Neural ordinary differential equations (ODEs) are widely recognized as the standard for modeling physical mechanisms, which help to perform approximate inference in unknown physical or biological environments. In partially observable (PO) environments, how to infer unseen information from raw observations puzzled the agents. By using a recurrent policy with a compact context, context-based reinforcement learning provides a flexible way to extract unobservable information from historical transitions. To help the agent extract more dynamics-related information, we present a novel ODE-based recurrent model combines with model-free reinforcement learning (RL) framework to solve partially observable Markov decision processes (POMDPs). We experimentally demonstrate the efficacy of our methods across various PO continuous control and meta-RL tasks. Furthermore, our experiments illustrate that our method is robust against irregular observations, owing to the ability of ODEs to model irregularly-sampled time series.
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Designing and analyzing model-based RL (MBRL) algorithms with guaranteed monotonic improvement has been challenging, mainly due to the interdependence between policy optimization and model learning. Existing discrepancy bounds generally ignore the impacts of model shifts, and their corresponding algorithms are prone to degrade performance by drastic model updating. In this work, we first propose a novel and general theoretical scheme for a non-decreasing performance guarantee of MBRL. Our follow-up derived bounds reveal the relationship between model shifts and performance improvement. These discoveries encourage us to formulate a constrained lower-bound optimization problem to permit the monotonicity of MBRL. A further example demonstrates that learning models from a dynamically-varying number of explorations benefit the eventual returns. Motivated by these analyses, we design a simple but effective algorithm CMLO (Constrained Model-shift Lower-bound Optimization), by introducing an event-triggered mechanism that flexibly determines when to update the model. Experiments show that CMLO surpasses other state-of-the-art methods and produces a boost when various policy optimization methods are employed.
The log-conformation formulation, although highly successful, was from the beginning formulated as a partial differential equation that contains an, for PDEs unusual, eigenvalue decomposition of the unknown field. To this day, most numerical implementations have been based on this or a similar eigenvalue decomposition, with Knechtges et al. (2014) being the only notable exception for two-dimensional flows. In this paper, we present an eigenvalue-free algorithm to compute the constitutive equation of the log-conformation formulation that works for two- and three-dimensional flows. Therefore, we first prove that the challenging terms in the constitutive equations are representable as a matrix function of a slightly modified matrix of the log-conformation field. We give a proof of equivalence of this term to the more common log-conformation formulations. Based on this formulation, we develop an eigenvalue-free algorithm to evaluate this matrix function. The resulting full formulation is first discretized using a finite volume method, and then tested on the confined cylinder and sedimenting sphere benchmarks.
An important dimension of pointer analysis is field-Sensitive, which has been proven to effectively enhance the accuracy of pointer analysis results. A crucial area of research within field-Sensitive is Structure-Sensitive. Structure-Sensitive has been shown to further enhance the precision of pointer analysis. However, existing structure-sensitive methods cannot handle cases where an object possesses multiple structures, even though it's common for an object to have multiple structures throughout its lifecycle. This paper introduces MTO-SS, a flow-sensitive pointer analysis method for objects with multiple structures. Our observation is that it's common for an object to possess multiple structures throughout its lifecycle. The novelty of MTO-SS lies in: MTO-SS introduces Structure-Flow-Sensitive. An object has different structure information at different locations in the program. To ensure the completeness of an object's structure information, MTO-SS always performs weak updates on the object's type. This means that once an object possesses a structure, this structure will accompany the object throughout its lifecycle. We evaluated our method of multi-structured object pointer analysis using the 12 largest programs in GNU Coreutils and compared the experimental results with sparse flow-sensitive method and another method, TYPECLONE, which only allows an object to have one structure information. Our experimental results confirm that MTO-SS is more precise than both sparse flow-sensitive pointer analysis and TYPECLONE, being able to answer, on average, over 22\% more alias queries with a no-alias result compared to the former, and over 3\% more compared to the latter. Additionally, the time overhead introduced by our method is very low.
Adjoint systems are widely used to inform control, optimization, and design in systems described by ordinary differential equations or differential-algebraic equations. In this paper, we explore the geometric properties and develop methods for such adjoint systems. In particular, we utilize symplectic and presymplectic geometry to investigate the properties of adjoint systems associated with ordinary differential equations and differential-algebraic equations, respectively. We show that the adjoint variational quadratic conservation laws, which are key to adjoint sensitivity analysis, arise from (pre)symplecticity of such adjoint systems. We discuss various additional geometric properties of adjoint systems, such as symmetries and variational characterizations. For adjoint systems associated with a differential-algebraic equation, we relate the index of the differential-algebraic equation to the presymplectic constraint algorithm of Gotay et al. [18]. As an application of this geometric framework, we discuss how the adjoint variational quadratic conservation laws can be used to compute sensitivities of terminal or running cost functions. Furthermore, we develop structure-preserving numerical methods for such systems using Galerkin Hamiltonian variational integrators (Leok and Zhang [23]) which admit discrete analogues of these quadratic conservation laws. We additionally show that such methods are natural, in the sense that reduction, forming the adjoint system, and discretization all commute, for suitable choices of these processes. We utilize this naturality to derive a variational error analysis result for the presymplectic variational integrator that we use to discretize the adjoint DAE system. Finally, we discuss the application of adjoint systems in the context of optimal control problems, where we prove a similar naturality result.
