Solving Partially Observable Markov Decision Processes (POMDPs) with continuous actions is challenging, particularly for high-dimensional action spaces. To alleviate this difficulty, we propose a new sampling-based online POMDP solver, called Adaptive Discretization using Voronoi Trees (ADVT). It uses Monte Carlo Tree Search in combination with an adaptive discretization of the action space as well as optimistic optimization to efficiently sample high-dimensional continuous action spaces and compute the best action to perform. Specifically, we adaptively discretize the action space for each sampled belief using a hierarchical partition which we call a Voronoi tree. A Voronoi tree is a Binary Space Partitioning (BSP) that implicitly maintains the partition of a cell as the Voronoi diagram of two points sampled from the cell. This partitioning strategy keeps the cost of partitioning and estimating the size of each cell low, even in high-dimensional spaces where many sampled points are required to cover the space well. ADVT uses the estimated sizes of the cells to form an upper-confidence bound of the action values of the cell, and in turn uses the upper-confidence bound to guide the Monte Carlo Tree Search expansion and further discretization of the action space. This strategy enables ADVT to better exploit local information in the action space, leading to an action space discretization that is more adaptive, and hence more efficient in computing good POMDP solutions, compared to existing solvers. Experiments on simulations of four types of benchmark problems indicate that ADVT outperforms and scales substantially better to high-dimensional continuous action spaces, compared to state-of-the-art continuous action POMDP solvers.
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Score matching (SM) is a convenient method for training flexible probabilistic models, which is often preferred over the traditional maximum-likelihood (ML) approach. However, these models are less interpretable than normalized models; as such, training robustness is in general difficult to assess. We present a critical study of existing variational SM objectives, showing catastrophic failure on a wide range of datasets and network architectures. Our theoretical insights on the objectives emerge directly from their equivalent autoencoding losses when optimizing variational autoencoder (VAE) models. First, we show that in the Fisher autoencoder, SM produces far worse models than maximum-likelihood, and approximate inference by Fisher divergence can lead to low-density local optima. However, with important modifications, this objective reduces to a regularized autoencoding loss that resembles the evidence lower bound (ELBO). This analysis predicts that the modified SM algorithm should behave very similarly to ELBO on Gaussian VAEs. We then review two other FD-based objectives from the literature and show that they reduce to uninterpretable autoencoding losses, likely leading to poor performance. The experiments verify our theoretical predictions and suggest that only ELBO and the baseline objective robustly produce expected results, while previously proposed SM methods do not.
Meta-learning aims to extract useful inductive biases from a set of related datasets. In Bayesian meta-learning, this is typically achieved by constructing a prior distribution over neural network parameters. However, specifying families of computationally viable prior distributions over the high-dimensional neural network parameters is difficult. As a result, existing approaches resort to meta-learning restrictive diagonal Gaussian priors, severely limiting their expressiveness and performance. To circumvent these issues, we approach meta-learning through the lens of functional Bayesian neural network inference, which views the prior as a stochastic process and performs inference in the function space. Specifically, we view the meta-training tasks as samples from the data-generating process and formalize meta-learning as empirically estimating the law of this stochastic process. Our approach can seamlessly acquire and represent complex prior knowledge by meta-learning the score function of the data-generating process marginals instead of parameter space priors. In a comprehensive benchmark, we demonstrate that our method achieves state-of-the-art performance in terms of predictive accuracy and substantial improvements in the quality of uncertainty estimates.
