Variational inference has recently emerged as a popular alternative to the classical Markov chain Monte Carlo (MCMC) in large-scale Bayesian inference. The core idea of variational inference is to trade statistical accuracy for computational efficiency. It aims to approximate the posterior, reducing computation costs but potentially compromising its statistical accuracy. In this work, we study this statistical and computational trade-off in variational inference via a case study in inferential model selection. Focusing on Gaussian inferential models (a.k.a. variational approximating families) with diagonal plus low-rank precision matrices, we initiate a theoretical study of the trade-offs in two aspects, Bayesian posterior inference error and frequentist uncertainty quantification error. From the Bayesian posterior inference perspective, we characterize the error of the variational posterior relative to the exact posterior. We prove that, given a fixed computation budget, a lower-rank inferential model produces variational posteriors with a higher statistical approximation error, but a lower computational error; it reduces variances in stochastic optimization and, in turn, accelerates convergence. From the frequentist uncertainty quantification perspective, we consider the precision matrix of the variational posterior as an uncertainty estimate. We find that, relative to the true asymptotic precision, the variational approximation suffers from an additional statistical error originating from the sampling uncertainty of the data. Moreover, this statistical error becomes the dominant factor as the computation budget increases. As a consequence, for small datasets, the inferential model need not be full-rank to achieve optimal estimation error. We finally demonstrate these statistical and computational trade-offs inference across empirical studies, corroborating the theoretical findings.
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Investigators are increasingly using novel methods for extending (generalizing or transporting) causal inferences from a trial to a target population. In many generalizability and transportability analyses, the trial and the observational data from the target population are separately sampled, following a non-nested trial design. In practical implementations of this design, non-randomized individuals from the target population are often identified by conditioning on the use of a particular treatment, while individuals who used other candidate treatments for the same indication or individuals who did not use any treatment are excluded. In this paper, we argue that conditioning on treatment in the target population changes the estimand of generalizability and transportability analyses and potentially introduces serious bias in the estimation of causal estimands in the target population or the subset of the target population using a specific treatment. Furthermore, we argue that the naive application of marginalization-based or weighting-based standardization methods does not produce estimates of any reasonable causal estimand. We use causal graphs and counterfactual arguments to characterize the identification problems induced by conditioning on treatment in the target population and illustrate the problems using simulated data. We conclude by considering the implications of our findings for applied work.
Reaching a consensus in a swarm of robots is one of the fundamental problems in swarm robotics, examining the possibility of reaching an agreement within the swarm members. The recently-introduced contamination problem offers a new perspective of the problem, in which swarm members should reach a consensus in spite of the existence of adversarial members that intentionally act to divert the swarm members towards a different consensus. In this paper, we search for a consensus-reaching algorithm under the contamination problem setting by taking a top-down approach: We transform the problem to a centralized two-player game in which each player controls the behavior of a subset of the swarm, trying to force the entire swarm to converge to an agreement on its own value. We define a performance metric for each players performance, proving a correlation between this metric and the chances of the player to win the game. We then present the globally optimal solution to the game and prove that unfortunately it is unattainable in a distributed setting, due to the challenging characteristics of the swarm members. We therefore examine the problem on a simplified swarm model, and compare the performance of the globally optimal strategy with locally optimal strategies, demonstrating its superiority in rigorous simulation experiments.
This paper deals with the grouped variable selection problem. A widely used strategy is to augment the negative log-likelihood function with a sparsity-promoting penalty. Existing methods include the group Lasso, group SCAD, and group MCP. The group Lasso solves a convex optimization problem but is plagued by underestimation bias. The group SCAD and group MCP avoid this estimation bias but require solving a nonconvex optimization problem that may be plagued by suboptimal local optima. In this work, we propose an alternative method based on the generalized minimax concave (GMC) penalty, which is a folded concave penalty that maintains the convexity of the objective function. We develop a new method for grouped variable selection in linear regression, the group GMC, that generalizes the strategy of the original GMC estimator. We present an efficient algorithm for computing the group GMC estimator and also prove properties of the solution path to guide its numerical computation and tuning parameter selection in practice. We establish error bounds for both the group GMC and original GMC estimators. A rich set of simulation studies and a real data application indicate that the proposed group GMC approach outperforms existing methods in several different aspects under a wide array of scenarios.
This work considers Gaussian process interpolation with a periodized version of the Mat{\'e}rn covariance function (Stein, 1999, Section 6.7) with Fourier coefficients $\phi$($\alpha$^2 + j^2)^(--$\nu$--1/2). Convergence rates are studied for the joint maximum likelihood estimation of $\nu$ and $\phi$ when the data is sampled according to the model. The mean integrated squared error is also analyzed with fixed and estimated parameters, showing that maximum likelihood estimation yields asymptotically the same error as if the ground truth was known. Finally, the case where the observed function is a ''deterministic'' element of a continuous Sobolev space is also considered, suggesting that bounding assumptions on some parameters can lead to different estimates.