We are concerned with the arithmetic of solutions to ordinary or partial nonlinear differential equations which are algebraic in the indeterminates and their derivatives. We call these solutions D-algebraic functions, and their equations are algebraic (ordinary or partial) differential equations (ADEs). The general purpose is to find ADEs whose solutions contain specified rational expressions of solutions to given ADEs. For univariate D-algebraic functions, we show how to derive an ADE of smallest possible order. In the multivariate case, we introduce a general algorithm for these computations and derive conclusions on the order bound of the resulting algebraic PDE. Using our accompanying Maple software, we discuss applications in physics, statistics, and symbolic integration.
Data depth is an efficient tool for robustly summarizing the distribution of functional data and detecting potential magnitude and shape outliers. Commonly used functional data depth notions, such as the modified band depth and extremal depth, are estimated from pointwise depth for each observed functional observation. However, these techniques require calculating one single depth value for each functional observation, which may not be sufficient to characterize the distribution of the functional data and detect potential outliers. This paper presents an innovative approach to make the best use of pointwise depth. We propose using the pointwise depth distribution for magnitude outlier visualization and the correlation between pairwise depth for shape outlier detection. Furthermore, a bootstrap-based testing procedure has been introduced for the correlation to test whether there is any shape outlier. The proposed univariate methods are then extended to bivariate functional data. The performance of the proposed methods is examined and compared to conventional outlier detection techniques by intensive simulation studies. In addition, the developed methods are applied to simulated solar energy datasets from a photovoltaic system. Results revealed that the proposed method offers superior detection performance over conventional techniques. These findings will benefit engineers and practitioners in monitoring photovoltaic systems by detecting unnoticed anomalies and outliers.
The computing in the network (COIN) paradigm has emerged as a potential solution for computation-intensive applications like the metaverse by utilizing unused network resources. The blockchain (BC) guarantees task-offloading privacy, but cost reduction, queueing delays, and redundancy elimination remain open problems. This paper presents a redundancy-aware BC-based approach for the metaverse's partial computation offloading (PCO). Specifically, we formulate a joint BC redundancy factor (BRF) and PCO problem to minimize computation costs, maximize incentives, and meet delay and BC offloading constraints. We proved this problem is NP-hard and transformed it into two subproblems based on their temporal correlation: real-time PCO and Markov decision process-based BRF. We formulated the PCO problem as a multiuser game, proposed a decentralized algorithm for Nash equilibrium under any BC redundancy state, and designed a double deep Q-network-based algorithm for the optimal BRF policy. The BRF strategy is updated periodically based on user computation demand and network status to assist the PCO algorithm. The experimental results suggest that the proposed approach outperforms existing schemes, resulting in a remarkable 47% reduction in cost overhead, delivering approximately 64% higher rewards, and achieving convergence in just a few training episodes.
Classical analysis of convex and non-convex optimization methods often requires the Lipshitzness of the gradient, which limits the analysis to functions bounded by quadratics. Recent work relaxed this requirement to a non-uniform smoothness condition with the Hessian norm bounded by an affine function of the gradient norm, and proved convergence in the non-convex setting via gradient clipping, assuming bounded noise. In this paper, we further generalize this non-uniform smoothness condition and develop a simple, yet powerful analysis technique that bounds the gradients along the trajectory, thereby leading to stronger results for both convex and non-convex optimization problems. In particular, we obtain the classical convergence rates for (stochastic) gradient descent and Nesterov's accelerated gradient method in the convex and/or non-convex setting under this general smoothness condition. The new analysis approach does not require gradient clipping and allows heavy-tailed noise with bounded variance in the stochastic setting.
Due to their inherent capability in semantic alignment of aspects and their context words, attention mechanism and Convolutional Neural Networks (CNNs) are widely applied for aspect-based sentiment classification. However, these models lack a mechanism to account for relevant syntactical constraints and long-range word dependencies, and hence may mistakenly recognize syntactically irrelevant contextual words as clues for judging aspect sentiment. To tackle this problem, we propose to build a Graph Convolutional Network (GCN) over the dependency tree of a sentence to exploit syntactical information and word dependencies. Based on it, a novel aspect-specific sentiment classification framework is raised. Experiments on three benchmarking collections illustrate that our proposed model has comparable effectiveness to a range of state-of-the-art models, and further demonstrate that both syntactical information and long-range word dependencies are properly captured by the graph convolution structure.
Recently, ensemble has been applied to deep metric learning to yield state-of-the-art results. Deep metric learning aims to learn deep neural networks for feature embeddings, distances of which satisfy given constraint. In deep metric learning, ensemble takes average of distances learned by multiple learners. As one important aspect of ensemble, the learners should be diverse in their feature embeddings. To this end, we propose an attention-based ensemble, which uses multiple attention masks, so that each learner can attend to different parts of the object. We also propose a divergence loss, which encourages diversity among the learners. The proposed method is applied to the standard benchmarks of deep metric learning and experimental results show that it outperforms the state-of-the-art methods by a significant margin on image retrieval tasks.
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