  We investigate the renormalization group (RG) structure of the gradient flow.
Instead of using the original bare action to generate the flow, we propose to
use the effective action at each flow time. We write down the basic equation
for scalar field theory that determines the evolution of the action, and argue
that the equation can be regarded as a RG equation if one makes a
field-variable transformation at every step such that the kinetic term is kept
to take the canonical form. We consider a local potential approximation (LPA)
to our equation, and show that the result has a natural interpretation with
Feynman diagrams. We make an $\varepsilon$ expansion of the LPA and show that
it reproduces the eigenvalues of the linearized RG transformation around both
the Gaussian and the Wilson-Fisher fixed points to the order of $\varepsilon$.
