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A Universal Law of Robustness via Isoperimetry [article]

Sébastien Bubeck, Mark Sellke
2021 arXiv   pre-print
We prove this universal law of robustness for any smoothly parametrized function class with polynomial size weights, and any covariate distribution verifying isoperimetry.  ...  In the case of two-layers neural networks and Gaussian covariates, this law was conjectured in prior work by Bubeck, Li and Nagaraj.  ...  We view our universal law of robustness as putting this empirical observation on a more solid footing: scale is actually necessary to achieve robustness.  ... 
arXiv:2105.12806v3 fatcat:jmeikybuung7tpnqzukaufisdu

Optimal search patterns in honeybee orientation flights are robust against emerging infectious diseases

Stephan Wolf, Elizabeth Nicholls, Andrew M. Reynolds, Patricia Wells, Ka S. Lim, Robert J. Paxton, Juliet L. Osborne
2016 Scientific Reports  
Lévy flights are scale-free (fractal) search patterns found in a wide range of animals.  ...  of the decision-making process, making them robust against diverse physiological effects of pathogen infection and possibly other stressors.  ...  in the understanding of how universal laws of optimal searches arise and shape animal behaviour 20, 27 .  ... 
doi:10.1038/srep32612 pmid:27615605 pmcid:PMC5018844 fatcat:tzovvipk6rcpllarquf3qg3ogy

Sampling for Bayesian Mixture Models: MCMC with Polynomial-Time Mixing [article]

Wenlong Mou, Nhat Ho, Martin J. Wainwright, Peter L. Bartlett, Michael I. Jordan
2019 arXiv   pre-print
We study the problem of sampling from the power posterior distribution in Bayesian Gaussian mixture models, a robust version of the classical posterior.  ...  Notably, this result requires no conditions on the separation of the two means.  ...  due to the symmetry of the law of z ξ −1 .  ... 
arXiv:1912.05153v1 fatcat:sm7w5zvknfem7pshm5rir7koni

Anomalous heat-kernel decay for random walk among bounded random conductances

N. Berger, M. Biskup, C. E. Hoffman, G. Kozma
2008 Annales de l'I.H.P. Probabilites et statistiques  
The conductance law is i.i.d. subject to the condition that the probability of ω_xy>0 exceeds the threshold for bond percolation on ^d.  ...  We consider the nearest-neighbor simple random walk on ^d, d>2, driven by a field of bounded random conductances ω_xy∈[0,1].  ...  The research of C.H. was supported by the NSF grant DMS-0100445 and a grant from the University of Washington Royalty Research Fund.  ... 
doi:10.1214/07-aihp126 fatcat:aa6wspwkgnbtfdcbjm35d5ayiq

Generalized No Free Lunch Theorem for Adversarial Robustness [article]

Elvis Dohmatob
2019 arXiv   pre-print
This manuscript presents some new impossibility results on adversarial robustness in machine learning, a very important yet largely open problem.  ...  We show that if conditioned on a class label the data distribution satisfies the W_2 Talagrand transportation-cost inequality (for example, this condition is satisfied if the conditional distribution has  ...  On the other hand, if X is a random variable with law µ| B and X is a random variable with law µ| X \B , then the definition of B ensures that d(X, X ) ≥ µ-a.s, and so by definition (7) , one has W 2  ... 
arXiv:1810.04065v8 fatcat:agboga5gqzdo5cxm7tj6dvtrdu

Functional Inequalities for Heavy Tailed Distributions and Application to Isoperimetry

Patrick Cattiaux, Nathael Gozlan, Arnaud Guillin, Cyril Roberto
2010 Electronic Journal of Probability  
A typical example in our mind is V(x) = |x|, and 0 < p < 1, which yileds to sub-exponential type law.  ...  like law".  ...  . ♦ Weighted inequalities and isoperimetry via mass transport In this section, we present another method to obtain weighted functional inequalities.  ... 
doi:10.1214/ejp.v15-754 fatcat:dz7bvdgpz5gujmgkkgwetru25a

Functional inequalities for heavy tails distributions and application to isoperimetry [article]

Patrick Cattiaux, Arnaud Guillin
2008 arXiv   pre-print
This paper is devoted to the study of probability measures with heavy tails.  ...  Using the Lyapunov function approach we prove that such measures satisfy different kind of functional inequalities such as weak Poincaré and weak Cheeger, weighted Poincaré and weighted Cheeger inequalities  ...  µ n is (up a to universal, hence dimension free constants) equal to I µ .  ... 
arXiv:0807.3112v1 fatcat:btuz5iugfzggbmak4cxvj6z37y

A Token-Based Approach for Distributed Computation in Sensor Networks

Venkatesh Saligrama, Murat Alanyali
2011 IEEE Journal on Selected Topics in Signal Processing  
The algorithms perform pairwise computations along edges of an underlying communication graph. A token is associated with each sensor node, which acts as a transmission permit.  ...  Examples of such algorithms include Simple-Random Walk (SRW), Coalescent-Random-Walk (CRW), and Controlled Flooding (CFLD) and their hybrid variants.  ...  To obtain an upper bound we need to construct a flow along the lines of [4] that satisfies the Kirchoff current law.  ... 
doi:10.1109/jstsp.2011.2125776 fatcat:hwkdhse5qrcrnb6y6opthkxjl4

