Langevin Diffusion: An Almost Universal Algorithm for Private Euclidean (Convex) Optimization
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In this paper we revisit the problem of differentially private empirical risk minimization (DP-ERM) and stochastic convex optimization (DP-SCO). We show that a well-studied continuous time algorithm from statistical physics called Langevin diffusion (LD) simultaneously provides optimal privacy/utility tradeoffs for both DP-ERM and DP-SCO under ϵ-DP and (ϵ,δ)-DP. Using the uniform stability properties of LD, we provide the optimal excess population risk guarantee for ℓ_2-Lipschitz convex losses under ϵ-DP (even up to log n factors), thus improving on Asi et al. Along the way we provide various technical tools which can be of independent interest: i) A new Rényi divergence bound for LD when run on loss functions over two neighboring data sets, ii) Excess empirical risk bounds for last-iterate LD analogous to that of Shamir and Zhang for noisy stochastic gradient descent (SGD), and iii) A two phase excess risk analysis of LD, where the first phase is when the diffusion has not converged in any reasonable sense to a stationary distribution, and in the second phase when the diffusion has converged to a variant of Gibbs distribution. Our universality results crucially rely on the dynamics of LD. When it has converged to a stationary distribution, we obtain the optimal bounds under ϵ-DP. When it is run only for a very short time ∝ 1/p, we obtain the optimal bounds under (ϵ,δ)-DP. Here, p is the dimensionality of the model space. Our work initiates a systematic study of DP continuous time optimization. We believe this may have ramifications in the design of discrete time DP optimization algorithms analogous to that in the non-private setting, where continuous time dynamical viewpoints have helped in designing new algorithms, including the celebrated mirror-descent and Polyak's momentum method.
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