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In this work, we improve upon the guarantees for sparse random embeddings, as they were recently provided and analyzed by Freksen at al. (NIPS'18) and Jagadeesan (NIPS'19). Specifically, we show that (a) our bounds are explicit as opposed to the asymptotic guarantees provided previously, and (b) our bounds are guaranteed to be sharper by practically significant constants across a wide range of parameters, including the dimensionality, sparsity and dispersion of the data. Moreover, we empirically demonstrate that our bounds significantly outperform prior works on a wide range of real-world datasets, such as collections of images, text documents represented as bags-of-words, and text sequences vectorized by neural embeddings. Behind our numerical improvements are techniques of broader interest, which improve upon key steps of previous analyses in terms of (c) tighter estimates for certain types of quadratic chaos, (d) establishing extreme properties of sparse linear forms, and (e) improvements on bounds for the estimation of sums of independent random variables.

This work studies an experimental design problem where {the values of a predictor variable, denoted by $x$}, are to be determined with the goal of estimating a function $m(x)$, which is observed with noise. A linear model is fitted to $m(x)$ but it is not assumed that the model is correctly specified. It follows that the quantity of interest is the best linear approximation of $m(x)$, which is denoted by $\ell(x)$. It is shown that in this framework the ordinary least squares estimator typically leads to an inconsistent estimation of $\ell(x)$, and rather weighted least squares should be considered. An asymptotic minimax criterion is formulated for this estimator, and a design that minimizes the criterion is constructed. An important feature of this problem is that the $x$'s should be random, rather than fixed. Otherwise, the minimax risk is infinite. It is shown that the optimal random minimax design is different from its deterministic counterpart, which was studied previously, and a simulation study indicates that it generally performs better when $m(x)$ is a quadratic or a cubic function. Another finding is that when the variance of the noise goes to infinity, the random and deterministic minimax designs coincide. The results are illustrated for polynomial regression models and the general case is also discussed.

We introduce a notion of \emph{generic local algorithm} which strictly generalizes existing frameworks of local algorithms such as \emph{factors of i.i.d.} by capturing local \emph{quantum} algorithms such as the Quantum Approximate Optimization Algorithm (QAOA). Motivated by a question of Farhi et al. [arXiv:1910.08187, 2019] we then show limitations of generic local algorithms including QAOA on random instances of constraint satisfaction problems (CSPs). Specifically, we show that any generic local algorithm whose assignment to a vertex depends only on a local neighborhood with $o(n)$ other vertices (such as the QAOA at depth less than $\epsilon\log(n)$) cannot arbitrarily-well approximate boolean CSPs if the problem satisfies a geometric property from statistical physics called the coupled overlap-gap property (OGP) [Chen et al., Annals of Probability, 47(3), 2019]. We show that the random MAX-k-XOR problem has this property when $k\geq4$ is even by extending the corresponding result for diluted $k$-spin glasses. Our concentration lemmas confirm a conjecture of Brandao et al. [arXiv:1812.04170, 2018] asserting that the landscape independence of QAOA extends to logarithmic depth -- in other words, for every fixed choice of QAOA angle parameters, the algorithm at logarithmic depth performs almost equally well on almost all instances. One of these concentration lemmas is a strengthening of McDiarmid's inequality, applicable when the random variables have a highly biased distribution, and may be of independent interest.

Given a point set $P\subset \mathbb{R}^d$, the kernel density estimate of $P$ is defined as \[ \overline{\mathcal{G}}_P(x) = \frac{1}{\left|P\right|}\sum_{p\in P}e^{-\left\lVert x-p \right\rVert^2} \] for any $x\in\mathbb{R}^d$. We study how to construct a small subset $Q$ of $P$ such that the kernel density estimate of $P$ is approximated by the kernel density estimate of $Q$. This subset $Q$ is called a coreset. The main technique in this work is constructing a $\pm 1$ coloring on the point set $P$ by discrepancy theory and we leverage Banaszczyk's Theorem. When $d>1$ is a constant, our construction gives a coreset of size $O\left(\frac{1}{\varepsilon}\right)$ as opposed to the best-known result of $O\left(\frac{1}{\varepsilon}\sqrt{\log\frac{1}{\varepsilon}}\right)$. It is the first result to give a breakthrough on the barrier of $\sqrt{\log}$ factor even when $d=2$.

