Daogao Liu

Yair Carmon, Arun Jambulapati, Yujia Jin et al.

We introduce a new tool for stochastic convex optimization (SCO): a Reweighted Stochastic Query (ReSQue) estimator for the gradient of a function convolved with a (Gaussian) probability density. Combining ReSQue with recent advances in ball oracle acceleration [CJJJLST20, ACJJS21], we develop algorithms achieving state-of-the-art complexities for SCO in parallel and private settings. For a SCO objective constrained to the unit ball in $\mathbb{R}^d$, we obtain the following results (up to polylogarithmic factors). We give a parallel algorithm obtaining optimization error $ε_{\text{opt}}$ with $d^{1/3}ε_{\text{opt}}^{-2/3}$ gradient oracle query depth and $d^{1/3}ε_{\text{opt}}^{-2/3} + ε_{\text{opt}}^{-2}$ gradient queries in total, assuming access to a bounded-variance stochastic gradient estimator. For $ε_{\text{opt}} \in [d^{-1}, d^{-1/4}]$, our algorithm matches the state-of-the-art oracle depth of [BJLLS19] while maintaining the optimal total work of stochastic gradient descent. Given $n$ samples of Lipschitz loss functions, prior works [BFTT19, BFGT20, AFKT21, KLL21] established that if $n \gtrsim d ε_{\text{dp}}^{-2}$, $(ε_{\text{dp}}, δ)$-differential privacy is attained at no asymptotic cost to the SCO utility. However, these prior works all required a superlinear number of gradient queries. We close this gap for sufficiently large $n \gtrsim d^2 ε_{\text{dp}}^{-3}$, by using ReSQue to design an algorithm with near-linear gradient query complexity in this regime.

Andrew Lowy, Daogao Liu, Hilal Asi

We study private stochastic convex optimization (SCO) under user-level differential privacy (DP) constraints. In this setting, there are $n$ users (e.g., cell phones), each possessing $m$ data items (e.g., text messages), and we need to protect the privacy of each user's entire collection of data items. Existing algorithms for user-level DP SCO are impractical in many large-scale machine learning scenarios because: (i) they make restrictive assumptions on the smoothness parameter of the loss function and require the number of users to grow polynomially with the dimension of the parameter space; or (ii) they are prohibitively slow, requiring at least $(mn)^{3/2}$ gradient computations for smooth losses and $(mn)^3$ computations for non-smooth losses. To address these limitations, we provide novel user-level DP algorithms with state-of-the-art excess risk and runtime guarantees, without stringent assumptions. First, we develop a linear-time algorithm with state-of-the-art excess risk (for a non-trivial linear-time algorithm) under a mild smoothness assumption. Our second algorithm applies to arbitrary smooth losses and achieves optimal excess risk in $\approx (mn)^{9/8}$ gradient computations. Third, for non-smooth loss functions, we obtain optimal excess risk in $n^{11/8} m^{5/4}$ gradient computations. Moreover, our algorithms do not require the number of users to grow polynomially with the dimension.