And at the second instant, the weight may change in the opposite direction by a large amount because of the negative gradient and would thus keep oscillating with a large variance about the optimal weights.
22.
So I'm going to assume your requirement is satisfied if f ( x ) has strictly positive gradient when " x " is negative and has strictly negative gradient when " x " is positive.
23.
In this example, the time-derivative of the momentum " p " equals the " Newtonian force ", and so the first Hamilton equation means that the force equals the negative gradient of potential energy.
24.
With respect to positive and negative gradients, it appears to be the case that negative events are thought to be perceived as increasingly more negative than positive events are increasingly positive the closer one gets ( spatially or temporally ) to the affective event itself.
25.
A quantitative description of the nuclear force relies on partially empirical equations that model the internucleon potential energies, or potentials . ( Generally, forces within a system of particles can be more simply modeled by describing the system's potential energy; the negative gradient of a quantum mechanical properties of the nucleon system.
26.
A generalization of this idea to other loss functions than squared error ( and to classification and ranking problems ) follows from the observation that residuals y-F ( x ) are the negative gradients of the squared error loss function \ frac { 1 } { 2 } ( y-F ( x ) ) ^ 2.
27.
Gradient descent is based on the observation that if the multi-variable function F ( \ mathbf { x } ) is differentiable in a neighborhood of a point \ mathbf { a }, then F ( \ mathbf { x } ) decreases " fastest " if one goes from \ mathbf { a } in the direction of the negative gradient of F at \ mathbf { a },-\ nabla F ( \ mathbf { a } ).
