The stochastic block model (SBM) and degree-corrected block model (DCBM) are network models often selected as the fundamental setting in which to analyze the theoretical properties of community detection methods. We consider the problem of spectral clustering of SBM and DCBM networks under a local form of edge differential privacy. Using a randomized response privacy mechanism called the edge-flip mechanism, we develop theoretical guarantees for differentially private community detection, demonstrating conditions under which this strong privacy guarantee can be upheld while achieving spectral clustering convergence rates that match the known rates without privacy. We prove the strongest theoretical results are achievable for dense networks (those with node degree linear in the number of nodes), while weak consistency is achievable under mild sparsity (node degree greater than $n^{-1/2}$). We empirically demonstrate our results on a number of network examples.
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Spectral methods are an important part of scientific computing's arsenal for solving partial differential equations (PDEs). However, their applicability and effectiveness depend crucially on the choice of basis functions used to expand the solution of a PDE. The last decade has seen the emergence of deep learning as a strong contender in providing efficient representations of complex functions. In the current work, we present an approach for combining deep neural networks with spectral methods to solve PDEs. In particular, we use a deep learning technique known as the Deep Operator Network (DeepONet), to identify candidate functions on which to expand the solution of PDEs. We have devised an approach which uses the candidate functions provided by the DeepONet as a starting point to construct a set of functions which have the following properties: i) they constitute a basis, 2) they are orthonormal, and 3) they are hierarchical i.e., akin to Fourier series or orthogonal polynomials. We have exploited the favorable properties of our custom-made basis functions to both study their approximation capability and use them to expand the solution of linear and nonlinear time-dependent PDEs.
Model extraction attacks have renewed interest in the classic problem of learning neural networks from queries. In this work we give the first polynomial-time algorithm for learning arbitrary one hidden layer neural networks activations provided black-box access to the network. Formally, we show that if $F$ is an arbitrary one hidden layer neural network with ReLU activations, there is an algorithm with query complexity and running time that is polynomial in all parameters that outputs a network $F'$ achieving low square loss relative to $F$ with respect to the Gaussian measure. While a number of works in the security literature have proposed and empirically demonstrated the effectiveness of certain algorithms for this problem, ours is the first with fully polynomial-time guarantees of efficiency even for worst-case networks (in particular our algorithm succeeds in the overparameterized setting).
Deep Markov models (DMM) are generative models that are scalable and expressive generalization of Markov models for representation, learning, and inference problems. However, the fundamental stochastic stability guarantees of such models have not been thoroughly investigated. In this paper, we provide sufficient conditions of DMM's stochastic stability as defined in the context of dynamical systems and propose a stability analysis method based on the contraction of probabilistic maps modeled by deep neural networks. We make connections between the spectral properties of neural network's weights and different types of used activation functions on the stability and overall dynamic behavior of DMMs with Gaussian distributions. Based on the theory, we propose a few practical methods for designing constrained DMMs with guaranteed stability. We empirically substantiate our theoretical results via intuitive numerical experiments using the proposed stability constraints.
Global spectral methods offer the potential to compute solutions of partial differential equations numerically to very high accuracy. In this work, we develop a novel global spectral method for linear partial differential equations on cubes by extending ideas of Chebop2 [Townsend and Olver, J. Comput. Phys., 299 (2015)] to the three-dimensional setting utilizing expansions in tensorized polynomial bases. Solving the discretized PDE involves a linear system that can be recast as a linear tensor equation. Under suitable additional assumptions, the structure of these equations admits for an efficient solution via the blocked recursive solver [Chen and Kressner, Numer. Algorithms, 84 (2020)]. In the general case, when these assumptions are not satisfied, this solver is used as a preconditioner to speed up computations.
