),
tape_logJacdet(),
tape_smd(),
tape_swap(),
tape_uld()