Differentially Private Worst-Case Ratio Analysis via Entropic Smoothing

Discriminative dimensionality reduction seeks a subspace in which the classes remain separated, and the pairs of classes that lie closest together are the hardest to keep apart. In this paper we investigate pairwise worst-case ratio analysis (PWCRA), a criterion that protects those pairs by maximizing, over all class pairs, the smallest ratio of the between-class scatter of a pair to its own within-class scatter. The resulting problem is a non-convex, non-smooth max–min fractional program. A second difficulty is that the learned projection is a function of class means and covariances, so releasing it can reveal information about individual samples. We address both with entropic smoothing. Reversing the sign of the criterion turns it into the minimization of a pointwise maximum, and replacing that maximum by a log-sum-exp function gives a smooth surrogate whose gap to the exact maximum is additive, independent of the ambient and the target dimension, and vanishing as the temperature tends to zero. Built on majorization–minimization principles, the resulting single-loop algorithm, ESRA, assembles at every iteration a gradient with closed-form softmax weights and then takes one polar-factor update, with no inner solver. We then make the same pipeline differentially private by releasing the per-class sufficient statistics once through a Gaussian mechanism, so that the whole optimization is post-processing and the privacy budget does not grow with the number of iterations. We prove privacy, monotone descent, stationary convergence and an end-to-end utility bound, together with a stability bound for the softmax weights that sets the temperature from the noise level of the mechanism, so the tighter the budget, the warmer the smoothing at which we stop. The private variant follows the privacy budget closely, and in terms of the nearest-neighbour, nearest-mean and quadratic discriminant error rates, supported by paired significance tests, ESRA outperforms the state-of-the-art and benchmark methods.