Search ResultsShowing 1-4 of 4
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arXiv:2104.07629 (Published 2021-04-15)
Spin glass to paramagnetic transition in Spherical Sherrington-Kirkpatrick model with ferromagnetic interaction
Comments: 31 pages, 3 figuresCategories: math.PRThis paper studies fluctuations of the free energy of the Spherical Sherrington-Kirkpatrick model with ferromagnetic Curie-Weiss interaction with coupling constant $J \in [0,1)$ and inverse temperature $\beta$. We consider the critical temperature regime $\beta=1+bN^{-1/3}\sqrt{\log N}$, $b\in\mathbb{R}$. For $b \leq 0$, the limiting distribution of the free energy is Gaussian. As $b$ increases from $0$ to $\infty$, we describe the transition of the limiting distribution from Gaussian to Tracy-Widom.
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arXiv:2011.13723 (Published 2020-11-27)
An edge CLT for the log determinant of Gaussian ensembles
Comments: 39 pages, 3 figuresCategories: math.PRWe derive a Central Limit Theorem (CLT) for $\log \left\vert\det \left( M_{N}/\sqrt{N}-2\theta_{N}\right)\right\vert,$ where $M_{N}$ is from the Gaussian Unitary or Gaussian Orthogonal Ensemble (GUE and GOE), and $2\theta_{N}$ is local to the edge of the semicircle law. Precisely, $2\theta_{N}=2+N^{-2/3}\sigma_N$ with $\sigma_N$ being either a constant (possibly negative), or a sequence of positive real numbers, slowly diverging to infinity so that $\sigma_N \ll \log^{2} N$. For slowly growing $\sigma_N$, our proofs hold for general Gaussian $\beta$-ensembles. We also extend our CLT to cover spiked GUE and GOE.
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arXiv:1910.07833 (Published 2019-10-17)
Sharper bounds for uniformly stable algorithms
Comments: 14 pagesThe generalization bounds for stable algorithms is a classical question in learning theory taking its roots in the early works of Vapnik and Chervonenkis and Rogers and Wagner. In a series of recent breakthrough papers, Feldman and Vondrak have shown that the best known high probability upper bounds for uniformly stable learning algorithms due to Bousquet and Elisseeff are sub-optimal in some natural regimes. To do so, they proved two generalization bounds that significantly outperform the original generalization bound. Feldman and Vondrak also asked if it is possible to provide sharper bounds and prove corresponding high probability lower bounds. This paper is devoted to these questions: firstly, inspired by the original arguments of, we provide a short proof of the moment bound that implies the generalization bound stronger than both recent results. Secondly, we prove general lower bounds, showing that our moment bound is sharp (up to a logarithmic factor) unless some additional properties of the corresponding random variables are used. Our main probabilistic result is a general concentration inequality for weakly correlated random variables, which may be of independent interest.
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arXiv:1812.03548 (Published 2018-12-09)
Uniform Hanson-Wright type concentration inequalities for unbounded entries via the entropy method
Comments: 28 pagesThis paper is devoted to uniform versions of the Hanson-Wright inequality for a random vector $X \in \mathbb{R}^n$ with independent subgaussian components. The core technique of the paper is based on the entropy method combined with truncations of both gradients of functions of interest and of the coordinates of $X$ itself. Our results recover, in particular, the classic uniform bound of Talagrand (1996) for Rademacher chaoses and the more recent uniform result of Adamczak (2015), which holds under certain rather strong assumptions on the distribution of $X$. We provide several applications of our techniques: we establish a version of the standard Hanson-Wright inequality, which is tighter in some regimes. Extending our techniques we show a version of the dimension-free matrix Bernstein inequality that holds for random matrices with a subexponential spectral norm. We apply the derived inequality to the problem of covariance estimation with missing observations and prove an almost optimal high probability version of the recent result of Lounici (2014). Finally, we show a uniform Hanson-Wright type inequality in the Ising model under Dobrushin's condition. A closely related question was posed by Marton (2003).