Is this discrete log generalization a well known cryptographic assumption?
Actually, that problem is exactly equivalent to the standard DLOG problem (assuming that you know the group order, and that it is prime).

Here's the reduction: suppose that we have an Oracle that can solve your problem with nontrivial probability.  Then, given a value $g$ and $h$, we can find $x$ with $g^x = h$ with nontrivial probability by:


Create random values $r_1, r_2, ..., r_n$ and $s$ between 0 and the group order (except that $s$ cannot be 0).
Compute $g_i = g^{r_i}$ and $t = h^s$.  Note that the values $g_i$ and $t$ are uniformly distributed (except that $t$ is not the identity).
Give the values $g_i$ and $t$ to your Oracle, which computes the values $a_i$ with nontrivial probability.
$x = s^{-1}\ \sum\  a_i r_i$


The only reason we needed to assume that the group order was prime was to be precise in the assumption that the elements be chosen randomly; if we relax that restriction on $t$, we can handle composite group orders as well.