A learning-based safety filter is developed for discrete-time linear time-invariant systems with unknown models subject to Gaussian noises with unknown covariance. Safety is characterized using polytopic constraints on the states and control inputs. The empirically learned model and process noise covariance with their confidence bounds are used to construct a robust optimization problem for minimally modifying nominal control actions to ensure safety with high probability. The optimization problem relies on tightening the original safety constraints. The magnitude of the tightening is larger at the beginning since there is little information to construct reliable models, but shrinks with time as more data becomes available.
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Deep Metric Learning (DML) models rely on strong representations and similarity-based measures with specific loss functions. Proxy-based losses have shown great performance compared to pair-based losses in terms of convergence speed. However, proxies that are assigned to different classes may end up being closely located in the embedding space and hence having a hard time to distinguish between positive and negative items. Alternatively, they may become highly correlated and hence provide redundant information with the model. To address these issues, we propose a novel approach that introduces Soft Orthogonality (SO) constraint on proxies. The constraint ensures the proxies to be as orthogonal as possible and hence control their positions in the embedding space. Our approach leverages Data-Efficient Image Transformer (DeiT) as an encoder to extract contextual features from images along with a DML objective. The objective is made of the Proxy Anchor loss along with the SO regularization. We evaluate our method on four public benchmarks for category-level image retrieval and demonstrate its effectiveness with comprehensive experimental results and ablation studies. Our evaluations demonstrate the superiority of our proposed approach over state-of-the-art methods by a significant margin.
In many modern statistical problems, the limited available data must be used both to develop the hypotheses to test, and to test these hypotheses-that is, both for exploratory and confirmatory data analysis. Reusing the same dataset for both exploration and testing can lead to massive selection bias, leading to many false discoveries. Selective inference is a framework that allows for performing valid inference even when the same data is reused for exploration and testing. In this work, we are interested in the problem of selective inference for data clustering, where a clustering procedure is used to hypothesize a separation of the data points into a collection of subgroups, and we then wish to test whether these data-dependent clusters in fact represent meaningful differences within the data. Recent work by Gao et al. [2022] provides a framework for doing selective inference for this setting, where a hierarchical clustering algorithm is used for producing the cluster assignments, which was then extended to k-means clustering by Chen and Witten [2022]. Both these works rely on assuming a known covariance structure for the data, but in practice, the noise level needs to be estimated-and this is particularly challenging when the true cluster structure is unknown. In our work, we extend this work to the setting of noise with unknown variance, and provide a selective inference method for this more general setting. Empirical results show that our new method is better able to maintain high power while controlling Type I error when the true noise level is unknown.
Reinforcement learning algorithms commonly seek to optimize policies for solving one particular task. How should we explore an unknown dynamical system such that the estimated model allows us to solve multiple downstream tasks in a zero-shot manner? In this paper, we address this challenge, by developing an algorithm -- OPAX -- for active exploration. OPAX uses well-calibrated probabilistic models to quantify the epistemic uncertainty about the unknown dynamics. It optimistically -- w.r.t. to plausible dynamics -- maximizes the information gain between the unknown dynamics and state observations. We show how the resulting optimization problem can be reduced to an optimal control problem that can be solved at each episode using standard approaches. We analyze our algorithm for general models, and, in the case of Gaussian process dynamics, we give a sample complexity bound and show that the epistemic uncertainty converges to zero. In our experiments, we compare OPAX with other heuristic active exploration approaches on several environments. Our experiments show that OPAX is not only theoretically sound but also performs well for zero-shot planning on novel downstream tasks.
Most works on joint state and unknown input (UI) estimation require the assumption that the UIs are linear; this is potentially restrictive as it does not hold in many intelligent autonomous systems. To overcome this restriction and circumvent the need to linearize the system, we propose a derivative-free Unknown Input Sigma-point Kalman Filter (SPKF-nUI) where the SPKF is interconnected with a general nonlinear UI estimator that can be implemented via nonlinear optimization and data-driven approaches. The nonlinear UI estimator uses the posterior state estimate which is less susceptible to state prediction error. In addition, we introduce a joint sigma-point transformation scheme to incorporate both the state and UI uncertainties in the estimation of SPKF-nUI. An in-depth stochastic stability analysis proves that the proposed SPKF-nUI yields exponentially converging estimation error bounds under reasonable assumptions. Finally, two case studies are carried out on a simulation-based rigid robot and a physical soft robot, i.e., robots made of soft materials with complex dynamics to validate effectiveness of the proposed filter on nonlinear dynamic systems. Our results demonstrate that the proposed SPKF-nUI achieves the lowest state and UI estimation errors when compared to the existing nonlinear state-UI filters.
