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We study the problem of the nonparametric estimation for the density $\pi$ of the stationary distribution of a $d$-dimensional stochastic differential equation $(X_t)_{t \in [0, T]}$ with possibly unbounded drift. From the continuous observation of the sampling path on $[0, T]$, we study the rate of estimation of $\pi(x)$ as $T$ goes to infinity. One finding is that, for $d \ge 3$, the rate of estimation depends on the smoothness $\beta = (\beta_1, ... , \beta_d)$ of $\pi$. In particular, having ordered the smoothness such that $\beta_1 \le ... \le \beta_d$, it depends on the fact that $\beta_2 < \beta_3$ or $\beta_2 = \beta_3$. We show that kernel density estimators achieve the rate $(\frac{\log T}{T})^\gamma$ in the first case and $(\frac{1}{T})^\gamma$ in the second, for an explicit exponent $\gamma$ depending on the dimension and on $\bar{\beta}_3$, the harmonic mean of the smoothness over the $d$ directions after having removed $\beta_1$ and $\beta_2$, the smallest ones. Moreover, we obtain a minimax lower bound on the $\mathbf{L}^2$-risk for the pointwise estimation with the same rates $(\frac{\log T}{T})^\gamma$ or $(\frac{1}{T})^\gamma$, depending on the value of $\beta_2$ and $\beta_3$.

We introduce the Weak-form Estimation of Nonlinear Dynamics (WENDy) method for estimating model parameters for non-linear systems of ODEs. The core mathematical idea involves an efficient conversion of the strong form representation of a model to its weak form, and then solving a regression problem to perform parameter inference. The core statistical idea rests on the Errors-In-Variables framework, which necessitates the use of the iteratively reweighted least squares algorithm. Further improvements are obtained by using orthonormal test functions, created from a set of $C^{\infty}$ bump functions of varying support sizes. We demonstrate that WENDy is a highly robust and efficient method for parameter inference in differential equations. Without relying on any numerical differential equation solvers, WENDy computes accurate estimates and is robust to large (biologically relevant) levels of measurement noise. For low dimensional systems with modest amounts of data, WENDy is competitive with conventional forward solver-based nonlinear least squares methods in terms of speed and accuracy. For both higher dimensional systems and stiff systems, WENDy is typically both faster (often by orders of magnitude) and more accurate than forward solver-based approaches. We illustrate the method and its performance in some common population and neuroscience models, including logistic growth, Lotka-Volterra, FitzHugh-Nagumo, Hindmarsh-Rose, and a Protein Transduction Benchmark model. Software and code for reproducing the examples is available at (//github.com/MathBioCU/WENDy).

We adopt an information-theoretic framework to analyze the generalization behavior of the class of iterative, noisy learning algorithms. This class is particularly suitable for study under information-theoretic metrics as the algorithms are inherently randomized, and it includes commonly used algorithms such as Stochastic Gradient Langevin Dynamics (SGLD). Herein, we use the maximal leakage (equivalently, the Sibson mutual information of order infinity) metric, as it is simple to analyze, and it implies both bounds on the probability of having a large generalization error and on its expected value. We show that, if the update function (e.g., gradient) is bounded in $L_2$-norm, then adding isotropic Gaussian noise leads to optimal generalization bounds: indeed, the input and output of the learning algorithm in this case are asymptotically statistically independent. Furthermore, we demonstrate how the assumptions on the update function affect the optimal (in the sense of minimizing the induced maximal leakage) choice of the noise. Finally, we compute explicit tight upper bounds on the induced maximal leakage for several scenarios of interest.

This paper considers distributed optimization (DO) where multiple agents cooperate to minimize a global objective function, expressed as a sum of local objectives, subject to some constraints. In DO, each agent iteratively solves a local optimization model constructed by its own data and communicates some information (e.g., a local solution) with its neighbors until a global solution is obtained. Even though locally stored data are not shared with other agents, it is still possible to reconstruct the data from the information communicated among agents, which could limit the practical usage of DO in applications with sensitive data. To address this issue, we propose a privacy-preserving DO algorithm for constrained convex optimization models, which provides a statistical guarantee of data privacy, known as differential privacy, and a sequence of iterates that converges to an optimal solution in expectation. The proposed algorithm generalizes a linearized alternating direction method of multipliers by introducing a multiple local updates technique to reduce communication costs and incorporating an objective perturbation method in the local optimization models to compute and communicate randomized feasible local solutions that cannot be utilized to reconstruct the local data, thus preserving data privacy. Under the existence of convex constraints, we show that, while both algorithms provide the same level of data privacy, the objective perturbation used in the proposed algorithm can provide better solutions than does the widely adopted output perturbation method that randomizes the local solutions by adding some noise. We present the details of privacy and convergence analyses and numerically demonstrate the effectiveness of the proposed algorithm by applying it in two different applications, namely, distributed control of power flow and federated learning, where data privacy is of concern.

