We develop a theory of asymptotic efficiency in regular parametric models when data confidentiality is ensured by local differential privacy (LDP). Even though efficient parameter estimation is a classical and well-studied problem in mathematical statistics, it leads to several non-trivial obstacles that need to be tackled when dealing with the LDP case. Starting from a standard parametric model $\mathcal P=(P_\theta)_{\theta\in\Theta}$, $\Theta\subseteq\mathbb R^p$, for the iid unobserved sensitive data $X_1,\dots, X_n$, we establish local asymptotic mixed normality (along subsequences) of the model $$Q^{(n)}\mathcal P=(Q^{(n)}P_\theta^n)_{\theta\in\Theta}$$ generating the sanitized observations $Z_1,\dots, Z_n$, where $Q^{(n)}$ is an arbitrary sequence of sequentially interactive privacy mechanisms. This result readily implies convolution and local asymptotic minimax theorems. In case $p=1$, the optimal asymptotic variance is found to be the inverse of the supremal Fisher-Information $\sup_{Q\in\mathcal Q_\alpha} I_\theta(Q\mathcal P)\in\mathbb R$, where the supremum runs over all $\alpha$-differentially private (marginal) Markov kernels. We present an algorithm for finding a (nearly) optimal privacy mechanism $\hat{Q}$ and an estimator $\hat{\theta}_n(Z_1,\dots, Z_n)$ based on the corresponding sanitized data that achieves this asymptotically optimal variance.
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We consider decentralized optimization problems in which a number of agents collaborate to minimize the average of their local functions by exchanging over an underlying communication graph. Specifically, we place ourselves in an asynchronous model where only a random portion of nodes perform computation at each iteration, while the information exchange can be conducted between all the nodes and in an asymmetric fashion. For this setting, we propose an algorithm that combines gradient tracking with a network-level variance reduction (in contrast to variance reduction within each node). This enables each node to track the average of the gradients of the objective functions. Our theoretical analysis shows that the algorithm converges linearly, when the local objective functions are strongly convex, under mild connectivity conditions on the expected mixing matrices. In particular, our result does not require the mixing matrices to be doubly stochastic. In the experiments, we investigate a broadcast mechanism that transmits information from computing nodes to their neighbors, and confirm the linear convergence of our method on both synthetic and real-world datasets.
Autoencoders have demonstrated remarkable success in learning low-dimensional latent features of high-dimensional data across various applications. Assuming that data are sampled near a low-dimensional manifold, we employ chart autoencoders, which encode data into low-dimensional latent features on a collection of charts, preserving the topology and geometry of the data manifold. Our paper establishes statistical guarantees on the generalization error of chart autoencoders, and we demonstrate their denoising capabilities by considering $n$ noisy training samples, along with their noise-free counterparts, on a $d$-dimensional manifold. By training autoencoders, we show that chart autoencoders can effectively denoise the input data with normal noise. We prove that, under proper network architectures, chart autoencoders achieve a squared generalization error in the order of $\displaystyle n^{-\frac{2}{d+2}}\log^4 n$, which depends on the intrinsic dimension of the manifold and only weakly depends on the ambient dimension and noise level. We further extend our theory on data with noise containing both normal and tangential components, where chart autoencoders still exhibit a denoising effect for the normal component. As a special case, our theory also applies to classical autoencoders, as long as the data manifold has a global parametrization. Our results provide a solid theoretical foundation for the effectiveness of autoencoders, which is further validated through several numerical experiments.
