  In this paper we provide a detailed description of the inchworm Monte Carlo
formalism for the exact study of real-time non-adiabatic dynamics. This method
optimally recycles Monte Carlo information from earlier times to greatly
suppress the dynamical sign problem. Using the example of the spin-boson model,
we formulate the inchworm expansion in two distinct ways: The first with
respect to an expansion in the system-bath coupling and the second as an
expansion in the diabatic coupling. The latter approach motivates the
development of a cumulant version of the inchworm Monte Carlo method, which has
the benefit of improved scaling. This paper deals completely with methodology,
while Paper II provides a comprehensive comparison of the performance of the
inchworm Monte Carlo algorithms to other exact methodologies as well as a
discussion of the relative advantages and disadvantages of each.
