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A practical challenge for structural estimation is the requirement to accurately minimize a sample objective function which is often non-smooth, non-convex, or both. This paper proposes a simple algorithm designed to find accurate solutions without performing an exhaustive search. It augments each iteration from a new Gauss-Newton algorithm with a grid search step. A finite sample analysis derives its optimization and statistical properties simultaneously using only econometric assumptions. After a finite number of iterations, the algorithm automatically transitions from global to fast local convergence, producing accurate estimates with high probability. Simulated examples and an empirical application illustrate the results.

Recently, we have classified Hermitian random matrix ensembles that are invariant under the conjugate action of the unitary group and stable with respect to matrix addition. Apart from a scaling and a shift, the whole information of such an ensemble is encoded in the stability exponent determining the ``heaviness'' of the tail and the spectral measure that describes the anisotropy of the probability distribution. In the present work, we address the question how these ensembles can be generated by the knowledge of the latter two quantities. We consider a sum of a specific construction of identically and independently distributed random matrices that are based on Haar distributed unitary matrices and a stable random vectors. For this construction, we derive the rate of convergence in the supremums norm and show that this rate is optimal in the class of all stable invariant random matrices for a fixed stability exponent. As a consequence we also give the rate of convergence in the total variation distance.

This paper is motivated by the need to quantify human immune responses to environmental challenges. Specifically, the genome of the selected cell population from a blood sample is amplified by the well-known PCR process of successive heating and cooling, producing a large number of reads. They number roughly 30,000 to 300,000. Each read corresponds to a particular rearrangement of so-called V(D)J sequences. In the end, the observation consists of a set of numbers of reads corresponding to different V(D)J sequences. The underlying relative frequencies of distinct V(D)J sequences can be summarized by a probability vector, with the cardinality being the number of distinct V(D)J rearrangements present in the blood. Statistical question is to make inferences on a summary parameter of the probability vector based on a single multinomial-type observation of a large dimension. Popular summary of the diversity of a cell population includes clonality and entropy, or more generally, is a suitable function of the probability vector. A point estimator of the clonality based on multiple replicates from the same blood sample has been proposed previously. After obtaining a point estimator of a particular function, the remaining challenge is to construct a confidence interval of the parameter to appropriately reflect its uncertainty. In this paper, we have proposed to couple the empirical Bayes method with a resampling-based calibration procedure to construct a robust confidence interval for different population diversity parameters. The method has been illustrated via extensive numerical study and real data examples.

In building practical applications of evolutionary computation (EC), two optimizations are essential. First, the parameters of the search method need to be tuned to the domain in order to balance exploration and exploitation effectively. Second, the search method needs to be distributed to take advantage of parallel computing resources. This paper presents BLADE (BLAnket Distributed Evolution) as an approach to achieving both goals simultaneously. BLADE uses blankets (i.e., masks on the genetic representation) to tune the evolutionary operators during the search, and implements the search through hub-and-spoke distribution. In the paper, (1) the blanket method is formalized for the (1 + 1)EA case as a Markov chain process. Its effectiveness is then demonstrated by analyzing dominant and subdominant eigenvalues of stochastic matrices, suggesting a generalizable theory; (2) the fitness-level theory is used to analyze the distribution method; and (3) these insights are verified experimentally on three benchmark problems, showing that both blankets and distribution lead to accelerated evolution. Moreover, a surprising synergy emerges between them: When combined with distribution, the blanket approach achieves more than $n$-fold speedup with $n$ clients in some cases. The work thus highlights the importance and potential of optimizing evolutionary computation in practical applications.

In a recent paper, Ling et al. investigated the over-parametrized Deep Equilibrium Model (DEQ) with ReLU activation and proved that the gradient descent converges to a globally optimal solution at a linear convergence rate for the quadratic loss function. In this paper, we show that this fact still holds for DEQs with any general activation which has bounded first and second derivatives. Since the new activation function is generally non-linear, a general population Gram matrix is designed, and a new form of dual activation with Hermite polynomial expansion is developed.

We prove a convergence theorem for U-statistics of degree two, where the data dimension $d$ is allowed to scale with sample size $n$. We find that the limiting distribution of a U-statistic undergoes a phase transition from the non-degenerate Gaussian limit to the degenerate limit, regardless of its degeneracy and depending only on a moment ratio. A surprising consequence is that a non-degenerate U-statistic in high dimensions can have a non-Gaussian limit with a larger variance and asymmetric distribution. Our bounds are valid for any finite $n$ and $d$, independent of individual eigenvalues of the underlying function, and dimension-independent under a mild assumption. As an application, we apply our theory to two popular kernel-based distribution tests, MMD and KSD, whose high-dimensional performance has been challenging to study. In a simple empirical setting, our results correctly predict how the test power at a fixed threshold scales with $d$ and the bandwidth.