The Bayesian paradigm provides a rigorous framework for estimating the whole probability distribution over unknown parameters, but due to high computational costs, its online application can be difficult. We propose the Adaptive Recursive Markov Chain Monte Carlo (ARMCMC) method, which calculates the complete probability density function of model parameters while alleviating the drawbacks of traditional online methods. These flaws include being limited to Gaussian noise, being solely applicable to linear in the parameters (LIP) systems, and having persisting excitation requirements (PE). A variable jump distribution based on a temporal forgetting factor (TFF) is proposed in ARMCMC. The TFF can be utilized in many dynamical systems as an effective way to adaptively present the forgetting factor instead of a constant hyperparameter. The particular jump distribution has tailored towards hybrid/multi-modal systems that enables inferences among modes by providing a trade-off between exploitation and exploration. These trade-off are adjusted based on parameter evolution rate. In comparison to traditional MCMC techniques, we show that ARMCMC requires fewer samples to obtain the same accuracy and reliability. We show our method on two challenging benchmarks: parameter estimation in a soft bending actuator and the Hunt-Crossley dynamic model. We also compare our method with recursive least squares and the particle filter, and show that our technique has significantly more accurate point estimates as well as a decrease in tracking error of the value of interest.
Diffusion models can be viewed as mapping points in a high-dimensional latent space onto a low-dimensional learned manifold, typically an image manifold. The intermediate values between the latent space and image manifold can be interpreted as noisy images which are determined by the noise scheduling scheme employed during pre-training. We exploit this interpretation to introduce Boomerang, a local image manifold sampling approach using the dynamics of diffusion models. We call it Boomerang because we first add noise to an input image, moving it closer to the latent space, then bring it back to the image space through diffusion dynamics. We use this method to generate images which are similar, but nonidentical, to the original input images on the image manifold. We are able to set how close the generated image is to the original based on how much noise we add. Additionally, the generated images have a degree of stochasticity, allowing us to locally sample as many times as we want without repetition. We show three applications for which Boomerang can be used. First, we provide a framework for constructing privacy-preserving datasets having controllable degrees of anonymity. Second, we show how to use Boomerang for data augmentation while staying on the image manifold. Third, we introduce a framework for image super-resolution with 8x upsampling. Boomerang does not require any modification to the training of diffusion models and can be used with pretrained models on a single, inexpensive GPU.
The selection of time step plays a crucial role in improving stability and efficiency in the Discontinuous Galerkin (DG) solution of hyperbolic conservation laws on adaptive moving meshes that typically employs explicit stepping. A commonly used selection of time step is a direct extension based on Courant-Friedrichs-Levy (CFL) conditions established for fixed and uniform meshes. In this work, we provide a mathematical justification for those time step selection strategies used in practical adaptive DG computations. A stability analysis is presented for a moving mesh DG method for linear scalar conservation laws. Based on the analysis, a new selection strategy of the time step is proposed, which takes into consideration the coupling of the $\alpha$-function (that is related to the eigenvalues of the Jacobian matrix of the flux and the mesh movement velocity) and the heights of the mesh elements. The analysis also suggests several stable combinations of the choices of the $\alpha$-function in the numerical scheme and in the time step selection. Numerical results obtained with a moving mesh DG method for Burgers' and Euler equations are presented. For comparison purpose, numerical results obtained with an error-based time step-size selection strategy are also given.
Feature selection is used to eliminate redundant features and keep relevant features, it can enhance machine learning algorithm's performance and accelerate computing speed. In various methods, mutual information has attracted increasingly more attention as it's an effective criterion to measure variable correlation. However, current works mainly focus on maximizing the feature relevancy with class label and minimizing the feature redundancy within selected features, we reckon that pursuing feature redundancy minimization is reasonable but not necessary because part of so-called redundant features also carries some useful information to promote performance. In terms of mutual information calculation, it may distort the true relationship between two variables without proper neighborhood partition. Traditional methods usually split the continuous variables into several intervals even ignore such influence. We theoretically prove how variable fluctuation negatively influences mutual information calculation. To remove the referred obstacles, for feature selection method, we propose a full conditional mutual information maximization method (FCMIM) which only considers the feature relevancy in two aspects. For obtaining a better partition effect and eliminating the negative influence of attribute fluctuation, we put up an adaptive neighborhood partition algorithm (ANP) with the feedback of mutual information maximization algorithm, the backpropagation process helps search for a proper neighborhood partition parameter. We compare our method with several mutual information methods on 17 benchmark datasets. Results of FCMIM are better than other methods based on different classifiers. Results show that ANP indeed promotes nearly all the mutual information methods' performance.
