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1. An Equivalence Between Private Classification and Online Prediction

   Bun, Mark; Livni, Roi; Moran, Shay (January 2020, 61st Annual IEEE Symposium on Foundations of Computer Science)

   null (Ed.)

   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. 

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2. Limits of Private Learning with Access to Public Data

   Alon, N.; Bassily, R.; and Moran, S. (October 2019, Advances in neural information processing systems)

   We consider learning problems where the training set consists of two types of examples: private and public. The goal is to design a learning algorithm that satisfies differential privacy only with respect to the private examples. This setting interpolates between private learning (where private) and classical learning (where all examples are public). We study the limits of learning in this setting in terms of private and public sample complexities. We show that any hypothesis class of VC-dimension d can be agnostically learned up to an excess error of α using only (roughly) d/α public examples and d/α2 private labeled examples. This result holds even when the public examples are unlabeled. This gives a quadratic improvement over the standard d/α2 upper bound on the public sample complexity (where private examples can be ignored altogether if the public examples are labeled). Furthermore, we give a nearly matching lower bound, which we prove via a generic reduction from this setting to the one of private learning without public data. 

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3. Smoothed Analysis of Online and Differentially Private Learning

   Haghtalab, Nika; Roughgarden, Tim; Shetty, Abhishek (October 2020, Advances in neural information processing systems)

   Practical and pervasive needs for robustness and privacy in algorithms have inspired the design of online adversarial and differentially private learning algorithms. The primary quantity that characterizes learnability in these settings is the Littlestone dimension of the class of hypotheses [Alon et al., 2019, Ben-David et al., 2009]. This characterization is often interpreted as an impossibility result because classes such as linear thresholds and neural networks have infinite Littlestone dimension. In this paper, we apply the framework of smoothed analysis [Spielman and Teng, 2004], in which adversarially chosen inputs are perturbed slightly by nature. We show that fundamentally stronger regret and error guarantees are possible with smoothed adversaries than with worst-case adversaries. In particular, we obtain regret and privacy error bounds that depend only on the VC dimension and the bracketing number of a hypothesis class, and on the magnitudes of the perturbations. 

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4. Robust Empirical Risk Minimization with Tolerance

   Bhattacharjee, Robi; Hopkins, Max; Kumar, Akash; Yu, Hantao; Chaudhuri, Kamalika (February 2023, Algorithmic Learning Theory)

   Developing simple, sample-efficient learning algorithms for robust classification is a pressing issue in today's tech-dominated world, and current theoretical techniques requiring exponential sample complexity and complicated improper learning rules fall far from answering the need. In this work we study the fundamental paradigm of (robust) empirical risk minimization (RERM), a simple process in which the learner outputs any hypothesis minimizing its training error. RERM famously fails to robustly learn VC classes (Montasser et al., 2019a), a bound we show extends even to `nice' settings such as (bounded) halfspaces. As such, we study a recent relaxation of the robust model called tolerant robust learning (Ashtiani et al., 2022) where the output classifier is compared to the best achievable error over slightly larger perturbation sets. We show that under geometric niceness conditions, a natural tolerant variant of RERM is indeed sufficient for γ-tolerant robust learning VC classes over ℝd, and requires only Õ (VC(H)dlogDγδϵ2) samples for robustness regions of (maximum) diameter D. 

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5. Efficient, Noise-Tolerant, and Private Learning via Boosting

   Bun, Mark; Carmosino, Marco L.; Sorrell, Jessica (January 2020, Proceedings of Thirty Third Conference on Learning Theory)

   null (Ed.)

   We introduce a simple framework for designing private boosting algorithms. We give natural conditions under which these algorithms are differentially private, efficient, and noise-tolerant PAC learners. To demonstrate our framework, we use it to construct noise-tolerant and private PAC learners for large-margin halfspaces whose sample complexity does not depend on the dimension. We give two sample complexity bounds for our large-margin halfspace learner. One bound is based only on differential privacy, and uses this guarantee as an asset for ensuring generalization. This first bound illustrates a general methodology for obtaining PAC learners from privacy, which may be of independent interest. The second bound uses standard techniques from the theory of large-margin classification (the fat-shattering dimension) to match the best known sample complexity for differentially private learning of large-margin halfspaces, while additionally tolerating random label noise. 

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