In this paper, we explore the theory and expand upon the practice of fitness landscape analysis for optimization problems over the space of permutations. Many of the computational and analytical tools for fitness landscape analysis, such as fitness distance correlation, require identifying a distance metric for measuring the similarity of different solutions to the problem. We begin with a survey of the available distance metrics for permutations, and then use principal component analysis to classify these metrics. The result of this analysis aligns with existing classifications of permutation problem types produced through less formal means, including the A-permutation, R-permutation, and P-permutation types, which classifies problems by whether absolute position of permutation elements, relative positions of elements, or general precedence of pairs of elements, is the dominant influence over solution fitness. Additionally, the formal analysis identifies subtypes within these problem categories. We see that the classification can assist in identifying appropriate metrics based on optimization problem feature for use in fitness landscape analysis. Using optimization problems of each class, we also demonstrate how the classification scheme can subsequently inform the choice of mutation operator within an evolutionary algorithm. From this, we present a classification of a variety of mutation operators as a counterpart to that of the metrics. Our implementations of the permutation metrics, permutation mutation operators, and associated evolutionary algorithm, are available in a pair of open source Java libraries. All of the code necessary to recreate our analysis and experimental results are also available as open source.
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We explore using neural operators, or neural network representations of nonlinear maps between function spaces, to accelerate infinite-dimensional Bayesian inverse problems (BIPs) with models governed by nonlinear parametric partial differential equations (PDEs). Neural operators have gained significant attention in recent years for their ability to approximate the parameter-to-solution maps defined by PDEs using as training data solutions of PDEs at a limited number of parameter samples. The computational cost of BIPs can be drastically reduced if the large number of PDE solves required for posterior characterization are replaced with evaluations of trained neural operators. However, reducing error in the resulting BIP solutions via reducing the approximation error of the neural operators in training can be challenging and unreliable. We provide an a priori error bound result that implies certain BIPs can be ill-conditioned to the approximation error of neural operators, thus leading to inaccessible accuracy requirements in training. To reliably deploy neural operators in BIPs, we consider a strategy for enhancing the performance of neural operators, which is to correct the prediction of a trained neural operator by solving a linear variational problem based on the PDE residual. We show that a trained neural operator with error correction can achieve a quadratic reduction of its approximation error, all while retaining substantial computational speedups of posterior sampling when models are governed by highly nonlinear PDEs. The strategy is applied to two numerical examples of BIPs based on a nonlinear reaction--diffusion problem and deformation of hyperelastic materials. We demonstrate that posterior representations of the two BIPs produced using trained neural operators are greatly and consistently enhanced by error correction.
While the theoretical analysis of evolutionary algorithms (EAs) has made significant progress for pseudo-Boolean optimization problems in the last 25 years, only sporadic theoretical results exist on how EAs solve permutation-based problems. To overcome the lack of permutation-based benchmark problems, we propose a general way to transfer the classic pseudo-Boolean benchmarks into benchmarks defined on sets of permutations. We then conduct a rigorous runtime analysis of the permutation-based $(1+1)$ EA proposed by Scharnow, Tinnefeld, and Wegener (2004) on the analogues of the \textsc{LeadingOnes} and \textsc{Jump} benchmarks. The latter shows that, different from bit-strings, it is not only the Hamming distance that determines how difficult it is to mutate a permutation $\sigma$ into another one $\tau$, but also the precise cycle structure of $\sigma \tau^{-1}$. For this reason, we also regard the more symmetric scramble mutation operator. We observe that it not only leads to simpler proofs, but also reduces the runtime on jump functions with odd jump size by a factor of $\Theta(n)$. Finally, we show that a heavy-tailed version of the scramble operator, as in the bit-string case, leads to a speed-up of order $m^{\Theta(m)}$ on jump functions with jump size~$m$.%
Producing statistics that respect the privacy of the samples while still maintaining their accuracy is an important topic of research. We study minimax lower bounds when the class of estimators is restricted to the differentially private ones. In particular, we show that characterizing the power of a distributional test under differential privacy can be done by solving a transport problem. With specific coupling constructions, this observation allows us to derivate Le Cam-type and Fano-type inequalities for both regular definitions of differential privacy and for divergence-based ones (based on Renyi divergence). We then proceed to illustrate our results on three simple, fully worked out examples. In particular, we show that the problem class has a huge importance on the provable degradation of utility due to privacy. For some problems, privacy leads to a provable degradation only when the rate of the privacy parameters is small enough whereas for other problem, the degradation systematically occurs under much looser hypotheses on the privacy parametters. Finally, we show that the known privacy guarantees of DP-SGLD, a private convex solver, when used to perform maximum likelihood, leads to an algorithm that is near-minimax optimal in both the sample size and the privacy tuning parameters of the problem for a broad class of parametric estimation procedures that includes exponential families.
