High dimensional black-box optimization has broad applications but remains a challenging problem to solve. Given a set of samples $\{\vx_i, y_i\}$, building a global model (like Bayesian Optimization (BO)) suffers from the curse of dimensionality in the high-dimensional search space, while a greedy search may lead to sub-optimality. By recursively splitting the search space into regions with high/low function values, recent works like LaNAS shows good performance in Neural Architecture Search (NAS), reducing the sample complexity empirically. In this paper, we coin LA-MCTS that extends LaNAS to other domains. Unlike previous approaches, LA-MCTS learns the partition of the search space using a few samples and their function values in an online fashion. While LaNAS uses linear partition and performs uniform sampling in each region, our LA-MCTS adopts a nonlinear decision boundary and learns a local model to pick good candidates. If the nonlinear partition function and the local model fits well with ground-truth black-box function, then good partitions and candidates can be reached with much fewer samples. LA-MCTS serves as a \emph{meta-algorithm} by using existing black-box optimizers (e.g., BO, TuRBO) as its local models, achieving strong performance in general black-box optimization and reinforcement learning benchmarks, in particular for high-dimensional problems.
JetBrains Research（JetBrains研究院）；HSE University（高等经济大学）；Saint Petersburg State University（圣彼得堡国立大学）
Black-box optimization is one of the vital tasks in machine learning, since it approximates real-world conditions, in that we do not always know all the properties of a given system, up to knowing almost nothing but the results. This paper describes our approach to solving the black-box optimization challenge at NeurIPS 2020 through learning search space partition for local Bayesian optimization. We describe the task of the challenge as well as our algorithm for low budget optimization that we named \texttt{SPBOpt}. We optimize the hyper-parameters of our algorithm for the competition finals u…
Discretization of continuous-time diffusion processes is a widely recognized method for sampling. However, the canonical Euler-Maruyama discretization of the Langevin diffusion process, also named as Langevin Monte Carlo (LMC), studied mostly in the context of smooth (gradient-Lipschitz) and strongly log-concave densities, a significant constraint for its deployment in many sciences, including computational statistics and statistical learning. In this paper, we establish several theoretical contributions to the literature on such sampling methods. Particularly, we generalize the Gaussian smoot…
We consider Markov chain Monte Carlo methods for calculating conditional p values of statistical models for count data arising in Box-Behnken designs. The statistical model we consider is a discrete version of the first-order model in the response surface methodology. For our models, the Markov basis, a key notion to construct a connected Markov chain on a given sample space, is characterized as generators of the toric ideals for the centrally symmetric configurations of root system D_n. We show the structure of the Groebner bases for these cases. A numerical example for an imaginary data set …
The paper proposes Monte Carlo algorithms for the computation of the information rate of two-dimensional source/channel models. The focus of the paper is on binary-input channels with constraints on the allowed input configurations. The problem of numerically computing the information rate, and even the noiseless capacity, of such channels has so far remained largely unsolved. Both problems can be reduced to computing a Monte Carlo estimate of a partition function. The proposed algorithms use tree-based Gibbs sampling and multilayer (multitemperature) importance sampling. The viability of the …
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Oak Ridge National Laboratory（橡树岭国家实验室）
We develop an Evolutionary Markov Chain Monte Carlo (EMCMC) algorithm for sampling spatial partitions that lie within a large and complex spatial state space. Our algorithm combines the advantages of evolutionary algorithms (EAs) as optimization heuristics for state space traversal and the theoretical convergence properties of Markov Chain Monte Carlo algorithms for sampling from unknown distributions. Local optimality information that is identified via a directed search by our optimization heuristic is used to adaptively update a Markov chain in a promising direction within the framework of a…
We consider the problem of adaptive stratified sampling for Monte Carlo integration of a noisy function, given a finite budget n of noisy evaluations to the function. We tackle in this paper the problem of adapting to the function at the same time the number of samples into each stratum and the partition itself. More precisely, it is interesting to refine the partition of the domain in area where the noise to the function, or where the variations of the function, are very heterogeneous. On the other hand, having a (too) refined stratification is not optimal. Indeed, the more refined the strati…