In this work we consider stochastic gradient descent (SGD) for solving linear inverse problems in Banach spaces. SGD and its variants have been established as one of the most successful optimisation methods in machine learning, imaging and signal processing, etc. At each iteration SGD uses a single datum, or a small subset of data, resulting in highly scalable methods that are very attractive for large-scale inverse problems. Nonetheless, the theoretical analysis of SGD-based approaches for inverse problems has thus far been largely limited to Euclidean and Hilbert spaces. In this work we present a novel convergence analysis of SGD for linear inverse problems in general Banach spaces: we show the almost sure convergence of the iterates to the minimum norm solution and establish the regularising property for suitable a priori stopping criteria. Numerical results are also presented to illustrate features of the approach.

We consider signal source localization from range-difference measurements. First, we give some readily-checked conditions on measurement noises and sensor deployment to guarantee the asymptotic identifiability of the model and show the consistency and asymptotic normality of the maximum likelihood (ML) estimator. Then, we devise an estimator that owns the same asymptotic property as the ML one. Specifically, we prove that the negative log-likelihood function converges to a function, which has a unique minimum and positive-definite Hessian at the true source's position. Hence, it is promising to execute local iterations, e.g., the Gauss-Newton (GN) algorithm, following a consistent estimate. The main issue involved is obtaining a preliminary consistent estimate. To this aim, we construct a linear least-squares problem via algebraic operation and constraint relaxation and obtain a closed-form solution. We then focus on deriving and eliminating the bias of the linear least-squares estimator, which yields an asymptotically unbiased (thus consistent) estimate. Noting that the bias is a function of the noise variance, we further devise a consistent noise variance estimator which involves $3$-order polynomial rooting. Based on the preliminary consistent location estimate, we prove that a one-step GN iteration suffices to achieve the same asymptotic property as the ML estimator. Simulation results demonstrate the superiority of our proposed algorithm in the large sample case.

In this work, we present a deterministic algorithm for computing the entire weight distribution of polar codes. As the first step, we derive an efficient recursive procedure to compute the weight distribution that arises in successive cancellation decoding of polar codes along any decoding path. This solves the open problem recently posed by Polyanskaya, Davletshin, and Polyanskii. Using this recursive procedure, at code length n, we can compute the weight distribution of any polar cosets in time O(n^2). We show that any polar code can be represented as a disjoint union of such polar cosets; moreover, this representation extends to polar codes with dynamically frozen bits. However, the number of polar cosets in such representation scales exponentially with a parameter introduced herein, which we call the mixing factor. To upper bound the complexity of our algorithm for polar codes being decreasing monomial codes, we study the range of their mixing factors. We prove that among all decreasing monomial codes with rates at most 1/2, self-dual Reed-Muller codes have the largest mixing factors. To further reduce the complexity of our algorithm, we make use of the fact that, as decreasing monomial codes, polar codes have a large automorphism group. That automorphism group includes the block lower-triangular affine group (BLTA), which in turn contains the lower-triangular affine group (LTA). We prove that a subgroup of LTA acts transitively on certain subsets of decreasing monomial codes, thereby drastically reducing the number of polar cosets that we need to evaluate. This complexity reduction makes it possible to compute the weight distribution of polar codes at length n = 128.

Sampling methods (e.g., node-wise, layer-wise, or subgraph) has become an indispensable strategy to speed up training large-scale Graph Neural Networks (GNNs). However, existing sampling methods are mostly based on the graph structural information and ignore the dynamicity of optimization, which leads to high variance in estimating the stochastic gradients. The high variance issue can be very pronounced in extremely large graphs, where it results in slow convergence and poor generalization. In this paper, we theoretically analyze the variance of sampling methods and show that, due to the composite structure of empirical risk, the variance of any sampling method can be decomposed into \textit{embedding approximation variance} in the forward stage and \textit{stochastic gradient variance} in the backward stage that necessities mitigating both types of variance to obtain faster convergence rate. We propose a decoupled variance reduction strategy that employs (approximate) gradient information to adaptively sample nodes with minimal variance, and explicitly reduces the variance introduced by embedding approximation. We show theoretically and empirically that the proposed method, even with smaller mini-batch sizes, enjoys a faster convergence rate and entails a better generalization compared to the existing methods.