The simulation of human neurons and neurotransmission mechanisms has been realized in deep neural networks based on the theoretical implementations of activation functions. However, recent studies have reported that the threshold potential of neurons exhibits different values according to the locations and types of individual neurons, and that the activation functions have limitations in terms of representing this variability. Therefore, this study proposes a simple yet effective activation function that facilitates different thresholds and adaptive activations according to the positions of units and the contexts of inputs. Furthermore, the proposed activation function mathematically exhibits a more generalized form of Swish activation function, and thus we denoted it as Adaptive SwisH (ASH). ASH highlights informative features that exhibit large values in the top percentiles in an input, whereas it rectifies low values. Most importantly, ASH exhibits trainable, adaptive, and context-aware properties compared to other activation functions. Furthermore, ASH represents general formula of the previously studied activation function and provides a reasonable mathematical background for the superior performance. To validate the effectiveness and robustness of ASH, we implemented ASH into many deep learning models for various tasks, including classification, detection, segmentation, and image generation. Experimental analysis demonstrates that our activation function can provide the benefits of more accurate prediction and earlier convergence in many deep learning applications.
In online learning problems, exploiting low variance plays an important role in obtaining tight performance guarantees yet is challenging because variances are often not known a priori. Recently, considerable progress has been made by Zhang et al. (2021) where they obtain a variance-adaptive regret bound for linear bandits without knowledge of the variances and a horizon-free regret bound for linear mixture Markov decision processes (MDPs). In this paper, we present novel analyses that improve their regret bounds significantly. For linear bandits, we achieve $\tilde O(\min\{d\sqrt{K}, d^{1.5}\sqrt{\sum_{k=1}^K \sigma_k^2}\} + d^2)$ where $d$ is the dimension of the features, $K$ is the time horizon, and $\sigma_k^2$ is the noise variance at time step $k$, and $\tilde O$ ignores polylogarithmic dependence, which is a factor of $d^3$ improvement. For linear mixture MDPs with the assumption of maximum cumulative reward in an episode being in $[0,1]$, we achieve a horizon-free regret bound of $\tilde O(d \sqrt{K} + d^2)$ where $d$ is the number of base models and $K$ is the number of episodes. This is a factor of $d^{3.5}$ improvement in the leading term and $d^7$ in the lower order term. Our analysis critically relies on a novel peeling-based regret analysis that leverages the elliptical potential `count' lemma.
Variable selection is crucial for sparse modeling in this age of big data. Missing values are common in data, and make variable selection more complicated. The approach of multiple imputation (MI) results in multiply imputed datasets for missing values, and has been widely applied in various variable selection procedures. However, directly performing variable selection on the whole MI data or bootstrapped MI data may not be worthy in terms of computation cost. To fast identify the active variables in the linear regression model, we propose the adaptive grafting procedure with three pooling rules on MI data. The proposed methods proceed iteratively, which starts from finding the active variables based on the complete case subset and then expand the working data matrix with both the number of active variables and available observations. A comprehensive simulation study shows the selection accuracy in different aspects and computational efficiency of the proposed methods. Two real-life examples illustrate the strength of the proposed methods.
Importance sampling (IS) is a powerful Monte Carlo methodology for the approximation of intractable integrals, very often involving a target probability distribution. The performance of IS heavily depends on the appropriate selection of the proposal distributions where the samples are simulated from. In this paper, we propose an adaptive importance sampler, called GRAMIS, that iteratively improves the set of proposals. The algorithm exploits geometric information of the target to adapt the location and scale parameters of those proposals. Moreover, in order to allow for a cooperative adaptation, a repulsion term is introduced that favors a coordinated exploration of the state space. This translates into a more diverse exploration and a better approximation of the target via the mixture of proposals. Moreover, we provide a theoretical justification of the repulsion term. We show the good performance of GRAMIS in two problems where the target has a challenging shape and cannot be easily approximated by a standard uni-modal proposal.
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