This paper investigates if a differential graded algebra can have more than
one $A_\infty$-structure extending the given differential graded algebra
structure. We give a sufficient condition for uniqueness of such an
$A_\infty$-structure up to quasi-isomorphism using Hochschild cohomology. We
then extend this condition to Sagave's notion of derived $A_\infty$-algebras
after introducing a notion of Hochschild cohomology that applies to this.