State-space models (SSMs) are a powerful statistical tool for modelling time-varying systems via a latent state. In these models, the latent state is never directly observed. Instead, a sequence of data points related to the state are obtained. The linear-Gaussian state-space model is widely used, since it allows for exact inference when all model parameters are known, however this is rarely the case. The estimation of these parameters is a very challenging but essential task to perform inference and prediction. In the linear-Gaussian model, the state dynamics are described via a state transition matrix. This model parameter is known to behard to estimate, since it encodes the relationships between the state elements, which are never observed. In many applications, this transition matrix is sparse since not all state components directly affect all other state components. However, most parameter estimation methods do not exploit this feature. In this work we propose SpaRJ, a fully probabilistic Bayesian approach that obtains sparse samples from the posterior distribution of the transition matrix. Our method explores sparsity by traversing a set of models that exhibit differing sparsity patterns in the transition matrix. Moreover, we also design new effective rules to explore transition matrices within the same level of sparsity. This novel methodology has strong theoretical guarantees, and unveils the latent structure of the data generating process, thereby enhancing interpretability. The performance of SpaRJ is showcased in example with dimension 144 in the parameter space, and in a numerical example with real data.
We present the formulation and optimization of a Runge-Kutta-type time-stepping scheme for solving the shallow water equations, aimed at substantially increasing the effective allowable time-step over that of comparable methods. This scheme, called FB-RK(3,2), uses weighted forward-backward averaging of thickness data to advance the momentum equation. The weights for this averaging are chosen with an optimization process that employs a von Neumann-type analysis, ensuring that the weights maximize the admittable Courant number. Through a simplified local truncation error analysis and numerical experiments, we show that the method is at least second order in time for any choice of weights and exhibits low dispersion and dissipation errors for well-resolved waves. Further, we show that an optimized FB-RK(3,2) can take time-steps up to 2.8 times as large as a popular three-stage, third-order strong stability preserving Runge-Kutta method in a quasi-linear test case. In fully nonlinear shallow water test cases relevant to oceanic and atmospheric flows, FB-RK(3,2) outperforms SSPRK3 in admittable time-step by factors between roughly between 1.6 and 2.2, making the scheme approximately twice as computationally efficient with little to no effect on solution quality.
Recommender systems have been widely applied in different real-life scenarios to help us find useful information. Recently, Reinforcement Learning (RL) based recommender systems have become an emerging research topic. It often surpasses traditional recommendation models even most deep learning-based methods, owing to its interactive nature and autonomous learning ability. Nevertheless, there are various challenges of RL when applying in recommender systems. Toward this end, we firstly provide a thorough overview, comparisons, and summarization of RL approaches for five typical recommendation scenarios, following three main categories of RL: value-function, policy search, and Actor-Critic. Then, we systematically analyze the challenges and relevant solutions on the basis of existing literature. Finally, under discussion for open issues of RL and its limitations of recommendation, we highlight some potential research directions in this field.
Deep learning is usually described as an experiment-driven field under continuous criticizes of lacking theoretical foundations. This problem has been partially fixed by a large volume of literature which has so far not been well organized. This paper reviews and organizes the recent advances in deep learning theory. The literature is categorized in six groups: (1) complexity and capacity-based approaches for analyzing the generalizability of deep learning; (2) stochastic differential equations and their dynamic systems for modelling stochastic gradient descent and its variants, which characterize the optimization and generalization of deep learning, partially inspired by Bayesian inference; (3) the geometrical structures of the loss landscape that drives the trajectories of the dynamic systems; (4) the roles of over-parameterization of deep neural networks from both positive and negative perspectives; (5) theoretical foundations of several special structures in network architectures; and (6) the increasingly intensive concerns in ethics and security and their relationships with generalizability.
A comprehensive artificial intelligence system needs to not only perceive the environment with different `senses' (e.g., seeing and hearing) but also infer the world's conditional (or even causal) relations and corresponding uncertainty. The past decade has seen major advances in many perception tasks such as visual object recognition and speech recognition using deep learning models. For higher-level inference, however, probabilistic graphical models with their Bayesian nature are still more powerful and flexible. In recent years, Bayesian deep learning has emerged as a unified probabilistic framework to tightly integrate deep learning and Bayesian models. In this general framework, the perception of text or images using deep learning can boost the performance of higher-level inference and in turn, the feedback from the inference process is able to enhance the perception of text or images. This survey provides a comprehensive introduction to Bayesian deep learning and reviews its recent applications on recommender systems, topic models, control, etc. Besides, we also discuss the relationship and differences between Bayesian deep learning and other related topics such as Bayesian treatment of neural networks.
This paper addresses the difficulty of forecasting multiple financial time series (TS) conjointly using deep neural networks (DNN). We investigate whether DNN-based models could forecast these TS more efficiently by learning their representation directly. To this end, we make use of the dynamic factor graph (DFG) from that we enhance by proposing a novel variable-length attention-based mechanism to render it memory-augmented. Using this mechanism, we propose an unsupervised DNN architecture for multivariate TS forecasting that allows to learn and take advantage of the relationships between these TS. We test our model on two datasets covering 19 years of investment funds activities. Our experimental results show that our proposed approach outperforms significantly typical DNN-based and statistical models at forecasting their 21-day price trajectory.
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