Zeroth-order (ZO) optimization is widely used to handle challenging tasks, such as query-based black-box adversarial attacks and reinforcement learning. Various attempts have been made to integrate prior information into the gradient estimation procedure based on finite differences, with promising empirical results. However, their convergence properties are not well understood. This paper makes an attempt to fill up this gap by analyzing the convergence of prior-guided ZO algorithms under a greedy descent framework with various gradient estimators. We provide a convergence guarantee for the prior-guided random gradient-free (PRGF) algorithms. Moreover, to further accelerate over greedy descent methods, we present a new accelerated random search (ARS) algorithm that incorporates prior information, together with a convergence analysis. Finally, our theoretical results are confirmed by experiments on several numerical benchmarks as well as adversarial attacks.
We present an asymptotically optimal $(\epsilon,\delta)$ differentially private mechanism for answering multiple, adaptively asked, $\Delta$-sensitive queries, settling the conjecture of Steinke and Ullman [2020]. Our algorithm has a significant advantage that it adds independent bounded noise to each query, thus providing an absolute error bound. Additionally, we apply our algorithm in adaptive data analysis, obtaining an improved guarantee for answering multiple queries regarding some underlying distribution using a finite sample. Numerical computations show that the bounded-noise mechanism outperforms the Gaussian mechanism in many standard settings.
Large organizations that collect data about populations (like the US Census Bureau) release summary statistics that are used by multiple stakeholders for resource allocation and policy making problems. These organizations are also legally required to protect the privacy of individuals from whom they collect data. Differential Privacy (DP) provides a solution to release useful summary data while preserving privacy. Most DP mechanisms are designed to answer a single set of queries. In reality, there are often multiple stakeholders that use a given data release and have overlapping but not-identical queries. This introduces a novel joint optimization problem in DP where the privacy budget must be shared among different analysts. We initiate study into the problem of DP query answering across multiple analysts. To capture the competing goals and priorities of multiple analysts, we formulate three desiderata that any mechanism should satisfy in this setting -- The Sharing Incentive, Non-Interference, and Adaptivity -- while still optimizing for overall error. We demonstrate how existing DP query answering mechanisms in the multi-analyst settings fail to satisfy at least one of the desiderata. We present novel DP algorithms that provably satisfy all our desiderata and empirically show that they incur low error on realistic tasks.
Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, one may obtain an alternative ODE limit, a stochastic differential equation or neither of these. These findings cast doubts on the validity of the neural ODE model as an adequate asymptotic description of deep ResNets and point to an alternative class of differential equations as a better description of the deep network limit.
Alternating Direction Method of Multipliers (ADMM) is a widely used tool for machine learning in distributed settings, where a machine learning model is trained over distributed data sources through an interactive process of local computation and message passing. Such an iterative process could cause privacy concerns of data owners. The goal of this paper is to provide differential privacy for ADMM-based distributed machine learning. Prior approaches on differentially private ADMM exhibit low utility under high privacy guarantee and often assume the objective functions of the learning problems to be smooth and strongly convex. To address these concerns, we propose a novel differentially private ADMM-based distributed learning algorithm called DP-ADMM, which combines an approximate augmented Lagrangian function with time-varying Gaussian noise addition in the iterative process to achieve higher utility for general objective functions under the same differential privacy guarantee. We also apply the moments accountant method to bound the end-to-end privacy loss. The theoretical analysis shows that DP-ADMM can be applied to a wider class of distributed learning problems, is provably convergent, and offers an explicit utility-privacy tradeoff. To our knowledge, this is the first paper to provide explicit convergence and utility properties for differentially private ADMM-based distributed learning algorithms. The evaluation results demonstrate that our approach can achieve good convergence and model accuracy under high end-to-end differential privacy guarantee.
We introduce a new family of deep neural network models. Instead of specifying a discrete sequence of hidden layers, we parameterize the derivative of the hidden state using a neural network. The output of the network is computed using a black-box differential equation solver. These continuous-depth models have constant memory cost, adapt their evaluation strategy to each input, and can explicitly trade numerical precision for speed. We demonstrate these properties in continuous-depth residual networks and continuous-time latent variable models. We also construct continuous normalizing flows, a generative model that can train by maximum likelihood, without partitioning or ordering the data dimensions. For training, we show how to scalably backpropagate through any ODE solver, without access to its internal operations. This allows end-to-end training of ODEs within larger models.
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