Models with intractable normalising functions have numerous applications. Because the normalising constants are functions of the parameters of interest, standard Markov chain Monte Carlo cannot be used for Bayesian inference for these models. Many algorithms have been developed for such models. Some have the posterior distribution as the asymptotic distribution. Other ``asymptotically inexact'' algorithms do not possess this property. There is limited guidance for evaluating approximations based on these algorithms. We propose two new diagnostics that address these problems. We provide theoretical justification for our methods and apply them to several algorithms in the context of challenging examples.
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Traditional hidden Markov models have been a useful tool to understand and model stochastic dynamic data; in the case of non-Gaussian data, models such as mixture of Gaussian hidden Markov models can be used. However, these suffer from the computation of precision matrices and have a lot of unnecessary parameters. As a consequence, such models often perform better when it is assumed that all variables are independent, a hypothesis that may be unrealistic. Hidden Markov models based on kernel density estimation are also capable of modeling non-Gaussian data, but they assume independence between variables. In this article, we introduce a new hidden Markov model based on kernel density estimation, which is capable of capturing kernel dependencies using context-specific Bayesian networks. The proposed model is described, together with a learning algorithm based on the expectation-maximization algorithm. Additionally, the model is compared to related HMMs on synthetic and real data. From the results, the benefits in likelihood and classification accuracy from the proposed model are quantified and analyzed.
Directional tests to compare incomplete undirected graphs are developed in the general context of covariance selection for Gaussian graphical models. The exactness of the underlying saddlepoint approximation is proved for chordal graphs and leads to exact control of the size of the tests, given that the only approximation error involved is due to the numerical calculation of two scalar integrals. Although exactness is not guaranteed for non-chordal graphs, the ability of the saddlepoint approximation to control the relative error leads the directional test to overperform its competitors even in these cases. The accuracy of our proposal is verified by simulation experiments under challenging scenarios, where inference via standard asymptotic approximations to the likelihood ratio test and some of its higher-order modifications fails. The directional approach is used to illustrate the assessment of Markovian dependencies in a dataset from a veterinary trial on cattle. A second example with microarray data shows how to select the graph structure related to genetic anomalies due to acute lymphocytic leukemia.
Nonlinear model predictive control (NMPC) solves a multivariate optimization problem to estimate the system's optimal control inputs in each control cycle. Such optimization is made more difficult by several factors, such as nonlinearities inherited in the system, highly coupled inputs, and various constraints related to the system's physical limitations. These factors make the optimization to be non-convex and hard to solve traditionally. Genetic algorithm (GA) is typically used extensively to tackle such optimization in several application domains because it does not involve differential calculation or gradient evaluation in its solution estimation. However, the size of the search space in which the GA searches for the optimal control inputs is crucial for the applicability of the GA with systems that require fast response. This paper proposes an approach to accelerate the genetic optimization of NMPC by learning optimal search space size. The proposed approach trains a multivariate regression model to adaptively predict the best smallest search space in every control cycle. The estimated best smallest size of search space is fed to the GA to allow for searching the optimal control inputs within this search space. The proposed approach not only reduces the GA's computational time but also improves the chance of obtaining the optimal control inputs in each cycle. The proposed approach was evaluated on two nonlinear systems and compared with two other genetic-based NMPC approaches implemented on the GPU of a Nvidia Jetson TX2 embedded platform in a processor-in-the-loop (PIL) fashion. The results show that the proposed approach provides a 39-53\% reduction in computational time. Additionally, it increases the convergence percentage to the optimal control inputs within the cycle's time by 48-56\%, resulting in a significant performance enhancement. The source code is available on GitHub.
Communication compression is an essential strategy for alleviating communication overhead by reducing the volume of information exchanged between computing nodes in large-scale distributed stochastic optimization. Although numerous algorithms with convergence guarantees have been obtained, the optimal performance limit under communication compression remains unclear. In this paper, we investigate the performance limit of distributed stochastic optimization algorithms employing communication compression. We focus on two main types of compressors, unbiased and contractive, and address the best-possible convergence rates one can obtain with these compressors. We establish the lower bounds for the convergence rates of distributed stochastic optimization in six different settings, combining strongly-convex, generally-convex, or non-convex functions with unbiased or contractive compressor types. To bridge the gap between lower bounds and existing algorithms' rates, we propose NEOLITHIC, a nearly optimal algorithm with compression that achieves the established lower bounds up to logarithmic factors under mild conditions. Extensive experimental results support our theoretical findings. This work provides insights into the theoretical limitations of existing compressors and motivates further research into fundamentally new compressor properties.
Inverse problems are in many cases solved with optimization techniques. When the underlying model is linear, first-order gradient methods are usually sufficient. With nonlinear models, due to nonconvexity, one must often resort to second-order methods that are computationally more expensive. In this work we aim to approximate a nonlinear model with a linear one and correct the resulting approximation error. We develop a sequential method that iteratively solves a linear inverse problem and updates the approximation error by evaluating it at the new solution. This treatment convexifies the problem and allows us to benefit from established convex optimization methods. We separately consider cases where the approximation is fixed over iterations and where the approximation is adaptive. In the fixed case we show theoretically under what assumptions the sequence converges. In the adaptive case, particularly considering the special case of approximation by first-order Taylor expansion, we show that with certain assumptions the sequence converges to a critical point of the original nonconvex functional. Furthermore, we show that with quadratic objective functions the sequence corresponds to the Gauss-Newton method. Finally, we showcase numerical results superior to the conventional model correction method. We also show, that a fixed approximation can provide competitive results with considerable computational speed-up.
