  The infinite reduction of couplings is a tool to consistently renormalize a
wide class of non-renormalizable theories with a reduced, eventually finite,
set of independent couplings, and classify the non-renormalizable interactions.
Several properties of the reduction of couplings, both in renormalizable and
non-renormalizable theories, can be better appreciated working at the
regularized level, using the dimensional-regularization technique. We show
that, when suitable invertibility conditions are fulfilled, the reduction
follows uniquely from the requirement that both the bare and renormalized
reduction relations be analytic in epsilon=D-d, where D and d are the physical
and continued spacetime dimensions, respectively. In practice, physically
independent interactions are distinguished by relatively non-integer powers of
epsilon. We discuss the main physical and mathematical properties of this
criterion for the reduction and compare it with other equivalent criteria. The
leading-log approximation is solved explicitly and contains sufficient
information for the existence and uniqueness of the reduction to all orders.
