BRACE: Differential Privacy for Dense Associative Memory with LSR Energy

Using large deviations theory, we solve and obtain a general expression for the free energy functional for a broad class of associative memories, including dense associative memories. We illustrate the method by reproducing classical results for the Hopfield model. For a finite number of patterns, we derive the temperature-dependent free energy functional for dense associative memories featuring polynomial interactions and Log-Sum-Exponential (LSE) activation. We also evaluate the disorder-averaged ground-state energy of these systems in the extensive limit. Our analytical framework reveals how memory retrieval depends on the initial state in higher-order dense networks, and gives the exact full-retrieval threshold for the LSE model. This method provides a systematic procedure for analyzing diverse, complex architectures in associative memory.

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.

Dense retrieval, the key component of Retrieval Augmented Generation (RAG), retrieves the most relevant documents by comparing dense vector representations of queries and passages from a large corpus. In privacy-sensitive applications, the server observes the query and controls which evidence is returned, creating both confidentiality and integrity risks. We formulate private dense retrieval as providing query privacy and retrieval integrity against a malicious server, and develop a two-round cryptographic protocol that provides both guarantees. Our protocol reduces private and verifiable retrieval to multiplication of a committed matrix by an encrypted vector and uses low-bit quantization to make this computation practical. We evaluate the resulting trade-off between cryptographic cost, retrieval quality, and downstream RAG accuracy across six embedding models, four language models, and corpora of up to 2.68 million passages. Our results show that, with a clipped quantizer, three-bit quantization largely preserves retrieval quality and downstream accuracy, while a private query over a corpus the size of a clinical reference requires one to three minutes of server time. These results suggest that private dense retrieval is already practical for moderately sized, privacy-sensitive corpora when minute-scale latency is acceptable.