Private optimization in the interpolation regime: faster rates and hardness results
This addresses privacy-preserving optimization for machine learning, particularly in settings with interpolating data, but is incremental as it builds on known private optimization frameworks.
The paper tackles the problem of private stochastic convex optimization in the interpolation regime, showing that while non-private methods achieve faster convergence, similar improvements are generally impossible privately, but near-exponential improvements occur with quadratic growth, reducing sample complexity from $\frac{d}{\varepsilon \sqrt{\alpha}}$ to $\frac{1}{\alpha^{\rho}} + \frac{d}{\varepsilon} \log\left(\frac{1}{\alpha}\right)$ for error $\alpha$.
In non-private stochastic convex optimization, stochastic gradient methods converge much faster on interpolation problems -- problems where there exists a solution that simultaneously minimizes all of the sample losses -- than on non-interpolating ones; we show that generally similar improvements are impossible in the private setting. However, when the functions exhibit quadratic growth around the optimum, we show (near) exponential improvements in the private sample complexity. In particular, we propose an adaptive algorithm that improves the sample complexity to achieve expected error $α$ from $\frac{d}{\varepsilon \sqrtα}$ to $\frac{1}{α^ρ} + \frac{d}{\varepsilon} \log\left(\frac{1}α\right)$ for any fixed $ρ>0$, while retaining the standard minimax-optimal sample complexity for non-interpolation problems. We prove a lower bound that shows the dimension-dependent term is tight. Furthermore, we provide a superefficiency result which demonstrates the necessity of the polynomial term for adaptive algorithms: any algorithm that has a polylogarithmic sample complexity for interpolation problems cannot achieve the minimax-optimal rates for the family of non-interpolation problems.