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We consider the optimization of a smooth and strongly convex objective using constant step-size stochastic gradient descent (SGD) and study its properties through the prism of Markov chains. We show that, for unbiased gradient estimates with mildly controlled variance, the iteration converges to an invariant distribution in total variation distance. We also establish this convergence in Wasserstein-2 distance under a relaxed assumption on the gradient noise distribution compared to previous work. Thanks to the invariance property of the limit distribution, our analysis shows that the latter inherits sub-Gaussian or sub-exponential concentration properties when these hold true for the gradient. This allows the derivation of high-confidence bounds for the final estimate. Finally, under such conditions in the linear case, we obtain a dimension-free deviation bound for the Polyak-Ruppert average of a tail sequence. All our results are non-asymptotic and their consequences are discussed through a few applications.

The convexity of a set can be generalized to the two weaker notions of reach and $r$-convexity; both describe the regularity of a set's boundary. For any compact subset of $\mathbb{R}^d$, we provide methods for computing upper bounds on these quantities from point cloud data. The bounds converge to the respective quantities as the point cloud becomes dense in the set, and the rate of convergence for the bound on the reach is given under a weak regularity condition. We also introduce the $\beta$-reach, a generalization of the reach that excludes small-scale features of size less than a parameter $\beta\in[0,\infty)$. Numerical studies suggest how the $\beta$-reach can be used in high-dimension to infer the reach and other geometric properties of smooth submanifolds.

Vector autoregressions (VARs) are popular in analyzing economic time series. However, VARs can be over-parameterized if the numbers of variables and lags are moderately large. Tensor VAR, a recent solution to over-parameterization, treats the coefficient matrix as a third-order tensor and estimates the corresponding tensor decomposition to achieve parsimony. In this paper, we employ the Tensor VAR structure with a CANDECOMP/PARAFAC (CP) decomposition and conduct Bayesian inference to estimate parameters. Firstly, we determine the rank by imposing the Multiplicative Gamma Prior to margins, i.e. elements in the decomposition, and accelerate the computation with an adaptive inferential scheme. Secondly, to obtain interpretable margins, we propose an interweaving algorithm to improve the mixing of margins. In the application of the US macroeconomic data, our models outperform standard VARs in point and density forecasting and yield a summary of the US economic dynamic.

Differentially private mean estimation is an important building block in privacy-preserving algorithms for data analysis and machine learning. Though the trade-off between privacy and utility is well understood in the worst case, many datasets exhibit structure that could potentially be exploited to yield better algorithms. In this paper we present $\textit{Private Limit Adapted Noise}$ (PLAN), a family of differentially private algorithms for mean estimation in the setting where inputs are independently sampled from a distribution $\mathcal{D}$ over $\mathbf{R}^d$, with coordinate-wise standard deviations $\boldsymbol{\sigma} \in \mathbf{R}^d$. Similar to mean estimation under Mahalanobis distance, PLAN tailors the shape of the noise to the shape of the data, but unlike previous algorithms the privacy budget is spent non-uniformly over the coordinates. Under a concentration assumption on $\mathcal{D}$, we show how to exploit skew in the vector $\boldsymbol{\sigma}$, obtaining a (zero-concentrated) differentially private mean estimate with $\ell_2$ error proportional to $\|\boldsymbol{\sigma}\|_1$. Previous work has either not taken $\boldsymbol{\sigma}$ into account, or measured error in Mahalanobis distance $\unicode{x2013}$ in both cases resulting in $\ell_2$ error proportional to $\sqrt{d}\|\boldsymbol{\sigma}\|_2$, which can be up to a factor $\sqrt{d}$ larger. To verify the effectiveness of PLAN, we empirically evaluate accuracy on both synthetic and real world data.

Machine learning methods are commonly evaluated and compared by their performance on data sets from public repositories. This allows for multiple methods, oftentimes several thousands, to be evaluated under identical conditions and across time. The highest ranked performance on a problem is referred to as state-of-the-art (SOTA) performance, and is used, among other things, as a reference point for publication of new methods. Using the highest-ranked performance as an estimate for SOTA is a biased estimator, giving overly optimistic results. The mechanisms at play are those of multiplicity, a topic that is well-studied in the context of multiple comparisons and multiple testing, but has, as far as the authors are aware of, been nearly absent from the discussion regarding SOTA estimates. The optimistic state-of-the-art estimate is used as a standard for evaluating new methods, and methods with substantial inferior results are easily overlooked. In this article, we provide a probability distribution for the case of multiple classifiers so that known analyses methods can be engaged and a better SOTA estimate can be provided. We demonstrate the impact of multiplicity through a simulated example with independent classifiers. We show how classifier dependency impacts the variance, but also that the impact is limited when the accuracy is high. Finally, we discuss a real-world example; a Kaggle competition from 2020.

