Learner-Private Online Convex Optimization

Online convex optimization is a framework where a learner sequentially queries an external data source in order to arrive at the optimal solution of a convex function. The paradigm has gained significant popularity recently thanks to its scalability in large-scale optimization and machine learning. The repeated interactions, however, expose the learner to privacy risks from eavesdropping adversary that observe the submitted queries. In this paper, we study how to optimally obfuscate the learner’s queries in first-order online convex optimization, so that their learned optimal value is provably difficult to estimate for the eavesdropping adversary. We consider two formulations of learner privacy: a Bayesian formulation in which the convex function is drawn randomly, and a minimax formulation in which the function is fixed and the adversary’s probability of error is measured with respect to a minimax criterion. We show that, if the learner wants to ensure the probability of accurate prediction by the adversary be kept below 1/L, then the overhead in query complexity is additive in L in the minimax formulation, but multiplicative in L in the Bayesian formulation. Compared to existing learnerprivate sequential learning models with binary feedback, our results apply to the significantly richer family of general convex functions with full-gradient feedback. Our proofs are largely enabled by tools from the theory of Dirichlet processes, as well as more sophisticated lines of analysis aimed at measuring the amount of information leakage under a full-gradient oracle.

[1]  Kuang Xu,et al.  Optimal query complexity for private sequential learning against eavesdropping , 2019 .

[2]  M. Abramowitz,et al.  Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables , 1966 .

[3]  Ian Goodfellow,et al.  Deep Learning with Differential Privacy , 2016, CCS.

[4]  Cynthia Dwork,et al.  Differential Privacy: A Survey of Results , 2008, TAMC.

[5]  L. Gordon,et al.  The Gamma Function , 1994, Series and Products in the Development of Mathematics.

[6]  Convex Optimization in Signal Processing and Communications , 2010 .

[7]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[8]  John N. Tsitsiklis,et al.  Private Sequential Learning , 2018, COLT.

[9]  Jingliang Duan,et al.  Improving Generalization of Reinforcement Learning with Minimax Distributional Soft Actor-Critic , 2020, 2020 IEEE 23rd International Conference on Intelligent Transportation Systems (ITSC).

[10]  Pramod Viswanath,et al.  Spy vs. Spy , 2014, SIGMETRICS.

[11]  Mine Su Erturk,et al.  Dynamically Protecting Privacy, under Uncertainty , 2019, ArXiv.

[12]  Pravesh Kothari,et al.  25th Annual Conference on Learning Theory Differentially Private Online Learning , 2022 .

[13]  Wuqiong Luo,et al.  Infection Spreading and Source Identification: A Hide and Seek Game , 2015, IEEE Transactions on Signal Processing.

[14]  Blaise Agüera y Arcas,et al.  Communication-Efficient Learning of Deep Networks from Decentralized Data , 2016, AISTATS.

[15]  Sanjiv Kumar,et al.  cpSGD: Communication-efficient and differentially-private distributed SGD , 2018, NeurIPS.

[16]  Sewoong Oh,et al.  Privacy-Utility Tradeoffs in Routing Cryptocurrency over Payment Channel Networks , 2020, Proc. ACM Meas. Anal. Comput. Syst..

[17]  Samuel Marchal,et al.  PRADA: Protecting Against DNN Model Stealing Attacks , 2018, 2019 IEEE European Symposium on Security and Privacy (EuroS&P).

[18]  Katja Ickstadt,et al.  Bayesian Nonparametric Estimation of Continuous Monotone Functions with Applications to Dose–Response Analysis , 2009, Biometrics.

[19]  Richard Nock,et al.  Advances and Open Problems in Federated Learning , 2021, Found. Trends Mach. Learn..

[20]  Rajesh Sundaresan,et al.  Separable Convex Optimization Problems with Linear Ascending Constraints , 2007, SIAM J. Optim..

[21]  Stephen P. Boyd,et al.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , 2011, Found. Trends Mach. Learn..

[22]  Anand D. Sarwate,et al.  Stochastic gradient descent with differentially private updates , 2013, 2013 IEEE Global Conference on Signal and Information Processing.

[23]  Wei Tang,et al.  Optimal Query Complexity of Secure Stochastic Convex Optimization , 2021, NeurIPS.

[24]  John N. Tsitsiklis,et al.  Delay-Predictability Trade-offs in Reaching a Secret Goal , 2018, Oper. Res..

[25]  Brian Neelon,et al.  Bayesian Isotonic Regression and Trend Analysis , 2004, Biometrics.

[26]  Kuang Xu,et al.  Query Complexity of Bayesian Private Learning , 2019, NeurIPS.

[27]  Audris Mockus,et al.  A nonparametric Bayes method for isotonic regression , 1995 .

[28]  Horst Alzer,et al.  On some inequalities for the gamma and psi functions , 1997, Math. Comput..

[29]  Anders Forsgren,et al.  Minimax optimization for handling range and setup uncertainties in proton therapy. , 2011, Medical physics.

[30]  Vitaly Shmatikov,et al.  Exploiting Unintended Feature Leakage in Collaborative Learning , 2018, 2019 IEEE Symposium on Security and Privacy (SP).