In this paper, we investigate the problem of strong approximation of the solution of SDEs in the case when the drift coefficient is given in the integral form. Such drift often appears when analyzing stochastic dynamics of optimization procedures in machine learning problems. We discuss connections of the defined randomized Euler approximation scheme with the perturbed version of the stochastic gradient descent (SGD) algorithm. We investigate its upper error bounds, in terms of the discretization parameter n and the size M of the random sample drawn at each step of the algorithm, in different subclasses of coefficients of the underlying SDE. Finally, the results of numerical experiments performed by using GPU architecture are also reported.
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In this study, we address the issue of enabling an artificial intelligence agent to execute complex language instructions within virtual environments. In our framework, we assume that these instructions involve intricate linguistic structures and multiple interdependent tasks that must be navigated successfully to achieve the desired outcomes. To effectively manage these complexities, we propose a hierarchical framework that combines the deep language comprehension of large language models with the adaptive action-execution capabilities of reinforcement learning agents. The language module (based on LLM) translates the language instruction into a high-level action plan, which is then executed by a pre-trained reinforcement learning agent. We have demonstrated the effectiveness of our approach in two different environments: in IGLU, where agents are instructed to build structures, and in Crafter, where agents perform tasks and interact with objects in the surrounding environment according to language commands.
Motivated by the application of saddlepoint approximations to resampling-based statistical tests, we prove that a Lugananni-Rice style approximation for conditional tail probabilities of averages of conditionally independent random variables has vanishing relative error. We also provide a general condition on the existence and uniqueness of the solution to the corresponding saddlepoint equation. The results are valid under a broad class of distributions involving no restrictions on the smoothness of the distribution function. The derived saddlepoint approximation formula can be directly applied to resampling-based hypothesis tests, including bootstrap, sign-flipping and conditional randomization tests. Our results extend and connect several classical saddlepoint approximation results. On the way to proving our main results, we prove a new conditional Berry-Esseen inequality for the sum of conditionally independent random variables, which may be of independent interest.
This paper investigates the application of mini-batch gradient descent to semiflows. Given a loss function, we introduce a continuous version of mini-batch gradient descent by randomly selecting sub-loss functions over time, defining a piecewise flow. We prove that, under suitable assumptions on the gradient flow, the mini-batch descent flow trajectory closely approximates the original gradient flow trajectory on average. Additionally, we propose a randomized minimizing movement scheme that also approximates the gradient flow of the loss function. We illustrate the versatility of this approach across various problems, including constrained optimization, sparse inversion, and domain decomposition. Finally, we validate our results with several numerical examples.
In this paper, we study an optimal control problem for a coupled non-linear system of reaction-diffusion equations with degenerate diffusion, consisting of two partial differential equations representing the density of cells and the concentration of the chemotactic agent. By controlling the concentration of the chemical substrates, this study can guide the optimal growth of cells. The novelty of this work lies on the direct and dual models that remain in a weak setting, which is uncommon in the recent literature for solving optimal control systems. Moreover, it is known that the adjoint problems offer a powerful approach to quantifying the uncertainty associated with model inputs. However, these systems typically lack closed-form solutions, making it challenging to obtain weak solutions. For that, the well-posedness of the direct problem is first well guaranteed. Then, the existence of an optimal control and the first-order optimality conditions are established. Finally, weak solutions for the adjoint system to the non-linear degenerate direct model, are introduced and investigated.
In this paper, we introduce the finite difference weighted essentially non-oscillatory (WENO) scheme based on the neural network for hyperbolic conservation laws. We employ the supervised learning and design two loss functions, one with the mean squared error and the other with the mean squared logarithmic error, where the WENO3-JS weights are computed as the labels. Each loss function consists of two components where the first component compares the difference between the weights from the neural network and WENO3-JS weights, while the second component matches the output weights of the neural network and the linear weights. The former of the loss function enforces the neural network to follow the WENO properties, implying that there is no need for the post-processing layer. Additionally the latter leads to better performance around discontinuities. As a neural network structure, we choose the shallow neural network (SNN) for computational efficiency with the Delta layer consisting of the normalized undivided differences. These constructed WENO3-SNN schemes show the outperformed results in one-dimensional examples and improved behavior in two-dimensional examples, compared with the simulations from WENO3-JS and WENO3-Z.