Bayesian bandit algorithms with approximate inference have been widely used in practice with superior performance. Yet, few studies regarding the fundamental understanding of their performances are available. In this paper, we propose a Bayesian bandit algorithm, which we call Generalized Bayesian Upper Confidence Bound (GBUCB), for bandit problems in the presence of approximate inference. Our theoretical analysis demonstrates that in Bernoulli multi-armed bandit, GBUCB can achieve $O(\sqrt{T}(\log T)^c)$ frequentist regret if the inference error measured by symmetrized Kullback-Leibler divergence is controllable. This analysis relies on a novel sensitivity analysis for quantile shifts with respect to inference errors. To our best knowledge, our work provides the first theoretical regret bound that is better than $o(T)$ in the setting of approximate inference. Our experimental evaluations on multiple approximate inference settings corroborate our theory, showing that our GBUCB is consistently superior to BUCB and Thompson sampling.

We propose a network architecture capable of reliably estimating uncertainty of regression based predictions without sacrificing accuracy. The current state-of-the-art uncertainty algorithms either fall short of achieving prediction accuracy comparable to the mean square error optimization or underestimate the variance of network predictions. We propose a decoupled network architecture that is capable of accomplishing both at the same time. We achieve this by breaking down the learning of prediction and prediction interval (PI) estimations into a two-stage training process. We use a custom loss function for learning a PI range around optimized mean estimation with a desired coverage of a proportion of the target labels within the PI range. We compare the proposed method with current state-of-the-art uncertainty quantification algorithms on synthetic datasets and UCI benchmarks, reducing the error in the predictions by 23 to 34% while maintaining 95% Prediction Interval Coverage Probability (PICP) for 7 out of 9 UCI benchmark datasets. We also examine the quality of our predictive uncertainty by evaluating on Active Learning and demonstrating 17 to 36% error reduction on UCI benchmarks.

We consider the well-studied problem of decomposing a vector time series signal into components with different characteristics, such as smooth, periodic, nonnegative, or sparse. We propose a simple and general framework in which the components are defined by loss functions (which include constraints), and the signal decomposition is carried out by minimizing the sum of losses of the components (subject to the constraints). When each loss function is the negative log-likelihood of a density for the signal component, our method coincides with maximum a posteriori probability (MAP) estimation; but it also includes many other interesting cases. We give two distributed optimization methods for computing the decomposition, which find the optimal decomposition when the component class loss functions are convex, and are good heuristics when they are not. Both methods require only the masked proximal operator of each of the component loss functions, a generalization of the well-known proximal operator that handles missing entries in its argument. Both methods are distributed, i.e., handle each component separately. We derive tractable methods for evaluating the masked proximal operators of some loss functions that, to our knowledge, have not appeared in the literature.

Ensemble methods based on subsampling, such as random forests, are popular in applications due to their high predictive accuracy. Existing literature views a random forest prediction as an infinite-order incomplete U-statistic to quantify its uncertainty. However, these methods focus on a small subsampling size of each tree, which is theoretically valid but practically limited. This paper develops an unbiased variance estimator based on incomplete U-statistics, which allows the tree size to be comparable with the overall sample size, making statistical inference possible in a broader range of real applications. Simulation results demonstrate that our estimators enjoy lower bias and more accurate confidence interval coverage without additional computational costs. We also propose a local smoothing procedure to reduce the variation of our estimator, which shows improved numerical performance when the number of trees is relatively small. Further, we investigate the ratio consistency of our proposed variance estimator under specific scenarios. In particular, we develop a new "double U-statistic" formulation to analyze the Hoeffding decomposition of the estimator's variance.

Implicit probabilistic models are models defined naturally in terms of a sampling procedure and often induces a likelihood function that cannot be expressed explicitly. We develop a simple method for estimating parameters in implicit models that does not require knowledge of the form of the likelihood function or any derived quantities, but can be shown to be equivalent to maximizing likelihood under some conditions. Our result holds in the non-asymptotic parametric setting, where both the capacity of the model and the number of data examples are finite. We also demonstrate encouraging experimental results.

Many problems on signal processing reduce to nonparametric function estimation. We propose a new methodology, piecewise convex fitting (PCF), and give a two-stage adaptive estimate. In the first stage, the number and location of the change points is estimated using strong smoothing. In the second stage, a constrained smoothing spline fit is performed with the smoothing level chosen to minimize the MSE. The imposed constraint is that a single change point occurs in a region about each empirical change point of the first-stage estimate. This constraint is equivalent to requiring that the third derivative of the second-stage estimate has a single sign in a small neighborhood about each first-stage change point. We sketch how PCF may be applied to signal recovery, instantaneous frequency estimation, surface reconstruction, image segmentation, spectral estimation and multivariate adaptive regression.