Xuelin Yang

Yuval Dagan, Michael I. Jordan, Xuelin Yang et al.

We present differentially private algorithms for high-dimensional mean estimation. Previous private estimators on distributions over $\mathbb{R}^d$ suffer from a curse of dimensionality, as they require $Ω(d^{1/2})$ samples to achieve non-trivial error, even in cases where $O(1)$ samples suffice without privacy. This rate is unavoidable when the distribution is isotropic, namely, when the covariance is a multiple of the identity matrix, or when accuracy is measured with respect to the affine-invariant Mahalanobis distance. Yet, real-world data is often highly anisotropic, with signals concentrated on a small number of principal components. We develop estimators that are appropriate for such signals$\unicode{x2013}$our estimators are $(\varepsilon,δ)$-differentially private and have sample complexity that is dimension-independent for anisotropic subgaussian distributions. Given $n$ samples from a distribution with known covariance-proxy $Σ$ and unknown mean $μ$, we present an estimator $\hatμ$ that achieves error $\|\hatμ-μ\|_2\leq α$, as long as $n\gtrsim\mathrm{tr}(Σ)/α^2+ \mathrm{tr}(Σ^{1/2})/(α\varepsilon)$. In particular, when $\pmbσ^2=(σ_1^2, \ldots, σ_d^2)$ are the singular values of $Σ$, we have $\mathrm{tr}(Σ)=\|\pmbσ\|_2^2$ and $\mathrm{tr}(Σ^{1/2})=\|\pmbσ\|_1$, and hence our bound avoids dimension-dependence when the signal is concentrated in a few principal components. We show that this is the optimal sample complexity for this task up to logarithmic factors. Moreover, for the case of unknown covariance, we present an algorithm whose sample complexity has improved dependence on the dimension, from $d^{1/2}$ to $d^{1/4}$.

Yi Zeng, Xuelin Yang, Li Chen et al.

To address issues of group-level fairness in machine learning, it is natural to adjust model parameters based on specific fairness objectives over a sensitive-attributed validation set. Such an adjustment procedure can be cast within a meta-learning framework. However, naive integration of fairness goals via meta-learning can cause hypergradient conflicts for subgroups, resulting in unstable convergence and compromising model performance and fairness. To navigate this issue, we frame the resolution of hypergradient conflicts as a multi-player cooperative bargaining game. We introduce a two-stage meta-learning framework in which the first stage involves the use of a Nash Bargaining Solution (NBS) to resolve hypergradient conflicts and steer the model toward the Pareto front, and the second stage optimizes with respect to specific fairness goals. Our method is supported by theoretical results, notably a proof of the NBS for gradient aggregation free from linear independence assumptions, a proof of Pareto improvement, and a proof of monotonic improvement in validation loss. We also show empirical effects across various fairness objectives in six key fairness datasets and two image classification tasks.