In this paper, we investigate the matrix estimation problem in the multi-response regression model with measurement errors. A nonconvex error-corrected estimator based on a combination of the amended loss function and the nuclear norm regularizer is proposed to estimate the matrix parameter. Then under the (near) low-rank assumption, we analyse statistical and computational theoretical properties of global solutions of the nonconvex regularized estimator from a general point of view. In the statistical aspect, we establish the nonasymptotic recovery bound for any global solution of the nonconvex estimator, under restricted strong convexity on the loss function. In the computational aspect, we solve the nonconvex optimization problem via the proximal gradient method. The algorithm is proved to converge to a near-global solution and achieve a linear convergence rate. In addition, we also verify sufficient conditions for the general results to be held, in order to obtain probabilistic consequences for specific types of measurement errors, including the additive noise and missing data. Finally, theoretical consequences are demonstrated by several numerical experiments on corrupted errors-in-variables multi-response regression models. Simulation results reveal excellent consistency with our theory under high-dimensional scaling.
We consider the stochastic gradient descent (SGD) algorithm driven by a general stochastic sequence, including i.i.d noise and random walk on an arbitrary graph, among others; and analyze it in the asymptotic sense. Specifically, we employ the notion of `efficiency ordering', a well-analyzed tool for comparing the performance of Markov Chain Monte Carlo (MCMC) samplers, for SGD algorithms in the form of Loewner ordering of covariance matrices associated with the scaled iterate errors in the long term. Using this ordering, we show that input sequences that are more efficient for MCMC sampling also lead to smaller covariance of the errors for SGD algorithms in the limit. This also suggests that an arbitrarily weighted MSE of SGD iterates in the limit becomes smaller when driven by more efficient chains. Our finding is of particular interest in applications such as decentralized optimization and swarm learning, where SGD is implemented in a random walk fashion on the underlying communication graph for cost issues and/or data privacy. We demonstrate how certain non-Markovian processes, for which typical mixing-time based non-asymptotic bounds are intractable, can outperform their Markovian counterparts in the sense of efficiency ordering for SGD. We show the utility of our method by applying it to gradient descent with shuffling and mini-batch gradient descent, reaffirming key results from existing literature under a unified framework. Empirically, we also observe efficiency ordering for variants of SGD such as accelerated SGD and Adam, open up the possibility of extending our notion of efficiency ordering to a broader family of stochastic optimization algorithms.
Point process models are of great importance in real world applications. In certain critical applications, estimation of point process models involves large amounts of sensitive personal data from users. Privacy concerns naturally arise which have not been addressed in the existing literature. To bridge this glaring gap, we propose the first general differentially private estimation procedure for point process models. Specifically, we take the Hawkes process as an example, and introduce a rigorous definition of differential privacy for event stream data based on a discretized representation of the Hawkes process. We then propose two differentially private optimization algorithms, which can efficiently estimate Hawkes process models with the desired privacy and utility guarantees under two different settings. Experiments are provided to back up our theoretical analysis.
We study the Bayesian density estimation of data living in the offset of an unknown submanifold of the Euclidean space. In this perspective, we introduce a new notion of anisotropic H\"older for the underlying density and obtain posterior rates that are minimax optimal and adaptive to the regularity of the density, to the intrinsic dimension of the manifold, and to the size of the offset, provided that the latter is not too small -- while still allowed to go to zero. Our Bayesian procedure, based on location-scale mixtures of Gaussians, appears to be convenient to implement and yields good practical results, even for quite singular data.
Variational Bayesian posterior inference often requires simplifying approximations such as mean-field parametrisation to ensure tractability. However, prior work has associated the variational mean-field approximation for Bayesian neural networks with underfitting in the case of small datasets or large model sizes. In this work, we show that invariances in the likelihood function of over-parametrised models contribute to this phenomenon because these invariances complicate the structure of the posterior by introducing discrete and/or continuous modes which cannot be well approximated by Gaussian mean-field distributions. In particular, we show that the mean-field approximation has an additional gap in the evidence lower bound compared to a purpose-built posterior that takes into account the known invariances. Importantly, this invariance gap is not constant; it vanishes as the approximation reverts to the prior. We proceed by first considering translation invariances in a linear model with a single data point in detail. We show that, while the true posterior can be constructed from a mean-field parametrisation, this is achieved only if the objective function takes into account the invariance gap. Then, we transfer our analysis of the linear model to neural networks. Our analysis provides a framework for future work to explore solutions to the invariance problem.
The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin. Traditional parameterised differential equations are a special case. Many popular neural network architectures, such as residual networks and recurrent networks, are discretisations. NDEs are suitable for tackling generative problems, dynamical systems, and time series (particularly in physics, finance, ...) and are thus of interest to both modern machine learning and traditional mathematical modelling. NDEs offer high-capacity function approximation, strong priors on model space, the ability to handle irregular data, memory efficiency, and a wealth of available theory on both sides. This doctoral thesis provides an in-depth survey of the field. Topics include: neural ordinary differential equations (e.g. for hybrid neural/mechanistic modelling of physical systems); neural controlled differential equations (e.g. for learning functions of irregular time series); and neural stochastic differential equations (e.g. to produce generative models capable of representing complex stochastic dynamics, or sampling from complex high-dimensional distributions). Further topics include: numerical methods for NDEs (e.g. reversible differential equations solvers, backpropagation through differential equations, Brownian reconstruction); symbolic regression for dynamical systems (e.g. via regularised evolution); and deep implicit models (e.g. deep equilibrium models, differentiable optimisation). We anticipate this thesis will be of interest to anyone interested in the marriage of deep learning with dynamical systems, and hope it will provide a useful reference for the current state of the art.
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