Our work explores the hardness of $3$SUM instances without certain additive structures, and its applications. As our main technical result, we show that solving $3$SUM on a size-$n$ integer set that avoids solutions to $a+b=c+d$ for $\{a, b\} \ne \{c, d\}$ still requires $n^{2-o(1)}$ time, under the $3$SUM hypothesis. Such sets are called Sidon sets and are well-studied in the field of additive combinatorics. - Combined with previous reductions, this implies that the All-Edges Sparse Triangle problem on $n$-vertex graphs with maximum degree $\sqrt{n}$ and at most $n^{k/2}$ $k$-cycles for every $k \ge 3$ requires $n^{2-o(1)}$ time, under the $3$SUM hypothesis. This can be used to strengthen the previous conditional lower bounds by Abboud, Bringmann, Khoury, and Zamir [STOC'22] of $4$-Cycle Enumeration, Offline Approximate Distance Oracle and Approximate Dynamic Shortest Path. In particular, we show that no algorithm for the $4$-Cycle Enumeration problem on $n$-vertex $m$-edge graphs with $n^{o(1)}$ delays has $O(n^{2-\varepsilon})$ or $O(m^{4/3-\varepsilon})$ pre-processing time for $\varepsilon >0$. We also present a matching upper bound via simple modifications of the known algorithms for $4$-Cycle Detection. - A slight generalization of the main result also extends the result of Dudek, Gawrychowski, and Starikovskaya [STOC'20] on the $3$SUM hardness of nontrivial 3-Variate Linear Degeneracy Testing (3-LDTs): we show $3$SUM hardness for all nontrivial 4-LDTs. The proof of our main technical result combines a wide range of tools: Balog-Szemer{\'e}di-Gowers theorem, sparse convolution algorithm, and a new almost-linear hash function with almost $3$-universal guarantee for integers that do not have small-coefficient linear relations.
Linear regression is a fundamental tool for statistical analysis. This has motivated the development of linear regression methods that also satisfy differential privacy and thus guarantee that the learned model reveals little about any one data point used to construct it. However, existing differentially private solutions assume that the end user can easily specify good data bounds and hyperparameters. Both present significant practical obstacles. In this paper, we study an algorithm which uses the exponential mechanism to select a model with high Tukey depth from a collection of non-private regression models. Given $n$ samples of $d$-dimensional data used to train $m$ models, we construct an efficient analogue using an approximate Tukey depth that runs in time $O(d^2n + dm\log(m))$. We find that this algorithm obtains strong empirical performance in the data-rich setting with no data bounds or hyperparameter selection required.
This paper is concerned with low-rank matrix optimization, which has found a wide range of applications in machine learning. This problem in the special case of matrix sensing has been studied extensively through the notion of Restricted Isometry Property (RIP), leading to a wealth of results on the geometric landscape of the problem and the convergence rate of common algorithms. However, the existing results can handle the problem in the case with a general objective function subject to noisy data only when the RIP constant is close to 0. In this paper, we develop a new mathematical framework to solve the above-mentioned problem with a far less restrictive RIP constant. We prove that as long as the RIP constant of the noiseless objective is less than $1/3$, any spurious local solution of the noisy optimization problem must be close to the ground truth solution. By working through the strict saddle property, we also show that an approximate solution can be found in polynomial time. We characterize the geometry of the spurious local minima of the problem in a local region around the ground truth in the case when the RIP constant is greater than $1/3$. Compared to the existing results in the literature, this paper offers the strongest RIP bound and provides a complete theoretical analysis on the global and local optimization landscapes of general low-rank optimization problems under random corruptions from any finite-variance family.
We consider a binary supervised learning classification problem where instead of having data in a finite-dimensional Euclidean space, we observe measures on a compact space $\mathcal{X}$. Formally, we observe data $D_N = (\mu_1, Y_1), \ldots, (\mu_N, Y_N)$ where $\mu_i$ is a measure on $\mathcal{X}$ and $Y_i$ is a label in $\{0, 1\}$. Given a set $\mathcal{F}$ of base-classifiers on $\mathcal{X}$, we build corresponding classifiers in the space of measures. We provide upper and lower bounds on the Rademacher complexity of this new class of classifiers that can be expressed simply in terms of corresponding quantities for the class $\mathcal{F}$. If the measures $\mu_i$ are uniform over a finite set, this classification task boils down to a multi-instance learning problem. However, our approach allows more flexibility and diversity in the input data we can deal with. While such a framework has many possible applications, this work strongly emphasizes on classifying data via topological descriptors called persistence diagrams. These objects are discrete measures on $\mathbb{R}^2$, where the coordinates of each point correspond to the range of scales at which a topological feature exists. We will present several classifiers on measures and show how they can heuristically and theoretically enable a good classification performance in various settings in the case of persistence diagrams.
