  We study the effects of the generic weak nonlinear loss on fast two-pulse
interactions in linear waveguides. The colliding pulses are described by a
system of coupled Schr\"odinger equations with a purely nonlinear coupling in
the presence of the weak $(2m+1)-$order of nonlinear loss, for any $m \geq 1$.
We derive the analytic expression for the collision-induced amplitude shift in
a fast two-pulse interaction. The analytic calculations are based on a
generalization of the perturbation technique for calculating the effects of
weak perturbations on fast collisions between solitons of the nonlinear
Schr\"odinger equation. The theoretical predictions are confirmed by the
numerical simulations with the full propagation model of coupled Schr\"odinger
equations.
