Continuous LWE

We introduce a continuous analogue of the Learning with Errors (LWE) problem, which we name CLWE. We give a polynomial-time quantum reduction from worst-case lattice problems to CLWE, showing that CLWE enjoys similar hardness guarantees to those of LWE. Alternatively, our result can also be seen as opening new avenues of (quantum) attacks on lattice problems. Our work resolves an open problem regarding the computational complexity of learning mixtures of Gaussians without separability assumptions (Diakonikolas 2016, Moitra 2018). As an additional motivation, (a slight variant of) CLWE was considered in the context of robust machine learning (Diakonikolas et al.~FOCS 2017), where hardness in the statistical query (SQ) model was shown; our work addresses the open question regarding its computational hardness (Bubeck et al.~ICML 2019).

[1]  Roman Vershynin,et al.  High-Dimensional Probability , 2018 .

[2]  Chris Peikert,et al.  Trapdoors for Lattices: Simpler, Tighter, Faster, Smaller , 2012, IACR Cryptol. ePrint Arch..

[3]  Aravindan Vijayaraghavan,et al.  On Learning Mixtures of Well-Separated Gaussians , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[4]  Joan Bruna,et al.  Intriguing properties of neural networks , 2013, ICLR.

[5]  Daniele Micciancio,et al.  On Bounded Distance Decoding, Unique Shortest Vectors, and the Minimum Distance Problem , 2009, CRYPTO.

[6]  Ankur Moitra,et al.  Algorithmic Aspects of Machine Learning , 2018 .

[7]  Sanjeev Arora,et al.  New Algorithms for Learning in Presence of Errors , 2011, ICALP.

[8]  Chris Peikert,et al.  Pseudorandomness of ring-LWE for any ring and modulus , 2017, STOC.

[9]  Santosh S. Vempala,et al.  Statistical Algorithms and a Lower Bound for Detecting Planted Cliques , 2012, J. ACM.

[10]  Prasad Raghavendra,et al.  List Decodable Learning via Sum of Squares , 2019, SODA.

[11]  László Lovász,et al.  Factoring polynomials with rational coefficients , 1982 .

[12]  Santosh S. Vempala,et al.  A spectral algorithm for learning mixtures of distributions , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[13]  Ilya P. Razenshteyn,et al.  Adversarial examples from computational constraints , 2018, ICML.

[14]  László Babai,et al.  On Lovász’ lattice reduction and the nearest lattice point problem , 1986, Comb..

[15]  Schrutka.  Geometrie der Zahlen , 1911 .

[16]  Pravesh Kothari,et al.  Robust moment estimation and improved clustering via sum of squares , 2018, STOC.

[17]  Chris Peikert,et al.  A Decade of Lattice Cryptography , 2016, Found. Trends Theor. Comput. Sci..

[18]  Cynthia Dwork,et al.  A public-key cryptosystem with worst-case/average-case equivalence , 1997, STOC '97.

[19]  K. Pearson.  Contributions to the Mathematical Theory of Evolution , 1894 .

[20]  Michael Kearns,et al.  Efficient noise-tolerant learning from statistical queries , 1993, STOC.

[21]  Sanjoy Dasgupta,et al.  A Probabilistic Analysis of EM for Mixtures of Separated, Spherical Gaussians , 2007, J. Mach. Learn. Res..

[22]  Ilias Diakonikolas,et al.  Learning Structured Distributions , 2016, Handbook of Big Data.

[23]  Daniel M. Kane,et al.  Statistical Query Lower Bounds for Robust Estimation of High-Dimensional Gaussians and Gaussian Mixtures , 2016, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[24]  Dorit Aharonov,et al.  Lattice problems in NP ∩ coNP , 2005, JACM.

[25]  Chris Peikert,et al.  An Efficient and Parallel Gaussian Sampler for Lattices , 2010, CRYPTO.

[26]  Sanjeev Arora,et al.  LEARNING MIXTURES OF SEPARATED NONSPHERICAL GAUSSIANS , 2005, math/0503457.

[27]  Ankur Moitra,et al.  Settling the Polynomial Learnability of Mixtures of Gaussians , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[28]  Rocco A. Servedio,et al.  Testing monotone high-dimensional distributions , 2005, STOC '05.

[29]  Oded Regev,et al.  New lattice based cryptographic constructions , 2003, STOC '03.

[30]  Daniele Micciancio,et al.  Worst-case to average-case reductions based on Gaussian measures , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[31]  Sanjoy Dasgupta,et al.  Learning mixtures of Gaussians , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[32]  Jerry Li,et al.  Mixture models, robustness, and sum of squares proofs , 2017, STOC.

[33]  Damien Stehlé,et al.  Classical hardness of learning with errors , 2013, STOC '13.

[34]  Daniel M. Kane,et al.  List-decodable robust mean estimation and learning mixtures of spherical gaussians , 2017, STOC.

[35]  Santosh S. Vempala,et al.  Isotropic PCA and Affine-Invariant Clustering , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[36]  Oded Regev,et al.  On lattices, learning with errors, random linear codes, and cryptography , 2005, STOC '05.

[37]  L. Devroye,et al.  The total variation distance between high-dimensional Gaussians , 2018, 1810.08693.

[38]  Adam R. Klivans,et al.  List-Decodable Linear Regression , 2019, NeurIPS.