We propose a novel alternative approach to our previous work (Ben Hammouda et al., 2023) to improve the efficiency of Monte Carlo (MC) estimators for rare event probabilities for stochastic reaction networks (SRNs). In the same spirit of (Ben Hammouda et al., 2023), an efficient path-dependent measure change is derived based on a connection between determining optimal importance sampling (IS) parameters within a class of probability measures and a stochastic optimal control formulation, corresponding to solving a variance minimization problem. In this work, we propose a novel approach to address the encountered curse of dimensionality by mapping the problem to a significantly lower-dimensional space via a Markovian projection (MP) idea. The output of this model reduction technique is a low-dimensional SRN (potentially even one dimensional) that preserves the marginal distribution of the original high-dimensional SRN system. The dynamics of the projected process are obtained by solving a related optimization problem via a discrete $L^2$ regression. By solving the resulting projected Hamilton-Jacobi-Bellman (HJB) equations for the reduced-dimensional SRN, we obtain projected IS parameters, which are then mapped back to the original full-dimensional SRN system, resulting in an efficient IS-MC estimator for rare events probabilities of the full-dimensional SRN. Our analysis and numerical experiments reveal that the proposed MP-HJB-IS approach substantially reduces the MC estimator variance, resulting in a lower computational complexity in the rare event regime than standard MC estimators.
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We consider the problem of computing the (two-sided) Hausdorff distance between the unit $\ell_{p_{1}}$ and $\ell_{p_{2}}$ norm balls in finite dimensional Euclidean space for $1 \leq p_1 < p_2 \leq \infty$, and derive a closed-form formula for the same. We also derive a closed-form formula for the Hausdorff distance between the $k_1$ and $k_2$ unit $D$-norm balls, which are certain polyhedral norm balls in $d$ dimensions for $1 \leq k_1 < k_2 \leq d$. When two different $\ell_p$ norm balls are transformed via a common linear map, we obtain several estimates for the Hausdorff distance between the resulting convex sets. These estimates upper bound the Hausdorff distance or its expectation, depending on whether the linear map is arbitrary or random. We then generalize the developments for the Hausdorff distance between two set-valued integrals obtained by applying a parametric family of linear maps to different $\ell_p$ unit norm balls, and then taking the Minkowski sums of the resulting sets in a limiting sense. To illustrate an application, we show that the problem of computing the Hausdorff distance between the reach sets of a linear dynamical system with different unit norm ball-valued input uncertainties, reduces to this set-valued integral setting.
Neural point estimators are neural networks that map data to parameter point estimates. They are fast, likelihood free and, due to their amortised nature, amenable to fast bootstrap-based uncertainty quantification. In this paper, we aim to increase the awareness of statisticians to this relatively new inferential tool, and to facilitate its adoption by providing user-friendly open-source software. We also give attention to the ubiquitous problem of making inference from replicated data, which we address in the neural setting using permutation-invariant neural networks. Through extensive simulation studies we show that these neural point estimators can quickly and optimally (in a Bayes sense) estimate parameters in weakly-identified and highly-parameterised models with relative ease. We demonstrate their applicability through an analysis of extreme sea-surface temperature in the Red Sea where, after training, we obtain parameter estimates and bootstrap-based confidence intervals from hundreds of spatial fields in a fraction of a second.
Recent works show that the data distribution in a network's latent space is useful for estimating classification uncertainty and detecting Out-of-distribution (OOD) samples. To obtain a well-regularized latent space that is conducive for uncertainty estimation, existing methods bring in significant changes to model architectures and training procedures. In this paper, we present a lightweight, fast, and high-performance regularization method for Mahalanobis distance-based uncertainty prediction, and that requires minimal changes to the network's architecture. To derive Gaussian latent representation favourable for Mahalanobis Distance calculation, we introduce a self-supervised representation learning method that separates in-class representations into multiple Gaussians. Classes with non-Gaussian representations are automatically identified and dynamically clustered into multiple new classes that are approximately Gaussian. Evaluation on standard OOD benchmarks shows that our method achieves state-of-the-art results on OOD detection with minimal inference time, and is very competitive on predictive probability calibration. Finally, we show the applicability of our method to a real-life computer vision use case on microorganism classification.
While reinforcement learning algorithms have had great success in the field of autonomous navigation, they cannot be straightforwardly applied to the real autonomous systems without considering the safety constraints. The later are crucial to avoid unsafe behaviors of the autonomous vehicle on the road. To highlight the importance of these constraints, in this study, we compare two learnable navigation policies: safe and unsafe. The safe policy takes the constraints into account, while the other does not. We show that the safe policy is able to generate trajectories with more clearance (distance to the obstacles) and makes less collisions while training without sacrificing the overall performance.
