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The certificate preserves the mean-dependent variance of multinomial batch averages, yielding the near-optimal robust rate $O(\eta\sqrt{\log(1/\eta)/m})$ and providing the score required by the privacy reduction.
We give a polynomial-time algorithm for approximating the partition function of mean-field mixed $p$-spin models to arbitrarily high accuracy throughout the second-moment regime. This improves the quasipolynomial-time algorithm of Bencs, Huang, Lee, Liu, and Regts (arXiv:2507.15616) and answers their open question. In...
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