Transference Principles for Log-Sobolev and Spectral-Gap with Applications to Conservative Spin Systems

Franck Barthe, Emanuel Milman
2013 Communications in Mathematical Physics  
When the self-interaction is a perturbation of a strongly convex potential, this partially recovers and partially extends previous results of Caputo, Chafaï, Grunewald, Landim, Lu, Menz, Otto, Panizo,  ...  We obtain new principles for transferring log-Sobolev and Spectral-Gap inequalities from a source metric-measure space to a target one, when the curvature of the target space is bounded from below.  ...  The asserted tail decay estimate now follows by a standard application of Chebyshev's inequality and optimization on p.  ... 
doi:10.1007/s00220-013-1782-2 fatcat:3xohukvwmvai7p4ef7645lfcu4

Concentration inequalities for log-concave distributions with applications to random surface fluctuations [article]

Alexander Magazinov, Ron Peled
2021 arXiv   pre-print
of marginals in a manner suitable for obtaining variance and tail probability bounds.  ...  We extend these bounds to the case of convex potentials whose second derivative vanishes only on a zero measure set, when the underlying graph is a d-dimensional discrete torus.  ...  We thank Jean-Dominique Deuschel for a discussion of the results of [21] . Lastly, we thank Shangjie Yang and an anonymous referee for helpful comments on the manuscript.  ... 
arXiv:2006.05393v2 fatcat:hoqvecbw3zasfmer5g6yhzpwei

The curse of overparametrization in adversarial training: Precise analysis of robust generalization for random features regression [article]

Hamed Hassani, Adel Javanmard
2022 arXiv   pre-print
In this paper, we will provide a precise characterization of the role of overparametrization on robustness by focusing on random features regression models (two-layer neural networks with random first  ...  , and the computational perspective via the structural properties of the optimization landscape.  ...  The research of H. Hassani is supported by the NSF CAREER award, AFOSR YIP, the Intel Rising Star award, as well as the AI Institute for Learning-Enabled Optimization at Scale (TILOS). A.  ... 
arXiv:2201.05149v1 fatcat:exxtlpgfqvajdgf4kxdc5jtrtm

On the mixing time of coordinate Hit-and-Run [article]

Hariharan Narayanan, Piyush Srivastava
2020 arXiv   pre-print
We obtain a polynomial upper bound on the mixing time T_CHR(ϵ) of the coordinate Hit-and-Run random walk on an n-dimensional convex body, where T_CHR(ϵ) is the number of steps needed in order to reach  ...  within ϵ of the uniform distribution with respect to the total variation distance, starting from a warm start (i.e., a distribution which has a density with respect to the uniform distribution on the convex  ...  a warm start follows via a result of Lovász and Simonovits [14] (see Theorem 2.1).  ... 
arXiv:2009.14004v2 fatcat:d3wjv2fplndbxnnywm524yf56m

Identifying and Exploiting Structures for Reliable Deep Learning [article]

Amartya Sanyal
2021 arXiv   pre-print
Deep learning research has recently witnessed an impressively fast-paced progress in a wide range of tasks including computer vision, natural language processing, and reinforcement learning.  ...  To do this, we identify structures in deep neural networks that can be exploited to mitigate the above causes of unreliability of deep learning algorithms.  ...  2) Further there exists an universal constant c such that for any , δ > 0 with be written Lemma 7 (Robustness of parity classifier).  ... 
arXiv:2108.07083v1 fatcat:lducrn5tlfeqvpxevz6gukfvse

Interactions of Computational Complexity Theory and Mathematics [article]

Avi Wigderson
2017 arXiv   pre-print
In my view, the breadth, depth, beauty and novelty of these connections is inspiring, and speaks to a great potential of future interactions (which indeed, are quickly expanding). We aim for variety.  ...  Each vignette focuses only on a single topic within a large mathematical filed.  ...  Another famous such algorithm is for computing the Legendre-Jacobi symbol ( m n ) via Gauss' law of quadratic reciprocity.  ... 
arXiv:1710.09780v1 fatcat:js6k4t7gefcutfus6te4vxx6ty

Densities for rough differential equations under Hörmander's condition

Thomas Cass, Peter Friz
2010 Annals of Mathematics  
Examples of driving signals include fractional Brownian motion with Hurst parameter H > 1=4, the Brownian bridge returning to zero after time T and the Ornstein-Uhlenbeck process. PROPOSITION 1.  ...  Let x be a geometric p-rough path over ‫ޒ‬ d and h 2 C q-var OE0; T ; ‫ޒ‬ d such that 1=p C 1=q > 1. Then  ...  The second author is partially supported by a Leverhulme Research Fellowship and EPSRC grant EP/E048609/1.  ... 
doi:10.4007/annals.2010.171.2115 fatcat:b5aadp2dxndgpe7glkfkbiz7g4
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