Privacy Amplification of Iterative Algorithms via Contraction Coefficients

Neural Stochastic Contraction Metrics for Learning-based Control and Estimation

We present Neural Stochastic Contraction Metrics (NSCM), a new design framework for provably-stable robust control and estimation for a class of stochastic nonlinear systems. It uses a spectrally-normalized deep neural network to construct a contraction metric, sampled via simplified convex optimization in the stochastic setting. Spectral normalization constrains the state-derivatives of the metric to be Lipschitz continuous, thereby ensuring exponential boundedness of the mean squared distance of system trajectories under stochastic disturbances. The NSCM framework allows autonomous agents to…

A Converse to Banach's Fixed Point Theorem and its CLS Completeness

Banach's fixed point theorem for contraction maps has been widely used to analyze the convergence of iterative methods in non-convex problems. It is a common experience, however, that iterative maps fail to be globally contracting under the natural metric in their domain, making the applicability of Banach's theorem limited. We explore how generally we can apply Banach's fixed point theorem to establish the convergence of iterative methods when pairing it with carefully designed metrics. Our first result is a strong converse of Banach's theorem, showing that it is a universal analysis tool for…

Iterative Decoding of Trellis-Constrained Codes inspired by Amplitude Amplification (Preliminary Version)

We propose a decoder for Trellis-Constrained Codes, a super-class of Turbo- and LDPC codes. Inspired by amplitude amplification from quantum computing, we attempt to amplify the relative likelihood of the most likely codeword until it stands out from all other codewords.

具有（强）凸目标函数的ADMM的迭代隐私放大

Privacy Amplification by Iteration for ADMM with (Strongly) Convex Objective Functions

该研究针对（强）凸目标的梯度型私有ADMM变体，通过定制范数、将连续两次迭代视为马尔可夫算子解决技术难题，证明其隐私保障可随迭代次数放大，强凸场景下呈指数增长。

A Fixed point view: A Model-Based Clustering Framework

With the inflation of the data, clustering analysis, as a branch of unsupervised learning, lacks unified understanding and application of its mathematical law. Based on the view of fixed point, this paper restates the model-based clustering and proposes a unified clustering framework. In order to find fixed points as cluster centers, the framework iteratively constructs the contraction map, which strongly reveals the convergence mechanism and interconnections among algorithms. By specifying a contraction map, Gaussian mixture model (GMM) can be mapped to the framework as an application. We hop…

迭代马尔可夫拟合的指数收敛保证

Exponential Convergence Guarantees for Iterative Markovian Fitting

本文首次为迭代马尔可夫拟合（IMF）在温和结构假设下提供非渐近指数收敛保证，涵盖对数凹与弱对数凹边际情形，并为扩散薛定谔桥匹配（DSBM）奠定理论基础。

Differentially Quantized Gradient Methods

Consider the following distributed optimization scenario. A worker has access to training data that it uses to compute the gradients while a server decides when to stop iterative computation based on its target accuracy or delay constraints. The server receives all its information about the problem instance from the worker via a rate-limited noiseless communication channel. We introduce the principle we call Differential Quantization (DQ) that prescribes compensating the past quantization errors to direct the descent trajectory of a quantized algorithm towards that of its unquantized counterpa…

这是我们的损失：非凸损失下隐藏状态DP-SGD不存在隐私放大效应

It's Our Loss: No Privacy Amplification for Hidden State DP-SGD With Non-Convex Loss

本研究针对隐藏状态DP-SGD是否存在通用隐私放大的开放问题，通过构造反例从理论和实证层面证明，非凸等一般损失函数下最终迭代泄露信息量与全迭代序列相当，现有DP-SGD隐私分析是紧的，不存在普遍隐私放大效应。

Learning Stabilizable Nonlinear Dynamics with Contraction-Based Regularization

We propose a novel framework for learning stabilizable nonlinear dynamical systems for continuous control tasks in robotics. The key contribution is a control-theoretic regularizer for dynamics fitting rooted in the notion of stabilizability, a constraint which guarantees the existence of robust tracking controllers for arbitrary open-loop trajectories generated with the learned system. Leveraging tools from contraction theory and statistical learning in Reproducing Kernel Hilbert Spaces, we formulate stabilizable dynamics learning as a functional optimization with convex objective and bi-conv…

散度不等式及其在遍历理论中的应用

Divergence Inequalities with Applications in Ergodic Theory

本文为二次可微的 $f$-散度提出了基于 $χ^{2}$-散度的一般界限，刻画了时齐 Markov 链的收缩速率与混合时间，并将结果扩展至量子信息论中的 Petz $f$-散度。

The MIMO Iterative Waterfilling Algorithm

This paper considers the non-cooperative maximization of mutual information in the vector Gaussian interference channel in a fully distributed fashion via game theory. This problem has been widely studied in a number of works during the past decade for frequency-selective channels, and recently for the more general MIMO case, for which the state-of-the art results are valid only for nonsingular square channel matrices. Surprisingly, these results do not hold true when the channel matrices are rectangular and/or rank deficient matrices. The goal of this paper is to provide a complete characteri…