In this paper we derive a PAC-Bayesian-Like error bound for a class of stochastic dynamical systems with inputs, namely, for linear time-invariant stochastic state-space models (stochastic LTI systems for short). This class of systems is widely used in control engineering and econometrics, in particular, they represent a special case of recurrent neural networks. In this paper we 1) formalize the learning problem for stochastic LTI systems with inputs, 2) derive a PAC-Bayesian-Like error bound for such systems, 3) discuss various consequences of this error bound.

Accurate uncertainty quantification is a major challenge in deep learning, as neural networks can make overconfident errors and assign high confidence predictions to out-of-distribution (OOD) inputs. The most popular approaches to estimate predictive uncertainty in deep learning are methods that combine predictions from multiple neural networks, such as Bayesian neural networks (BNNs) and deep ensembles. However their practicality in real-time, industrial-scale applications are limited due to the high memory and computational cost. Furthermore, ensembles and BNNs do not necessarily fix all the issues with the underlying member networks. In this work, we study principled approaches to improve the uncertainty property of a single network, based on a single, deterministic representation. By formalizing the uncertainty quantification as a minimax learning problem, we first identify distance awareness, i.e., the model's ability to quantify the distance of a testing example from the training data, as a necessary condition for a DNN to achieve highquality (i.e., minimax optimal) uncertainty estimation. We then propose Spectral-normalized Neural Gaussian Process (SNGP), a simple method that improves the distance-awareness ability of modern DNNs with two simple changes: (1) applying spectral normalization to hidden weights to enforce bi-Lipschitz smoothness in representations and (2) replacing the last output layer with a Gaussian process layer. On a suite of vision and language understanding benchmarks and on modern architectures (Wide-ResNet and BERT), SNGP consistently outperforms other single-model approaches in prediction, calibration and out-of-domain detection. Furthermore, SNGP provides complementary benefits to popular techniques such as deep ensembles and data augmentation, making it a simple and scalable building block for probabilistic deep learning.

You tagged this question with the tag "Maximum Likelihood". In maximum likelihood estimation you explicitly maximize an objective function (namely the likelihood). It just so happens that for an observation that we assume to be drawn from a Gaussian random variable, the likelihood function usually takes a nice form after you take a logarithm. Then there is usually a leading negation, encouraging the entrepreneurial optimizer to switch away from maximizing the objective to minimizing the negative of objective, or roughly the "cost". For discrete maximum likelihood estimation the "cost" also has another meaningful name since it takes the same form as the euclidean distance in the observation space.

We provide a unifying approximate dynamic programming framework that applies to a broad variety of problems involving sequential estimation. We consider first the construction of surrogate cost functions for the purposes of optimization, and we focus on the special case of Bayesian optimization, using the rollout algorithm and some of its variations. We then discuss the more general case of sequential estimation of a random vector using optimal measurement selection, and its application to problems of stochastic and adaptive control. We distinguish between adaptive control of deterministic and stochastic systems: the former are better suited for the use of rollout, while the latter are well suited for the use of rollout with certainty equivalence approximations. As an example of the deterministic case, we discuss sequential decoding problems, and a rollout algorithm for the approximate solution of the Wordle and Mastermind puzzles, recently developed in the paper [BBB22].

Active tracking of space noncooperative object that merely relies on vision camera is greatly significant for autonomous rendezvous and debris removal. Considering its Partial Observable Markov Decision Process (POMDP) property, this paper proposes a novel tracker based on deep recurrent reinforcement learning, named as RAMAVT which drives the chasing spacecraft to follow arbitrary space noncooperative object with high-frequency and near-optimal velocity control commands. To further improve the active tracking performance, we introduce Multi-Head Attention (MHA) module and Squeeze-and-Excitation (SE) layer into RAMAVT, which remarkably improve the representative ability of neural network with almost no extra computational cost. Extensive experiments and ablation study implemented on SNCOAT benchmark show the effectiveness and robustness of our method compared with other state-of-the-art algorithm. The source codes are available on https://github.com/Dongzhou-1996/RAMAVT.

We study the problem of estimating latent population flows from aggregated count data. This problem arises when individual trajectories are not available due to privacy issues or measurement fidelity. Instead, the aggregated observations are measured over discrete-time points, for estimating the population flows among states. Most related studies tackle the problems by learning the transition parameters of a time-homogeneous Markov process. Nonetheless, most real-world population flows can be influenced by various uncertainties such as traffic jam and weather conditions. Thus, in many cases, a time-homogeneous Markov model is a poor approximation of the much more complex population flows. To circumvent this difficulty, we resort to a multi-marginal optimal transport (MOT) formulation that can naturally represent aggregated observations with constrained marginals, and encode time-dependent transition matrices by the cost functions. In particular, we propose to estimate the transition flows from aggregated data by learning the cost functions of the MOT framework, which enables us to capture time-varying dynamic patterns. The experiments demonstrate the improved accuracy of the proposed algorithms than the related methods in estimating several real-world transition flows.

