  We extend the definition of Lagrangian local bias proposed by Matsubara
(2008) to include curvature and higher-derivative bias operators. Evolution of
initially biased tracers using perturbation theory (PT) generates multivariate
bias parameters as soon as nonlinear fluctuations become important. We present
a procedure that reparametrize a set of spectral parameters, the arguments of
the Fourier transformed Lagrangian bias function, from which multivariate
renormalized biases can be derived at any order in bias expansion and PT. We
find our method simpler than previous renormalization schemes because it only
relies on the definition of bias, fixed from the beginning, and in one equation
relating renormalized and unrenormalized spectral parameters. We also show that
our multivariate biases can be obtained within the peak background split
framework, in that sense this work extends that of Schmidt, Jeong and
Desjacques (2013); however, we restrict our method to Gaussian initial
conditions. Non-linear evolution also leads to the appearance of products of
correlators evaluated at the same point, commonly named contact terms, yielding
divergent contributions to the power spectrum, in this work we present an
explicit method to remove these divergences by introducing stochastic fields.