We consider random sample splitting for estimation and inference in high dimensional generalized linear models, where we first apply the lasso to select a submodel using one subsample and then apply the debiased lasso to fit the selected model using the remaining subsample. We show that, no matter including a prespecified subset of regression coefficients or not, the debiased lasso estimation of the selected submodel after a single splitting follows a normal distribution asymptotically. Furthermore, for a set of prespecified regression coefficients, we show that a multiple splitting procedure based on the debiased lasso can address the loss of efficiency associated with sample splitting and produce asymptotically normal estimates under mild conditions. Our simulation results indicate that using the debiased lasso instead of the standard maximum likelihood estimator in the estimation stage can vastly reduce the bias and variance of the resulting estimates. We illustrate the proposed multiple splitting debiased lasso method with an analysis of the smoking data of the Mid-South Tobacco Case-Control Study.

The additive model is a popular nonparametric regression method due to its ability to retain modeling flexibility while avoiding the curse of dimensionality. The backfitting algorithm is an intuitive and widely used numerical approach for fitting additive models. However, its application to large datasets may incur a high computational cost and is thus infeasible in practice. To address this problem, we propose a novel approach called independence-encouraging subsampling (IES) to select a subsample from big data for training additive models. Inspired by the minimax optimality of an orthogonal array (OA) due to its pairwise independent predictors and uniform coverage for the range of each predictor, the IES approach selects a subsample that approximates an OA to achieve the minimax optimality. Our asymptotic analyses demonstrate that an IES subsample converges to an OA and that the backfitting algorithm over the subsample converges to a unique solution even if the predictors are highly dependent in the original big data. The proposed IES method is also shown to be numerically appealing via simulations and a real data application.

We investigate how to efficiently compute the difference result of two (or multiple) conjunctive queries, which is the last operator in relational algebra to be unraveled. The standard approach in practical database systems is to materialize the results for every input query as a separate set, and then compute the difference of two (or multiple) sets. This approach is bottlenecked by the complexity of evaluating every input query individually, which could be very expensive, particularly when there are only a few results in the difference. In this paper, we introduce a new approach by exploiting the structural property of input queries and rewriting the original query by pushing the difference operator down as much as possible. We show that for a large class of difference queries, this approach can lead to a linear-time algorithm, in terms of the input size and (final) output size, i.e., the number of query results that survive from the difference operator. We complete this result by showing the hardness of computing the remaining difference queries in linear time. Although a linear-time algorithm is hard to achieve in general, we also provide some heuristics that can provably improve the standard approach. At last, we compare our approach with standard SQL engines over graph and benchmark datasets. The experiment results demonstrate order-of-magnitude speedups achieved by our approach over the vanilla SQL.

In this paper, we target the problem of sufficient dimension reduction with symmetric positive definite matrices valued responses. We propose the intrinsic minimum average variance estimation method and the intrinsic outer product gradient method which fully exploit the geometric structure of the Riemannian manifold where responses lie. We present the algorithms for our newly developed methods under the log-Euclidean metric and the log-Cholesky metric. Each of the two metrics is linked to an abelian Lie group structure that transforms our model defined on a manifold into a Euclidean one. The proposed methods are then further extended to general Riemannian manifolds. We establish rigourous asymptotic results for the proposed estimators, including the rate of convergence and the asymptotic normality. We also develop a cross validation algorithm for the estimation of the structural dimension with theoretical guarantee Comprehensive simulation studies and an application to the New York taxi network data are performed to show the superiority of the proposed methods.

Strict stationarity is a common assumption used in the time series literature in order to derive asymptotic distributional results for second-order statistics, like sample autocovariances and sample autocorrelations. Focusing on weak stationarity, this paper derives the asymptotic distribution of the maximum of sample autocovariances and sample autocorrelations under weak conditions by using Gaussian approximation techniques. The asymptotic theory for parameter estimation obtained by fitting a (linear) autoregressive model to a general weakly stationary time series is revisited and a Gaussian approximation theorem for the maximum of the estimators of the autoregressive coefficients is derived. To perform statistical inference for the second order parameters considered, a bootstrap algorithm, the so-called second-order wild bootstrap, is applied. Consistency of this bootstrap procedure is proven. In contrast to existing bootstrap alternatives, validity of the second-order wild bootstrap does not require the imposition of strict stationary conditions or structural process assumptions, like linearity. The good finite sample performance of the second-order wild bootstrap is demonstrated by means of simulations.

Data imbalance is a common problem in machine learning that can have a critical effect on the performance of a model. Various solutions exist but their impact on the convergence of the learning dynamics is not understood. Here, we elucidate the significant negative impact of data imbalance on learning, showing that the learning curves for minority and majority classes follow sub-optimal trajectories when training with a gradient-based optimizer. This slowdown is related to the imbalance ratio and can be traced back to a competition between the optimization of different classes. Our main contribution is the analysis of the convergence of full-batch (GD) and stochastic gradient descent (SGD), and of variants that renormalize the contribution of each per-class gradient. We find that GD is not guaranteed to decrease the loss for each class but that this problem can be addressed by performing a per-class normalization of the gradient. With SGD, class imbalance has an additional effect on the direction of the gradients: the minority class suffers from a higher directional noise, which reduces the effectiveness of the per-class gradient normalization. Our findings not only allow us to understand the potential and limitations of strategies involving the per-class gradients, but also the reason for the effectiveness of previously used solutions for class imbalance such as oversampling.