Reinforcement Learning (RL) algorithms are known to scale poorly to environments with many available actions, requiring numerous samples to learn an optimal policy. The traditional approach of considering the same fixed action space in every possible state implies that the agent must understand, while also learning to maximize its reward, to ignore irrelevant actions such as $\textit{inapplicable actions}$ (i.e. actions that have no effect on the environment when performed in a given state). Knowing this information can help reduce the sample complexity of RL algorithms by masking the inapplicable actions from the policy distribution to only explore actions relevant to finding an optimal policy. While this technique has been formalized for quite some time within the Automated Planning community with the concept of precondition in the STRIPS language, RL algorithms have never formally taken advantage of this information to prune the search space to explore. This is typically done in an ad-hoc manner with hand-crafted domain logic added to the RL algorithm. In this paper, we propose a more systematic approach to introduce this knowledge into the algorithm. We (i) standardize the way knowledge can be manually specified to the agent; and (ii) present a new framework to autonomously learn the partial action model encapsulating the precondition of an action jointly with the policy. We show experimentally that learning inapplicable actions greatly improves the sample efficiency of the algorithm by providing a reliable signal to mask out irrelevant actions. Moreover, we demonstrate that thanks to the transferability of the knowledge acquired, it can be reused in other tasks and domains to make the learning process more efficient.
High-dimensional linear regression under heavy-tailed noise or outlier corruption is challenging, both computationally and statistically. Convex approaches have been proven statistically optimal but suffer from high computational costs, especially since the robust loss functions are usually non-smooth. More recently, computationally fast non-convex approaches via sub-gradient descent are proposed, which, unfortunately, fail to deliver a statistically consistent estimator even under sub-Gaussian noise. In this paper, we introduce a projected sub-gradient descent algorithm for both the sparse linear regression and low-rank linear regression problems. The algorithm is not only computationally efficient with linear convergence but also statistically optimal, be the noise Gaussian or heavy-tailed with a finite 1 + epsilon moment. The convergence theory is established for a general framework and its specific applications to absolute loss, Huber loss and quantile loss are investigated. Compared with existing non-convex methods, ours reveals a surprising phenomenon of two-phase convergence. In phase one, the algorithm behaves as in typical non-smooth optimization that requires gradually decaying stepsizes. However, phase one only delivers a statistically sub-optimal estimator, which is already observed in the existing literature. Interestingly, during phase two, the algorithm converges linearly as if minimizing a smooth and strongly convex objective function, and thus a constant stepsize suffices. Underlying the phase-two convergence is the smoothing effect of random noise to the non-smooth robust losses in an area close but not too close to the truth. Numerical simulations confirm our theoretical discovery and showcase the superiority of our algorithm over prior methods.
The Importance Markov chain is a novel algorithm bridging the gap between rejection sampling and importance sampling, moving from one to the other through a tuning parameter. Based on a modified sample of an instrumental Markov chain targeting an instrumental distribution (typically via a MCMC kernel), the Importance Markov chain produces an extended Markov chain where the marginal distribution of the first component converges to the target distribution. For example, when targeting a multimodal distribution, the instrumental distribution can be chosen as a tempered version of the target which allows the algorithm to explore its modes more efficiently. We obtain a Law of Large Numbers and a Central Limit Theorem as well as geometric ergodicity for this extended kernel under mild assumptions on the instrumental kernel. Computationally, the algorithm is easy to implement and preexisting librairies can be used to sample from the instrumental distribution.
Model selection is critical in the modern statistics and machine learning community. However, most existing works do not apply to heavy-tailed data, which are commonly encountered in real applications, such as the single-cell multiomics data. In this paper, we propose a rank-sum based approach that outputs a confidence set containing the optimal model with guaranteed probability. Motivated by conformal inference, we developed a general method that is applicable without moment or tail assumptions on the data. We demonstrate the advantage of the proposed method through extensive simulation and a real application on the COVID-19 genomics dataset (Stephenson et al., 2021). To perform the inference on rank-sum statistics, we derive a general Gaussian approximation theory for high dimensional two-sample U-statistics, which may be of independent interest to the statistics and machine learning community.
When and why can a neural network be successfully trained? This article provides an overview of optimization algorithms and theory for training neural networks. First, we discuss the issue of gradient explosion/vanishing and the more general issue of undesirable spectrum, and then discuss practical solutions including careful initialization and normalization methods. Second, we review generic optimization methods used in training neural networks, such as SGD, adaptive gradient methods and distributed methods, and theoretical results for these algorithms. Third, we review existing research on the global issues of neural network training, including results on bad local minima, mode connectivity, lottery ticket hypothesis and infinite-width analysis.
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