This paper addresses the kinodynamic motion planning for non-holonomic robots in dynamic environments with both static and dynamic obstacles -- a challenging problem that lacks a universal solution yet. One of the promising approaches to solve it is decomposing the problem into the smaller sub problems and combining the local solutions into the global one. The crux of any planning method for non-holonomic robots is the generation of motion primitives that generates solutions to local planning sub-problems. In this work we introduce a novel learnable steering function (policy), which takes into account kinodynamic constraints of the robot and both static and dynamic obstacles. This policy is efficiently trained via the policy optimization. Empirically, we show that our steering function generalizes well to unseen problems. We then plug in the trained policy into the sampling-based and lattice-based planners, and evaluate the resultant POLAMP algorithm (Policy Optimization that Learns Adaptive Motion Primitives) in a range of challenging setups that involve a car-like robot operating in the obstacle-rich parking-lot environments. We show that POLAMP is able to plan collision-free kinodynamic trajectories with success rates higher than 92%, when 50 simultaneously moving obstacles populate the environment showing better performance than the state-of-the-art competitors.

Model estimates obtained from traditional subspace identification methods may be subject to significant variance. This elevated variance is aggravated in the cases of large models or of a limited sample size. Common solutions to reduce the effect of variance are regularized estimators, shrinkage estimators and Bayesian estimation. In the current work we investigate the latter two solutions, which have not yet been applied to subspace identification. Our experimental results show that our proposed estimators may reduce the estimation risk up to $40\%$ of that of traditional subspace methods.

Rapid advancements in collection, processing, and dissemination of multi-platform molecular and genomics (multi-omics, in short) data has resulted in enormous opportunities to aggregate such data in order to understand, prevent, and treat diseases. This has catalyzed development of integrative methods that can collectively mine multiple types and scales of multi-omics data, in order to provide a more holistic view of human disease evolution and progression (Subramanian et al. 2020). Specifically, in the context of cancer, a disease driven predominantly by agglomerations of several molecular changes (Sun et al. 2021), the importance of synthesizing information from multi-platform omics and clinical sources to understand the cellular basis of the disease is even further underscored. Cellular oncological mechanisms, triggered at different molecular levels of the DNA RNA Protein path, can confer profound phenotypic advantages/disadvantages. While significant improvements have been made in multi-omics data integration methods to unveil such mechanisms, focused on both prognosis (Duan et al. 2021) and treatment (Finotello et al. 2020), the precise functions governing them need detailed and data-driven de-novo evaluations. Our work, in the same vein, aims at two different but inter-related scientific axes: (i) selection of biomarkers associated with cancer prognosis and clinical outcomes, and (ii) learning the mechanism of these biomarkers' effects upon such outcomes via integrating upstream molecular information - we provide some additional scientific context below. Classes of Integrative Omics Models First, we briefly discuss existing integrative omics approaches in order to contextualize the need for our framework. Broadly, most of the existing integrative statistical methods can be classified into two categories - horizontal (meta-analysis type) and vertical (multi-omics) integration procedures (Tseng et al. 2015).

Kernels are efficient in representing nonlocal dependence and they are widely used to design operators between function spaces. Thus, learning kernels in operators from data is an inverse problem of general interest. Due to the nonlocal dependence, the inverse problem can be severely ill-posed with a data-dependent singular inversion operator. The Bayesian approach overcomes the ill-posedness through a non-degenerate prior. However, a fixed non-degenerate prior leads to a divergent posterior mean when the observation noise becomes small, if the data induces a perturbation in the eigenspace of zero eigenvalues of the inversion operator. We introduce a data-adaptive prior to achieve a stable posterior whose mean always has a small noise limit. The data-adaptive prior's covariance is the inversion operator with a hyper-parameter selected adaptive to data by the L-curve method. Furthermore, we provide a detailed analysis on the computational practice of the data-adaptive prior, and demonstrate it on Toeplitz matrices and integral operators. Numerical tests show that a fixed prior can lead to a divergent posterior mean in the presence of any of the four types of errors: discretization error, model error, partial observation and wrong noise assumption. In contrast, the data-adaptive prior always attains posterior means with small noise limits.