Monte Carlo Tree Search based Space Transfer for Black-box Optimization
National Key Laboratory for Novel Software Technology, Nanjing University（南京大学计算机软件新技术国家重点实验室）；School of Artificial Intelligence, Nanjing University（南京大学人工智能学院）
Normal factor graph duality offers new possibilities for Monte Carlo algorithms in graphical models. Specifically, we consider the problem of estimating the partition function of the ferromagnetic Ising and Potts models by Monte Carlo methods, which are known to work well at high temperatures, but to fail at low temperatures. We propose Monte Carlo methods (uniform sampling and importance sampling) in the dual normal factor graph, and demonstrate that they behave differently: they work particularly well at low temperatures. By comparing the relative error in estimating the partition function, …
Efficient algorithms for $k$-means clustering frequently converge to suboptimal partitions, and given a partition, it is difficult to detect $k$-means optimality. In this paper, we develop an a posteriori certifier of approximate optimality for $k$-means clustering. The certifier is a sub-linear Monte Carlo algorithm based on Peng and Wei's semidefinite relaxation of $k$-means. In particular, solving the relaxation for small random samples of the dataset produces a high-confidence lower bound on the $k$-means objective, and being sub-linear, our algorithm is faster than $k$-means++ when the nu…
Central European University（中欧大学）；ISI Foundation（ISI基金会）；University of Bath（巴斯大学）
We present a Markov chain Monte Carlo scheme based on merges and splits of groups that is capable of efficiently sampling from the posterior distribution of network partitions, defined according to the stochastic block model (SBM). We demonstrate how schemes based on the move of single nodes between groups systematically fail at correctly sampling from the posterior distribution even on small networks, and how our merge-split approach behaves significantly better, and improves the mixing time of the Markov chain by several orders of magnitude in typical cases. We also show how the scheme can b…
We study approximations of the partition function of dense graphical models. Partition functions of graphical models play a fundamental role is statistical physics, in statistics and in machine learning. Two of the main methods for approximating the partition function are Markov Chain Monte Carlo and Variational Methods. An impressive body of work in mathematics, physics and theoretical computer science provides conditions under which Markov Chain Monte Carlo methods converge in polynomial time. These methods often lead to polynomial time approximation algorithms for the partition function in …
Quasi-Monte Carlo Feature Maps for Shift-Invariant Kernels
IBM T.J. Watson Research Center（IBM托马斯·J·沃森研究中心）；Google Research（谷歌研究院）；Stanford University（斯坦福大学）；International Computer Science Institute（国际计算机科学研究所）；University of California at Berkeley（加州大学伯克利分校）
The partition function of a factor graph can sometimes be accurately estimated by Monte Carlo methods. In this paper, such methods are extended to factor graphs with negative and complex factors.
Many probabilistic models of interest in scientific computing and machine learning have expensive, black-box likelihoods that prevent the application of standard techniques for Bayesian inference, such as MCMC, which would require access to the gradient or a large number of likelihood evaluations. We introduce here a novel sample-efficient inference framework, Variational Bayesian Monte Carlo (VBMC). VBMC combines variational inference with Gaussian-process based, active-sampling Bayesian quadrature, using the latter to efficiently approximate the intractable integral in the variational object…
Divide-and-conquer strategies for Monte Carlo algorithms are an increasingly popular approach to making Bayesian inference scalable to large data sets. In its simplest form, the data are partitioned across multiple computing cores and a separate Markov chain Monte Carlo algorithm on each core targets the associated partial posterior distribution, which we refer to as a sub-posterior, that is the posterior given only the data from the segment of the partition associated with that core. Divide-and-conquer techniques reduce computational, memory and disk bottle-necks, but make it difficult to rec…
The partition function of a factor graph and the partition function of the dual factor graph are related to each other by the normal factor graph duality theorem. We apply this result to the classical problem of computing the partition function of the Ising model. In the one-dimensional case, we thus obtain an alternative derivation of the (well-known) analytical solution. In the two-dimensional case, we find that Monte Carlo methods are much more efficient on the dual graph than on the original graph, especially at low temperature.
