Characterizing Online and Private Learnability under Distributional Constraints via Generalized Smoothness

We show that every class of distributions that is learnable up to arbitrarily small target error in total variation distance with finitely many samples is also learnable under ($\varepsilon, δ$)-differential privacy up to arbitrarily small target error with finitely many samples. This proves a conjecture of Ashtiani. Previous work by Bun, Kamath, Mouzakis, and Singhal [arXiv:2402.00267] gave a weak refutation of this conjecture, showing that this implication is not true when learnability is defined with respect to a fixed target error. In contrast, we show that, under the standard definition of learnability, the conjecture is true.

Online learning is an attractive framework in many domains because it permits well-defined learning even when data are dependent or chosen adversarially. This generality, however, comes at a steep price, introducing significant statistical and computational barriers. Recently, smoothed online learning has emerged as a promising framework that interpolates between the fully adversarial and fully stochastic settings by assuming that the conditional law of each covariate has density at most $1/σ$ with respect to some fixed base measure $μ$, and it is known to match the statistical and computational guarantees of classical learning while still allowing for much of the flexibility of online learning. However, existing oracle-efficient algorithms require either (i) sampling access to the base measure $μ$ or (ii) labels that are perfectly predicted by a fixed hypothesis. Both assumptions limit the applicability of these algorithms, in contrast to statistical learning, where empirical risk minimization (ERM) learns efficiently in the agnostic setting without any knowledge of the data distribution. We show that neither assumption is necessary, giving the first oracle-efficient algorithm that achieves sublinear regret in the agnostic setting without knowledge of $μ$. Our algorithm, based on Gaussian Follow-The-Perturbed-Leader, is parameter-free: it requires no knowledge of $μ$, the smoothing parameter $σ$, or the horizon $T$, and it achieves regret $\widetilde O(d\sqrt{T/σ})$ for binary classes of VC dimension $d$ with a single call to an ERM oracle per round, which is optimal up to a $\sqrt{d}$ factor. En route to establishing the regret bound, we introduce several new techniques that may be of independent interest.

We study mistake bounds for differentially private online learning and online prediction under oblivious realisable adversaries. Online learning requires the learner to release a hypothesis at each time step whereas in online prediction, the learner only needs to make predictions without releasing a hypothesis. Using a novel lower bound for private online learning and an upper bound for private prediction, we show that the sample complexity of these two problems are separated by a factor that grows with the time horizon for every class of finite Littlestone dimension $d$. First, we prove that every $(ε,δ)$-private online learner has a deterministic realisable stream of length $T$ on which the mistake bound is at least $\mathbb E [M_T]= O\left(\frac dε\log^{2/3} T\right)$. In particular, this is the first non-trivial lower in the range $1/T<δ<1/\log T)$ left open in earlier works[SR22,DSS24,LWY24]. Second, we prove that for every class of of Littlestone dimension $d$, there exists an $(ε,δ)$-jointly private predictor with at most $2^{2^{cd^2}}ε^{-2}\log^2(2/(εδ))$ expected mistakes, independently of $T$, for some absolute constant $c>0$. Thus, for every fixed class of finite Littlestone dimension when $δ=Θ(1/\log T)$, private learning requires $O\left(\log^{2/3} T\right)$ expected mistakes, whereas private prediction admits $O(\log\log T^2)$.