  We provide an algorithm that maintains, against an adaptive adversary, a
$(1-\varepsilon)$-approximate maximum matching in $n$-node $m$-edge general
(not necessarily bipartite) undirected graph undergoing edge deletions with
high probability with (amortized) $O(\mathrm{poly}(\varepsilon^{-1}, \log n))$
time per update. We also obtain the same update time for maintaining a
fractional approximate weighted matching (and hence an approximation to the
value of the maximum weight matching) and an integral approximate weighted
matching in dense graphs. Our unweighted result improves upon the prior
state-of-the-art which includes a $\mathrm{poly}(\log{n}) \cdot
2^{O(1/\varepsilon^2)}$ update time [Assadi-Bernstein-Dudeja 2022] and an
$O(\sqrt{m} \varepsilon^{-2})$ update time [Gupta-Peng 2013], and our weighted
result improves upon the $O(\sqrt{m}\varepsilon^{-O(1/\varepsilon)}\log{n})$
update time due to [Gupta-Peng 2013].
  To obtain our results, we generalize a recent optimization approach to
dynamic algorithms from [Jambulapati-Jin-Sidford-Tian 2022]. We show that
repeatedly solving entropy-regularized optimization problems yields a lazy
updating scheme for fractional decremental problems with a near-optimal number
of updates. To apply this framework we develop optimization methods compatible
with it and new dynamic rounding algorithms for the matching polytope.