This paper proposes a new approach to estimating the distribution of a response variable conditioned on observing some factors. The proposed approach possesses desirable properties of flexibility, interpretability, tractability and extendability. The conditional quantile function is modeled by a mixture (weighted sum) of basis quantile functions, with the weights depending on factors. The calibration problem is formulated as a convex optimization problem. It can be viewed as conducting quantile regressions for all confidence levels simultaneously while avoiding quantile crossing by definition. The calibration problem is equivalent to minimizing the continuous ranked probability score (CRPS). Based on the canonical polyadic (CP) decomposition of tensors, we propose a dimensionality reduction method that reduces the rank of the parameter tensor and propose an alternating algorithm for estimation. Additionally, based on Risk Quadrangle framework, we generalize the approach to conditional distributions defined by Conditional Value-at-Risk (CVaR), expectile and other functions of uncertainty measures. Although this paper focuses on using splines as the weight functions, it can be extended to neural networks. Numerical experiments demonstrate the effectiveness of our approach.
The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin. Traditional parameterised differential equations are a special case. Many popular neural network architectures, such as residual networks and recurrent networks, are discretisations. NDEs are suitable for tackling generative problems, dynamical systems, and time series (particularly in physics, finance, ...) and are thus of interest to both modern machine learning and traditional mathematical modelling. NDEs offer high-capacity function approximation, strong priors on model space, the ability to handle irregular data, memory efficiency, and a wealth of available theory on both sides. This doctoral thesis provides an in-depth survey of the field. Topics include: neural ordinary differential equations (e.g. for hybrid neural/mechanistic modelling of physical systems); neural controlled differential equations (e.g. for learning functions of irregular time series); and neural stochastic differential equations (e.g. to produce generative models capable of representing complex stochastic dynamics, or sampling from complex high-dimensional distributions). Further topics include: numerical methods for NDEs (e.g. reversible differential equations solvers, backpropagation through differential equations, Brownian reconstruction); symbolic regression for dynamical systems (e.g. via regularised evolution); and deep implicit models (e.g. deep equilibrium models, differentiable optimisation). We anticipate this thesis will be of interest to anyone interested in the marriage of deep learning with dynamical systems, and hope it will provide a useful reference for the current state of the art.
We consider the problem of discovering $K$ related Gaussian directed acyclic graphs (DAGs), where the involved graph structures share a consistent causal order and sparse unions of supports. Under the multi-task learning setting, we propose a $l_1/l_2$-regularized maximum likelihood estimator (MLE) for learning $K$ linear structural equation models. We theoretically show that the joint estimator, by leveraging data across related tasks, can achieve a better sample complexity for recovering the causal order (or topological order) than separate estimations. Moreover, the joint estimator is able to recover non-identifiable DAGs, by estimating them together with some identifiable DAGs. Lastly, our analysis also shows the consistency of union support recovery of the structures. To allow practical implementation, we design a continuous optimization problem whose optimizer is the same as the joint estimator and can be approximated efficiently by an iterative algorithm. We validate the theoretical analysis and the effectiveness of the joint estimator in experiments.
This paper focuses on the expected difference in borrower's repayment when there is a change in the lender's credit decisions. Classical estimators overlook the confounding effects and hence the estimation error can be magnificent. As such, we propose another approach to construct the estimators such that the error can be greatly reduced. The proposed estimators are shown to be unbiased, consistent, and robust through a combination of theoretical analysis and numerical testing. Moreover, we compare the power of estimating the causal quantities between the classical estimators and the proposed estimators. The comparison is tested across a wide range of models, including linear regression models, tree-based models, and neural network-based models, under different simulated datasets that exhibit different levels of causality, different degrees of nonlinearity, and different distributional properties. Most importantly, we apply our approaches to a large observational dataset provided by a global technology firm that operates in both the e-commerce and the lending business. We find that the relative reduction of estimation error is strikingly substantial if the causal effects are accounted for correctly.
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