We consider conditional tests for non-negative discrete exponential families. We develop two Markov Chain Monte Carlo (MCMC) algorithms which allow us to sample from the conditional space and to perform approximated tests. The first algorithm is based on the MCMC sampling described by Sturmfels. The second MCMC sampling consists in a more efficient algorithm which exploits the optimal partition of the conditional space into orbits of permutations. We thus establish a link between standard permutation and algebraic-statistics-based sampling. Through a simulation study we compare the exact cumul…
University of Bristol（布里斯托大学）；Spotify（声田）；University of Bath（巴斯大学）；University of Cambridge（剑桥大学）
We present a novel approach to Bayesian inference and general Bayesian computation that is defined through a sequential decision loop. Our method defines a recursive partitioning of the sample space. It neither relies on gradients nor requires any problem-specific tuning, and is asymptotically exact for any density function with a bounded domain. The output is an approximation to the whole density function including the normalisation constant, via partitions organised in efficient data structures. Such approximations may be used for evidence estimation or fast posterior sampling, but also as b…
The key to Black-Box Optimization is to efficiently search through input regions with potentially widely-varying numerical properties, to achieve low-regret descent and fast progress toward the optima. Monte Carlo Tree Search (MCTS) methods have recently been introduced to improve Bayesian optimization by computing better partitioning of the search space that balances exploration and exploitation. Extending this promising framework, we study how to further integrate sample-based descent for faster optimization. We design novel ways of expanding Monte Carlo search trees, with new descent method…
University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；Mount Holyoke College（曼荷莲学院）
Recent research has seen several advances relevant to black-box VI, but the current state of automatic posterior inference is unclear. One such advance is the use of normalizing flows to define flexible posterior densities for deep latent variable models. Another direction is the integration of Monte-Carlo methods to serve two purposes; first, to obtain tighter variational objectives for optimization, and second, to define enriched variational families through sampling. However, both flows and variational Monte-Carlo methods remain relatively unexplored for black-box VI. Moreover, on a pragmat…
In this work, we propose a simple but effective method to interpret black-box machine learning models globally. That is, we use a compact binary tree, the interpretation tree, to explicitly represent the most important decision rules that are implicitly contained in the black-box machine learning models. This tree is learned from the contribution matrix which consists of the contributions of input variables to predicted scores for each single prediction. To generate the interpretation tree, a unified process recursively partitions the input variable space by maximizing the difference in the av…
San José State University（圣何塞州立大学）；Johns Hopkins University（约翰斯·霍普金斯大学）；Universitá degli Studi di Napoli Federico II（那不勒斯费德里科二世大学）；Scuola Superiore Meridionale（南方高等学校）
We propose a machine learning framework for the data-driven discovery of macroscopic chemotactic Partial Differential Equations (PDEs) -- and the closures that lead to them -- from high-fidelity, individual-based stochastic simulations of E.coli bacterial motility. The fine scale, detailed, hybrid (continuum - Monte Carlo) simulation model embodies the underlying biophysics, and its parameters are informed from experimental observations of individual cells. We exploit Automatic Relevance Determination (ARD) within a Gaussian Process framework for the identification of a parsimonious set of col…
Fudan University（复旦大学）；The University of Melbourne（墨尔本大学）
Deep neural networks (DNNs) are known for their vulnerability to adversarial examples. These are examples that have undergone small, carefully crafted perturbations, and which can easily fool a DNN into making misclassifications at test time. Thus far, the field of adversarial research has mainly focused on image models, under either a white-box setting, where an adversary has full access to model parameters, or a black-box setting where an adversary can only query the target model for probabilities or labels. Whilst several white-box attacks have been proposed for video models, black-box vide…
We interpret the predictions of any black-box structured input-structured output model around a specific input-output pair. Our method returns an "explanation" consisting of groups of input-output tokens that are causally related. These dependencies are inferred by querying the black-box model with perturbed inputs, generating a graph over tokens from the responses, and solving a partitioning problem to select the most relevant components. We focus the general approach on sequence-to-sequence problems, adopting a variational autoencoder to yield meaningful input perturbations. We test our meth…
Department of Basic Neuroscience, University of Geneva（日内瓦大学基础神经科学系）
Variational Bayesian Monte Carlo (VBMC) is a recently introduced framework that uses Gaussian process surrogates to perform approximate Bayesian inference in models with black-box, non-cheap likelihoods. In this work, we extend VBMC to deal with noisy log-likelihood evaluations, such as those arising from simulation-based models. We introduce new `global' acquisition functions, such as expected information gain (EIG) and variational interquantile range (VIQR), which are robust to noise and can be efficiently evaluated within the VBMC setting. In a novel, challenging, noisy-inference benchmark …
