  Let $A$ be a bi-Koszul algebra, we describe all possible $A_\infty$-algebra
structures on the Ext-algebra $E(A)$, and prove that $E(A)$ must be $[m_2,
m_3]$-finitely generated. An equivalent description for a connected graded
algebra to be a bi-Koszul algebra is given in terms of $A_\infty$-language. The
case that $E(A)$ is endowed with minimal number of multiplications is discussed
for decomposition.
