Here, we investigate symmetric bi-derivations and their generalizations on L∞0 (G) ∗. For κ ∈ N, we show that if B : L∞0 (G) ∗×L∞0 (G)∗ → L∞0 (G)∗ is a symmetric bi-derivation such that [B(m,m),mκ] ∈ Z(L∞0 (G)∗) for all m ∈ L∞0 (G)∗, then B is the zero map. Furthermore, we characterize symmetric generalized bi- derivations on group algebras. We also prove that any symmetric Jordan bi-derivation on L∞0 (G) ∗ is a symmetric bi-derivation