# Boundary-Responsive Adaptive Correction Evolution

The authors propose BRACE, the first differentially private retrieval mechanism for log-sum-ReLU dense associative memory (LSR-DAM): it identifies memory points whose membership changes under privacy noise and applies an adaptive boundary correction before injecting calibrated noise, controlling error accumulation from boundary switching in compact-support retrieval. They prove minimax-optimal terminal and full-trajectory retrieval error rates and establish trajectory central limit theorems characterizing privacy-induced variability; synthetic experiments show a tiny perturbation changes 11 gate memberships, and MNIST retrieval reduces prediction MSE from 0.061 for the corrupted input to 0.011, nearly matching noiseless retrieval.

The authors propose BRACE, the first differentially private retrieval mechanism for log-sum-ReLU dense associative memory (LSR-DAM): it identifies memory points whose membership changes under privacy noise and applies an adaptive boundary correction before injecting calibrated noise, controlling error accumulation from boundary switching in compact-support retrieval. They prove minimax-optimal terminal and full-trajectory retrieval error rates and establish trajectory central limit theorems characterizing privacy-induced variability; synthetic experiments show a tiny perturbation changes 11 gate memberships, and MNIST retrieval reduces prediction MSE from 0.061 for the corrupted input to 0.011, nearly matching noiseless retrieval.