Modelling in biology must adapt to increasingly complex and massive data. The efficiency of the inference algorithms used to estimate model parameters is therefore questioned. Many of these are based on stochastic optimization processes which waste a significant part of the computation time due to their rejection sampling approaches. We introduce the Fixed Landscape Inference MethOd (flimo), a new likelihood-free inference method for continuous state-space stochastic models. It applies deterministic gradient-based optimization algorithms to obtain a point estimate of the parameters, minimizing the difference between the data and some simulations according to some prescribed summary statistics. In this sense, it is analogous to Approximate Bayesian Computation (ABC). Like ABC, it can also provide an approximation of the distribution of the parameters. Three applications are proposed: a usual theoretical example, namely the inference of the parameters of g-and-k distributions; a population genetics problem, not so simple as it seems, namely the inference of a selective value from time series in a Wright-Fisher model; and simulations from a Ricker model, representing chaotic population dynamics. In the two first applications, the results show a drastic reduction of the computational time needed for the inference phase compared to the other methods, despite an equivalent accuracy. Even when likelihood-based methods are applicable, the simplicity and efficiency of flimo make it a compelling alternative. Implementations in Julia and in R are available on //metabarcoding.org/flimo. To run flimo, the user must simply be able to simulate data according to the chosen model.
The growing demand for accurate control in varying and unknown environments has sparked a corresponding increase in the requirements for power supply components, including permanent magnet synchronous motors (PMSMs). To infer the unknown part of the system, machine learning techniques are widely employed, especially Gaussian process regression (GPR) due to its flexibility of continuous system modeling and its guaranteed performance. For practical implementation, distributed GPR is adopted to alleviate the high computational complexity. However, the study of distributed GPR from a control perspective remains an open problem. In this paper, a control-aware optimal aggregation strategy of distributed GPR for PMSMs is proposed based on the Lyapunov stability theory. This strategy exclusively leverages the posterior mean, thereby obviating the need for computationally intensive calculations associated with posterior variance in alternative approaches. Moreover, the straightforward calculation process of our proposed strategy lends itself to seamless implementation in high-frequency PMSM control. The effectiveness of the proposed strategy is demonstrated in the simulations.
Stochastic-gradient sampling methods are often used to perform Bayesian inference on neural networks. It has been observed that the methods in which notions of differential geometry are included tend to have better performances, with the Riemannian metric improving posterior exploration by accounting for the local curvature. However, the existing methods often resort to simple diagonal metrics to remain computationally efficient. This loses some of the gains. We propose two non-diagonal metrics that can be used in stochastic-gradient samplers to improve convergence and exploration but have only a minor computational overhead over diagonal metrics. We show that for fully connected neural networks (NNs) with sparsity-inducing priors and convolutional NNs with correlated priors, using these metrics can provide improvements. For some other choices the posterior is sufficiently easy also for the simpler metrics.
Representation learning plays a crucial role in automated feature selection, particularly in the context of high-dimensional data, where non-parametric methods often struggle. In this study, we focus on supervised learning scenarios where the pertinent information resides within a lower-dimensional linear subspace of the data, namely the multi-index model. If this subspace were known, it would greatly enhance prediction, computation, and interpretation. To address this challenge, we propose a novel method for linear feature learning with non-parametric prediction, which simultaneously estimates the prediction function and the linear subspace. Our approach employs empirical risk minimisation, augmented with a penalty on function derivatives, ensuring versatility. Leveraging the orthogonality and rotation invariance properties of Hermite polynomials, we introduce our estimator, named RegFeaL. By utilising alternative minimisation, we iteratively rotate the data to improve alignment with leading directions and accurately estimate the relevant dimension in practical settings. We establish that our method yields a consistent estimator of the prediction function with explicit rates. Additionally, we provide empirical results demonstrating the performance of RegFeaL in various experiments.
We introduce a multifidelity estimator of covariance matrices formulated as the solution to a regression problem on the manifold of symmetric positive definite matrices. The estimator is positive definite by construction, and the Mahalanobis distance minimized to obtain it possesses properties which enable practical computation. We show that our manifold regression multifidelity (MRMF) covariance estimator is a maximum likelihood estimator under a certain error model on manifold tangent space. More broadly, we show that our Riemannian regression framework encompasses existing multifidelity covariance estimators constructed from control variates. We demonstrate via numerical examples that our estimator can provide significant decreases, up to one order of magnitude, in squared estimation error relative to both single-fidelity and other multifidelity covariance estimators. Furthermore, preservation of positive definiteness ensures that our estimator is compatible with downstream tasks, such as data assimilation and metric learning, in which this property is essential.
While many real-world problems that might benefit from reinforcement learning, these problems rarely fit into the MDP mold: interacting with the environment is often expensive and specifying reward functions is challenging. Motivated by these challenges, prior work has developed data-driven approaches that learn entirely from samples from the transition dynamics and examples of high-return states. These methods typically learn a reward function from high-return states, use that reward function to label the transitions, and then apply an offline RL algorithm to these transitions. While these methods can achieve good results on many tasks, they can be complex, often requiring regularization and temporal difference updates. In this paper, we propose a method for offline, example-based control that learns an implicit model of multi-step transitions, rather than a reward function. We show that this implicit model can represent the Q-values for the example-based control problem. Across a range of state-based and image-based offline control tasks, our method outperforms baselines that use learned reward functions; additional experiments demonstrate improved robustness and scaling with dataset size.
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