In this paper we discuss reduced order models for the approximation of parametric eigenvalue problems. In particular, we are interested in the presence of intersections or clusters of eigenvalues. The singularities originating by these phenomena make it hard a straightforward generalization of well known strategies normally used for standards PDEs. We investigate how the known results extend (or not) to higher order frequencies.
Robust and accurate localization in challenging environments is becoming crucial for SLAM. In this paper, we propose a unique sensor configuration for precise and robust odometry by integrating chip radar and a legged robot. Specifically, we introduce a tightly coupled radar-leg odometry algorithm for complementary drift correction. Adopting the 4-DoF optimization and decoupled RANSAC to mmWave chip radar significantly enhances radar odometry beyond the existing method, especially z-directional even when using a single radar. For the leg odometry, we employ rolling contact modeling-aided forward kinematics, accommodating scenarios with the potential possibility of contact drift and radar failure. We evaluate our method by comparing it with other chip radar odometry algorithms using real-world datasets with diverse environments while the datasets will be released for the robotics community. //github.com/SangwooJung98/Co-RaL-Dataset
In this study, we introduce the application of causal disparity analysis to unveil intricate relationships and causal pathways between sensitive attributes and the targeted outcomes within real-world observational data. Our methodology involves employing causal decomposition analysis to quantify and examine the causal interplay between sensitive attributes and outcomes. We also emphasize the significance of integrating heterogeneity assessment in causal disparity analysis to gain deeper insights into the impact of sensitive attributes within specific sub-groups on outcomes. Our two-step investigation focuses on datasets where race serves as the sensitive attribute. The results on two datasets indicate the benefit of leveraging causal analysis and heterogeneity assessment not only for quantifying biases in the data but also for disentangling their influences on outcomes. We demonstrate that the sub-groups identified by our approach to be affected the most by disparities are the ones with the largest ML classification errors. We also show that grouping the data only based on a sensitive attribute is not enough, and through these analyses, we can find sub-groups that are directly affected by disparities. We hope that our findings will encourage the adoption of such methodologies in future ethical AI practices and bias audits, fostering a more equitable and fair technological landscape.
In this article, we consider the sparse tensor singular value decomposition, which aims for dimension reduction on high-dimensional high-order data with certain sparsity structure. A method named Sparse Tensor Alternating Thresholding for Singular Value Decomposition (STAT-SVD) is proposed. The proposed procedure features a novel double projection \& thresholding scheme, which provides a sharp criterion for thresholding in each iteration. Compared with regular tensor SVD model, STAT-SVD permits more robust estimation under weaker assumptions. Both the upper and lower bounds for estimation accuracy are developed. The proposed procedure is shown to be minimax rate-optimal in a general class of situations. Simulation studies show that STAT-SVD performs well under a variety of configurations. We also illustrate the merits of the proposed procedure on a longitudinal tensor dataset on European country mortality rates.
In this paper, we focus on the self-supervised learning of visual correspondence using unlabeled videos in the wild. Our method simultaneously considers intra- and inter-video representation associations for reliable correspondence estimation. The intra-video learning transforms the image contents across frames within a single video via the frame pair-wise affinity. To obtain the discriminative representation for instance-level separation, we go beyond the intra-video analysis and construct the inter-video affinity to facilitate the contrastive transformation across different videos. By forcing the transformation consistency between intra- and inter-video levels, the fine-grained correspondence associations are well preserved and the instance-level feature discrimination is effectively reinforced. Our simple framework outperforms the recent self-supervised correspondence methods on a range of visual tasks including video object tracking (VOT), video object segmentation (VOS), pose keypoint tracking, etc. It is worth mentioning that our method also surpasses the fully-supervised affinity representation (e.g., ResNet) and performs competitively against the recent fully-supervised algorithms designed for the specific tasks (e.g., VOT and VOS).
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