  We perform an effective field theory analysis to correlate the charged lepton
flavor violating processes $\ell_i\to\ell_j\gamma\gamma$ and
$\ell_i\to\ell_j\gamma$. Using the current upper bounds on the rate for
$\ell_i\to\ell_j\gamma$, we derive model-independent upper limits on the rates
for $\ell_i\to\ell_j\gamma\gamma$. Our indirect limits are about three orders
of magnitude stronger than the direct bounds from current searches for $\mu\to
e\gamma\gamma$, and four orders of magnitude better than current bounds for
$\tau\to\ell\gamma\gamma$. We also stress the relevance of Belle II or a Super
Tau Charm Facility to discover the rare decay $\tau\to\ell\gamma\gamma$.