The use of expectiles in risk management contexts has recently gathered substantial momentum because of their excellent axiomatic and probabilistic properties. While expectile estimation at central levels already has a substantial history, expectile estimation at extreme levels has so far only been considered when the underlying distribution has a heavy right tail. This article focuses on the challenging short-tailed setting when the distribution of the variable of interest has a negative extreme value index and is bounded to the right. We derive an asymptotic expansion of extreme expectiles in this context under a general second-order extreme value condition. This asymptotic expansion makes it possible to study two semiparametric estimators of extreme expectiles, whose asymptotic properties we obtain in a general model of strictly stationary but weakly dependent observations. A simulation study and real data analysis illustrate the performance of the proposed statistical techniques.
Ordinary differential equation models are used to describe dynamic processes across biology. To perform likelihood-based parameter inference on these models, it is necessary to specify a statistical process representing the contribution of factors not explicitly included in the mathematical model. For this, independent Gaussian noise is commonly chosen, with its use so widespread that researchers typically provide no explicit justification for this choice. This noise model assumes `random' latent factors affect the system in ephemeral fashion resulting in unsystematic deviation of observables from their modelled counterparts. However, like the deterministically modelled parts of a system, these latent factors can have persistent effects on observables. Here, we use experimental data from dynamical systems drawn from cardiac physiology and electrochemistry to demonstrate that highly persistent differences between observations and modelled quantities can occur. Considering the case when persistent noise arises due only to measurement imperfections, we use the Fisher information matrix to quantify how uncertainty in parameter estimates is artificially reduced when erroneously assuming independent noise. We present a workflow to diagnose persistent noise from model fits and describe how to remodel accounting for correlated errors.
(Stochastic) bilevel optimization is a frequently encountered problem in machine learning with a wide range of applications such as meta-learning, hyper-parameter optimization, and reinforcement learning. Most of the existing studies on this problem only focused on analyzing the convergence or improving the convergence rate, while little effort has been devoted to understanding its generalization behaviors. In this paper, we conduct a thorough analysis on the generalization of first-order (gradient-based) methods for the bilevel optimization problem. We first establish a fundamental connection between algorithmic stability and generalization error in different forms and give a high probability generalization bound which improves the previous best one from $\bigO(\sqrt{n})$ to $\bigO(\log n)$, where $n$ is the sample size. We then provide the first stability bounds for the general case where both inner and outer level parameters are subject to continuous update, while existing work allows only the outer level parameter to be updated. Our analysis can be applied in various standard settings such as strongly-convex-strongly-convex (SC-SC), convex-convex (C-C), and nonconvex-nonconvex (NC-NC). Our analysis for the NC-NC setting can also be extended to a particular nonconvex-strongly-convex (NC-SC) setting that is commonly encountered in practice. Finally, we corroborate our theoretical analysis and demonstrate how iterations can affect the generalization error by experiments on meta-learning and hyper-parameter optimization.
In this paper, we study the lower complexity bounds for finite-sum optimization problems, where the objective is the average of $n$ individual component functions. We consider Proximal Incremental First-order (PIFO) algorithms which have access to the gradient and proximal oracles for each component function. To incorporate loopless methods, we also allow PIFO algorithms to obtain the full gradient infrequently. We develop a novel approach to constructing the hard instances, which partitions the tridiagonal matrix of classical examples into $n$ groups. This construction is friendly to the analysis of PIFO algorithms. Based on this construction, we establish the lower complexity bounds for finite-sum minimax optimization problems when the objective is convex-concave or nonconvex-strongly-concave and the class of component functions is $L$-average smooth. Most of these bounds are nearly matched by existing upper bounds up to log factors. We can also derive similar lower bounds for finite-sum minimization problems as previous work under both smoothness and average smoothness assumptions. Our lower bounds imply that proximal oracles for smooth functions are not much more powerful than gradient oracles.
