While various complexity measures for deep neural networks exist, specifying an appropriate measure capable of predicting and explaining generalization in deep networks has provenchallenging.

Buffered Linear Toeplitz (BLT) matrices are a family of parameterized lower-triangular matrices that play an important role in streaming differential privacy with correlated noise. Our main result is a BLT inversion theorem: the inverse of a BLT matrix is itself a BLT matrix with different parameters. We also present an efficient and differentiable $O(d^3)$ algorithm to compute the parameters of the inverse BLT matrix, where $d$ is the degree of the original BLT (typically $d < 10$). Our characterization enables direct optimization of BLT parameters for privacy mechanisms through automatic differentiation.

The state-of-the-art for training on-device language models for mobile keyboard applications combines federated learning (FL) with differential privacy (DP) via the DP-Follow-the-Regularized-Leader (DP-FTRL) algorithm. Two variants of DP-FTRL are used in practice, tree aggregation and matrix factorization. However, tree aggregation suffers from significantly suboptimal privacy/utility tradeoffs, while matrix mechanisms require expensive optimization parameterized by hard-to-estimate-in-advance constants, and high runtime memory costs.This paper extends the recently introduced Buffered Linear Toeplitz (BLT) mechanism to multi-participation scenarios. Our BLT-DP-FTRL maintains the ease-of-use advantages of tree aggregation, while essentially matching matrix factorization in terms of utility and privacy. We evaluate BLT-DP-FTRL on the StackOverflow dataset, serving as a re-producible simulation benchmark, and across four on-device language model tasks in a production FL system. Our empirical results highlight the advantages of the BLT mechanism and elevate the practicality and effectiveness of DP in real-world scenarios.

Objective: Transformers, born to remedy the inadequate receptive fields of CNNs, have drawn explosive attention recently. However, the daunting computational complexity of global representation learning, together with rigid window partitioning, hinders their deployment in medical image segmentation. This work aims to address the above two issues in transformers for better medical image segmentation. Methods: We propose a boundary-aware lightweight transformer (BATFormer) that can build cross-scale global interaction with lower computational complexity and generate windows flexibly under the guidance of entropy. Specifically, to fully explore the benefits of transformers in long-range dependency establishment, a cross-scale global transformer (CGT) module is introduced to jointly utilize multiple small-scale feature maps for richer global features with lower computational complexity. Given the importance of shape modeling in medical image segmentation, a boundary-aware local transformer (BLT) module is constructed. Different from rigid window partitioning in vanilla transformers which would produce boundary distortion, BLT adopts an adaptive window partitioning scheme under the guidance of entropy for both computational complexity reduction and shape preservation. Results: BATFormer achieves the best performance in Dice of 92.84%, 91.97%, 90.26%, and 96.30% for the average, right ventricle, myocardium, and left ventricle respectively on the ACDC dataset and the best performance in Dice, IoU, and ACC of 90.76%, 84.64%, and 96.76% respectively on the ISIC 2018 dataset. More importantly, BATFormer requires the least amount of model parameters and the lowest computational complexity compared to the state-of-the-art approaches. Conclusion and Significance: Our results demonstrate the necessity of developing customized transformers for efficient and better medical image segmentation.