  It is well-known that the Kontsevich formality [K97] for Hochschild cochains
of the polynomial algebra $A=S(V^*)$ fails if the vector space $V$ is
infinite-dimensional. In the present paper, we study the corresponding
obstructions. We construct an $L_\infty$ structure on polyvector fields on $V$
having the even degree Taylor components, with the degree 2 component given by
the Schouten-Nijenhuis bracket, but having as well higher non-vanishing Taylor
components. We prove that this $L_\infty$ algebra is quasi-isomorphic to the
corresponding Hochschild cochain complex. We prove that our $L_\infty$ algebra
is $L_\infty$ quasi-isomorphic to the Lie algebra of polyvector fields on $V$
with the Schouten-Nijenhuis bracket, if $V$ is finite-dimensional.
