  Privacy amplification exploits randomness in data selection to provide
tighter differential privacy (DP) guarantees. This analysis is key to DP-SGD's
success in machine learning, but, is not readily applicable to the newer
state-of-the-art algorithms. This is because these algorithms, known as
DP-FTRL, use the matrix mechanism to add correlated noise instead of
independent noise as in DP-SGD.
  In this paper, we propose "MMCC", the first algorithm to analyze privacy
amplification via sampling for any generic matrix mechanism. MMCC is nearly
tight in that it approaches a lower bound as $\epsilon\to0$. To analyze
correlated outputs in MMCC, we prove that they can be analyzed as if they were
independent, by conditioning them on prior outputs. Our "conditional
composition theorem" has broad utility: we use it to show that the noise added
to binary-tree-DP-FTRL can asymptotically match the noise added to DP-SGD with
amplification. Our amplification algorithm also has practical empirical
utility: we show it leads to significant improvement in the privacy-utility
trade-offs for DP-FTRL algorithms on standard benchmarks.
