Private Rate-Constrained Optimization with Applications to Fair Learning

Differentially private stochastic gradient descent (DP-SGD) enables private deep learning through per-example clipping and calibrated Gaussian noise, but its high-variance updates can reduce utility on challenging datasets. We propose \textbf{SMA-DP-SGD}, a \textbf{Spectral Memory-Aware Differentially Private Stochastic Gradient Descent} method that augments DP-SGD with a fractional memory branch built only from previously privatized noisy releases. WeightWatcher-inspired power-law spectral exponents provide group-wise reliability signals, instantiated layer-wise in our experiments, to adapt the decay and effective memory depth. Private-history alignment, norm matching, and warm-up activation stabilize the memory contribution. Privacy remains transparent: conditioned on the private release history, the memory branch is fixed, and the only newly data-dependent term is the current clipped sum scaled by a fixed coefficient $β$. Hence, SMA-DP-SGD preserves a clean conditional sensitivity structure and exactly recovers group-wise DP-SGD when $β=1$. Experiments on CIFAR-100, CIFAR-10, and MNIST show competitive or superior accuracy over several DP optimization baselines, with the largest gains on CIFAR-100 and CIFAR-10. CIFAR-10 ablations show that $β$ controls the privacy--utility trajectory, while spectral and memory diagnostics confirm a controlled short-to-moderate effective memory depth and a small memory-branch ratio. Runtime analysis shows that the mechanism incurs additional overhead, about $2.94\times$ DP-SGD in our CIFAR-10 implementation, revealing a practical trade-off between adaptive private memory and computational cost.

Differential privacy (DP) is increasingly deployed to limit membership inference risk in machine-learning systems. Prior work has shown that DP-SGD can widen accuracy disparities across demographic groups, but this framing treats fairness as a purely outcome-side concern. We argue that privacy cost, the information leakage borne by each group, is itself a form of harm, and adopt a compensatory-fairness framework in which a group that involuntarily bears greater privacy exposure is owed proportionally greater benefit from the system. From this principle we derive the \emph{Privacy-Cost Equity Ratio} (PCER), a group fairness metric defined as a group's positive prediction rate normalized by its per-group overfitting gap. By a standard membership inference bound, this overfitting gap upper-bounds each group's vulnerability to inference attacks, making PCER a conservative measure of benefit relative to exposure. PCER needs only per-group train and test accuracy (no shadow models), making it a practical post-hoc audit tool. We evaluate PCER alongside standard fairness metrics across six benchmark--attribute combinations spanning tabular and NLP domains, under DP-SGD at a range of privacy budgets, and validate the overfitting-gap proxy against a direct threshold membership-inference attack. The results reveal patterns that outcome-based metrics miss. On COMPAS, PCER uncovers a persistent double disadvantage: the protected group bears both greater privacy exposure and worse predictive outcomes, something demographic parity gap masks entirely. Sensitivity analysis shows very strong privacy guarantees collapse both groups' overfitting to a numerical floor, rendering exposure-based audits uninformative in that regime. Together, these findings show that fairness audits of privacy-preserving systems must account for who bears the cost of protection, not only who benefits from its outcomes.

Differentially private stochastic gradient descent (DP-SGD) is a standard approach to privacy-preserving learning based on per-example clipping, subsampling, Gaussian perturbation, and privacy accounting. Classical DP-SGD releases a noisy version of the current clipped subsampled gradient sum. We propose Fractional-Order Differentially Private Stochastic Gradient Descent (\textbf{FO-DP-SGD}), a mechanism-level extension that replaces this current-only query, before Gaussian noise is added, with a fractional recursive query combining the current clipped sum with a finite-window, power-law-weighted aggregation of previously released private sum-level outputs. This injects fractional memory into the release mechanism while preserving the standard \emph{sum-then-noise-then-divide} structure. Under add/remove adjacency with Poisson subsampling, the current-step sensitivity analysis shows that the only newly data-dependent term is the scaled current clipped sum. Hence, conditioned on the private history, the effective $\ell_2$-sensitivity is at most $βC$, where $C$ is the clipping threshold and $β\in(0,1]$ controls the current-step contribution. Thus, FO-DP-SGD admits standard per-step Rényi differential privacy accounting via a Poisson-subsampled Gaussian mechanism with effective noise-to-sensitivity ratio $σ/β$, and composes to yield overall $(\varepsilon,δ)$-differential privacy guarantees. FO-DP-SGD provides a framework for studying long-memory effects in private optimization. The fractional order, memory window, and mixing coefficient govern the trade-off among current-step sensitivity, signal retention, and private-history influence. Experiments on SVHN, CIFAR-10, and CIFAR-100 show improved test accuracy and privacy--utility performance over DP-SGD and private baselines including DP-Adam, DP-IS, SA-DP-SGD, ADP-AdamW, DP-SAT, and DP-Adam-AC.