Given a Boolean formula $\phi$ over $n$ variables, the problem of model counting is to compute the number of solutions of $\phi$. Model counting is a fundamental problem in computer science with wide-ranging applications. Owing to the \#P-hardness of the problems, Stockmeyer initiated the study of the complexity of approximate counting. Stockmeyer showed that $\log n$ calls to an NP oracle are necessary and sufficient to achieve $(\varepsilon,\delta)$ guarantees. The hashing-based framework proposed by Stockmeyer has been very influential in designing practical counters over the past decade, wherein the SAT solver substitutes the NP oracle calls in practice. It is well known that an NP oracle does not fully capture the behavior of SAT solvers, as SAT solvers are also designed to provide satisfying assignments when a formula is satisfiable, without additional overhead. Accordingly, the notion of SAT oracle has been proposed to capture the behavior of SAT solver wherein given a Boolean formula, an SAT oracle returns a satisfying assignment if the formula is satisfiable or returns unsatisfiable otherwise. Since the practical state-of-the-art approximate counting techniques use SAT solvers, a natural question is whether an SAT oracle is more powerful than an NP oracle in the context of approximate model counting. The primary contribution of this work is to study the relative power of the NP oracle and SAT oracle in the context of approximate model counting. The previous techniques proposed in the context of an NP oracle are weak to provide strong bounds in the context of SAT oracle since, in contrast to an NP oracle that provides only one bit of information, a SAT oracle can provide $n$ bits of information. We therefore develop a new methodology to achieve the main result: a SAT oracle is no more powerful than an NP oracle in the context of approximate model counting.

We consider the multiple testing of the general regression framework aiming at studying the relationship between a univariate response and a p-dimensional predictor. To test the hypothesis of the effect of each predictor, we construct an Angular Balanced Statistic (ABS) based on the estimator of the sliced inverse regression without assuming a model of the conditional distribution of the response. According to the developed limiting distribution results in this paper, we have shown that ABS is asymptotically symmetric with respect to zero under the null hypothesis. We then propose a Model-free multiple Testing procedure using Angular balanced statistics (MTA) and show theoretically that the false discovery rate of this method is less than or equal to a designated level asymptotically. Numerical evidence has shown that the MTA method is much more powerful than its alternatives, subject to the control of the false discovery rate.

We consider Shor's quantum factoring algorithm in the setting of noisy quantum gates. Under a generic model of random noise for (controlled) rotation gates, we prove that the algorithm does not factor integers of the form $pq$ when the noise exceeds a vanishingly small level in terms of $n$ -- the number of bits of the integer to be factored, where $p$ and $q$ are from a well-defined set of primes of positive density. We further prove that with probability $1 - o(1)$ over random prime pairs $(p,q)$, Shor's factoring algorithm does not factor numbers of the form $pq$, with the same level of random noise present.

Reliable probabilistic primality tests are fundamental in public-key cryptography. In adversarial scenarios, a composite with a high probability of passing a specific primality test could be chosen. In such cases, we need worst-case error estimates for the test. However, in many scenarios the numbers are randomly chosen and thus have significantly smaller error probability. Therefore, we are interested in average case error estimates. In this paper, we establish such bounds for the strong Lucas primality test, as only worst-case, but no average case error bounds, are currently available. This allows us to use this test with more confidence. We examine an algorithm that draws odd $k$-bit integers uniformly and independently, runs $t$ independent iterations of the strong Lucas test with randomly chosen parameters, and outputs the first number that passes all $t$ consecutive rounds. We attain numerical upper bounds on the probability on returing a composite. Furthermore, we consider a modified version of this algorithm that excludes integers divisible by small primes, resulting in improved bounds. Additionally, we classify the numbers that contribute most to our estimate.

In this brief note, we consider estimation of the bitwise combination $x_1 \lor \dots \lor x_n = \max_i x_i$ observing a set of noisy bits $\tilde x_i \in \{0, 1\}$ that represent the true, unobserved bits $x_i \in \{0, 1\}$ under randomized response. We demonstrate that various existing estimators for the extreme bit, including those based on computationally costly estimates of the sum of bits, can be reduced to a simple closed form computed in linear time (in $n$) and constant space, including in an online fashion as new $\tilde x_i$ are observed. In particular, we derive such an estimator and provide its variance using only elementary techniques.