We consider a contextual online learning (multi-armed bandit) problem with high-dimensional covariate $\mathbf{x}$ and decision $\mathbf{y}$. The reward function to learn, $f(\mathbf{x},\mathbf{y})$, does not have a particular parametric form. The literature has shown that the optimal regret is $\tilde{O}(T^{(d_x+d_y+1)/(d_x+d_y+2)})$, where $d_x$ and $d_y$ are the dimensions of $\mathbf x$ and $\mathbf y$, and thus it suffers from the curse of dimensionality. In many applications, only a small subset of variables in the covariate affect the value of $f$, which is referred to as \textit{sparsity} in statistics. To take advantage of the sparsity structure of the covariate, we propose a variable selection algorithm called \textit{BV-LASSO}, which incorporates novel ideas such as binning and voting to apply LASSO to nonparametric settings. Our algorithm achieves the regret $\tilde{O}(T^{(d_x^*+d_y+1)/(d_x^*+d_y+2)})$, where $d_x^*$ is the effective covariate dimension. The regret matches the optimal regret when the covariate is $d^*_x$-dimensional and thus cannot be improved. Our algorithm may serve as a general recipe to achieve dimension reduction via variable selection in nonparametric settings.
We study a general matrix optimization problem with a fixed-rank positive semidefinite (PSD) constraint. We perform the Burer-Monteiro factorization and consider a particular Riemannian quotient geometry in a search space that has a total space equipped with the Euclidean metric. When the original objective f satisfies standard restricted strong convexity and smoothness properties, we characterize the global landscape of the factorized objective under the Riemannian quotient geometry. We show the entire search space can be divided into three regions: (R1) the region near the target parameter of interest, where the factorized objective is geodesically strongly convex and smooth; (R2) the region containing neighborhoods of all strict saddle points; (R3) the remaining regions, where the factorized objective has a large gradient. To our best knowledge, this is the first global landscape analysis of the Burer-Monteiro factorized objective under the Riemannian quotient geometry. Our results provide a fully geometric explanation for the superior performance of vanilla gradient descent under the Burer-Monteiro factorization. When f satisfies a weaker restricted strict convexity property, we show there exists a neighborhood near local minimizers such that the factorized objective is geodesically convex. To prove our results we provide a comprehensive landscape analysis of a matrix factorization problem with a least squares objective, which serves as a critical bridge. Our conclusions are also based on a result of independent interest stating that the geodesic ball centered at Y with a radius 1/3 of the least singular value of Y is a geodesically convex set under the Riemannian quotient geometry, which as a corollary, also implies a quantitative bound of the convexity radius in the Bures-Wasserstein space. The convexity radius obtained is sharp up to constants.
Classic machine learning methods are built on the $i.i.d.$ assumption that training and testing data are independent and identically distributed. However, in real scenarios, the $i.i.d.$ assumption can hardly be satisfied, rendering the sharp drop of classic machine learning algorithms' performances under distributional shifts, which indicates the significance of investigating the Out-of-Distribution generalization problem. Out-of-Distribution (OOD) generalization problem addresses the challenging setting where the testing distribution is unknown and different from the training. This paper serves as the first effort to systematically and comprehensively discuss the OOD generalization problem, from the definition, methodology, evaluation to the implications and future directions. Firstly, we provide the formal definition of the OOD generalization problem. Secondly, existing methods are categorized into three parts based on their positions in the whole learning pipeline, namely unsupervised representation learning, supervised model learning and optimization, and typical methods for each category are discussed in detail. We then demonstrate the theoretical connections of different categories, and introduce the commonly used datasets and evaluation metrics. Finally, we summarize the whole literature and raise some future directions for OOD generalization problem. The summary of OOD generalization methods reviewed in this survey can be found at //out-of-distribution-generalization.com.
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