An Equivalence Between Private Classification and Online Prediction

We prove that every concept class with finite Littlestone dimension can be learned by an (approximate) differentially-private algorithm. This answers an open question of Alon et al. (STOC 2019) who proved the converse statement (this question was also asked by Neel et al. (FOCS 2019)). Together these two results yield an equivalence between online learnability and private PAC learnability. We introduce a new notion of algorithmic stability called “global stability” which is essential to our proof and may be of independent interest. We also discuss an application of our results to boosting the privacy and accuracy parameters of differentially-private learners.

[1]  Seth Neel,et al.  How to Use Heuristics for Differential Privacy , 2019, 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS).

[2]  Amos Beimel,et al.  Learning Privately with Labeled and Unlabeled Examples , 2014, Algorithmica.

[3]  Hunter Chase,et al.  Model Theory and Machine Learning , 2019, Bull. Symb. Log..

[4]  Elad Hazan,et al.  Introduction to Online Convex Optimization , 2016, Found. Trends Optim..

[5]  Salil P. Vadhan,et al.  The Complexity of Differential Privacy , 2017, Tutorials on the Foundations of Cryptography.

[6]  Cynthia Dwork,et al.  Calibrating Noise to Sensitivity in Private Data Analysis , 2006, TCC.

[7]  Sofya Raskhodnikova,et al.  What Can We Learn Privately? , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[8]  Shay Moran,et al.  Private Learning Implies Online Learning: An Efficient Reduction , 2019, NeurIPS.

[9]  Katrina Ligett,et al.  A necessary and sufficient stability notion for adaptive generalization , 2019 .

[10]  Ambuj Tewari,et al.  Online Learning via Differential Privacy , 2017, ArXiv.

[11]  Gábor Lugosi,et al.  Prediction, learning, and games , 2006 .

[12]  Kobbi Nissim,et al.  Differentially Private Release and Learning of Threshold Functions , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[13]  Elad Hazan,et al.  Competing in the Dark: An Efficient Algorithm for Bandit Linear Optimization , 2008, COLT.

[14]  Adam Tauman Kalai,et al.  Geometric algorithms for online optimization , 2002 .

[15]  Nina Mishra,et al.  Releasing search queries and clicks privately , 2009, WWW '09.

[16]  Badih Ghazi,et al.  Sample-efficient proper PAC learning with approximate differential privacy , 2021, STOC.

[17]  Seth Neel,et al.  The Role of Interactivity in Local Differential Privacy , 2019, 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS).

[18]  Noga Alon,et al.  Closure Properties for Private Classification and Online Prediction , 2020, COLT.

[19]  Shai Ben-David,et al.  Understanding Machine Learning: From Theory to Algorithms , 2014 .

[20]  André Elisseeff,et al.  Stability and Generalization , 2002, J. Mach. Learn. Res..

[21]  Thomas Steinke,et al.  Composable and versatile privacy via truncated CDP , 2018, STOC.

[22]  Tony Doyle,et al.  Weapons of Math Destruction: How Big Data Increases Inequality and Threatens Democracy , 2017, Inf. Soc..

[23]  Harris Mateen.  Weapons of Math Destruction: How Big Data Increases Inequality and Threatens Democracy , 2018 .

[24]  Shafi Goldwasser,et al.  Probabilistic Search Algorithms with Unique Answers and Their Cryptographic Applications , 2011, Electron. Colloquium Comput. Complex..

[25]  Yoram Singer,et al.  A primal-dual perspective of online learning algorithms , 2007, Machine Learning.

[26]  N. Littlestone.  Learning Quickly When Irrelevant Attributes Abound: A New Linear-Threshold Algorithm , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[27]  Haim Kaplan,et al.  Privately Learning Thresholds: Closing the Exponential Gap , 2019, COLT.

[28]  Amos Beimel,et al.  Private Learning and Sanitization: Pure vs. Approximate Differential Privacy , 2013, APPROX-RANDOM.

[29]  Aaron Roth,et al.  The Algorithmic Foundations of Differential Privacy , 2014, Found. Trends Theor. Comput. Sci..

[30]  Igor Carboni Oliveira,et al.  Pseudo-derandomizing learning and approximation , 2018, Electron. Colloquium Comput. Complex..

[31]  Amos Beimel,et al.  Bounds on the sample complexity for private learning and private data release , 2010, Machine Learning.

[32]  Varun Gupta,et al.  On the Compatibility of Privacy and Fairness , 2019, UMAP.

[33]  Kunal Talwar,et al.  Mechanism Design via Differential Privacy , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[34]  Shai Shalev-Shwartz,et al.  Online Learning and Online Convex Optimization , 2012, Found. Trends Mach. Learn..

[35]  Roi Livni,et al.  Passing Tests without Memorizing: Two Models for Fooling Discriminators , 2019, ArXiv.

[36]  Karan Singh,et al.  The Price of Differential Privacy for Online Learning , 2017, ICML.

[37]  Kobbi Nissim,et al.  Simultaneous Private Learning of Multiple Concepts , 2015, ITCS.

[38]  Shai Ben-David,et al.  Agnostic Online Learning , 2009, COLT.

[39]  Leslie G. Valiant,et al.  A theory of the learnable , 1984, STOC '84.

[40]  Mark Bun,et al.  Efficient, Noise-Tolerant, and Private Learning via Boosting , 2020, COLT 2020.

[41]  Toniann Pitassi,et al.  Fairness through awareness , 2011, ITCS '12.

[42]  Amos Beimel,et al.  Characterizing the Sample Complexity of Pure Private Learners , 2019, J. Mach. Learn. Res..

[43]  Noga Alon,et al.  Private PAC learning implies finite Littlestone dimension , 2018, STOC.

[44]  Santosh S. Vempala,et al.  Efficient algorithms for online decision problems , 2005, J. Comput. Syst. Sci..

[45]  Vitaly Feldman,et al.  Sample Complexity Bounds on Differentially Private Learning via Communication Complexity , 2014, SIAM J. Comput..

[46]  Raef Bassily,et al.  Privately Answering Classification Queries in the Agnostic PAC Model , 2019, ALT.

[47]  Guy N. Rothblum,et al.  Boosting and Differential Privacy , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.