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Register a new userWeak in the NEES?: Auto-tuning Kalman Filters with Bayesian Optimization
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Kalman filters are routinely used for many data fusion applications including navigation, tracking, and simultaneous localization and mapping problems. However, significant time and effort is frequently required to tune various Kalman filter model parameters, e.g. process noise covariance, pre-whitening filter models for non-white noise, etc. Conventional optimization techniques for tuning can get stuck in poor local minima and can be expensive to implement with real sensor data. To address these issues, a new black box Bayesian optimization strategy is developed for automatically tuning Kalman filters. In this approach, performance is characterized by one of two stochastic objective functions: normalized estimation error squared (NEES) when ground truth state models are available, or the normalized innovation error squared (NIS) when only sensor data is available. By intelligently sampling the parameter space to both learn and exploit a nonparametric Gaussian process surrogate function for the NEES/NIS costs, Bayesian optimization can efficiently identify multiple local minima and provide uncertainty quantification on its results.
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Data assimilation is concerned with sequentially estimating a temporally-evolving state. This task, which arises in a wide range of scientific and engineering applications, is particularly challenging when the state is high-dimensional and the state-space dynamics are unknown. This paper introduces a machine learning framework for learning dynamical systems in data assimilation. Our auto-differentiable ensemble Kalman filters (AD-EnKFs) blend ensemble Kalman filters for state recovery with machine learning tools for learning the dynamics. In doing so, AD-EnKFs leverage the ability of ensemble Kalman filters to scale to high-dimensional states and the power of automatic differentiation to train high-dimensional surrogate models for the dynamics. Numerical results using the Lorenz-96 model show that AD-EnKFs outperform existing methods that use expectation-maximization or particle filters to merge data assimilation and machine learning. In addition, AD-EnKFs are easy to implement and require minimal tuning.
Many state estimation algorithms must be tuned given the state space process and observation models, the process and observation noise parameters must be chosen. Conventional tuning approaches rely on heuristic hand-tuning or gradient-based optimization techniques to minimize a performance cost function. However, the relationship between tuned noise values and estimator performance is highly nonlinear and stochastic. Therefore, the tuning solutions can easily get trapped in local minima, which can lead to poor choices of noise parameters and suboptimal estimator performance. This paper describes how Bayesian Optimization (BO) can overcome these issues. BO poses optimization as a Bayesian search problem for a stochastic ``black box cost function, where the goal is to search the solution space to maximize the probability of improving the current best solution. As such, BO offers a principled approach to optimization-based estimator tuning in the presence of local minima and performance stochasticity. While extended Kalman filters (EKFs) are the main focus of this work, BO can be similarly used to tune other related state space filters. The method presented here uses performance metrics derived from normalized innovation squared (NIS) filter residuals obtained via sensor data, which renders knowledge of ground-truth states unnecessary. The robustness, accuracy, and reliability of BO-based tuning is illustrated on practical nonlinear state estimation problems,losed-loop aero-robotic control.
Kalman Filters are one of the most influential models of time-varying phenomena. They admit an intuitive probabilistic interpretation, have a simple functional form, and enjoy widespread adoption in a variety of disciplines. Motivated by recent variational methods for learning deep generative models, we introduce a unified algorithm to efficiently learn a broad spectrum of Kalman filters. Of particular interest is the use of temporal generative models for counterfactual inference. We investigate the efficacy of such models for counterfactual inference, and to that end we introduce the Healing MNIST dataset where long-term structure, noise and actions are applied to sequences of digits. We show the efficacy of our method for modeling this dataset. We further show how our model can be used for counterfactual inference for patients, based on electronic health record data of 8,000 patients over 4.5 years.
This paper presents novel mixed-type Bayesian optimization (BO) algorithms to accelerate the optimization of a target objective function by exploiting correlated auxiliary information of binary type that can be more cheaply obtained, such as in policy search for reinforcement learning and hyperparameter tuning of machine learning models with early stopping. To achieve this, we first propose a mixed-type multi-output Gaussian process (MOGP) to jointly model the continuous target function and binary auxiliary functions. Then, we propose information-based acquisition functions such as mixed-type entropy search (MT-ES) and mixed-type predictive ES (MT-PES) for mixed-type BO based on the MOGP predictive belief of the target and auxiliary functions. The exact acquisition functions of MT-ES and MT-PES cannot be computed in closed form and need to be approximated. We derive an efficient approximation of MT-PES via a novel mixed-type random features approximation of the MOGP model whose cross-correlation structure between the target and auxiliary functions can be exploited for improving the belief of the global target maximizer using observations from evaluating these functions. We propose new practical constraints to relate the global target maximizer to the binary auxiliary functions. We empirically evaluate the performance of MT-ES and MT-PES with synthetic and real-world experiments.
We propose a practical Bayesian optimization method over sets, to minimize a black-box function that takes a set as a single input. Because set inputs are permutation-invariant, traditional Gaussian process-based Bayesian optimization strategies which assume vector inputs can fall short. To address this, we develop a Bayesian optimization method with emph{set kernel} that is used to build surrogate functions. This kernel accumulates similarity over set elements to enforce permutation-invariance, but this comes at a greater computational cost. To reduce this burden, we propose two key components: (i) a more efficient approximate set kernel which is still positive-definite and is an unbiased estimator of the true set kernel with upper-bounded variance in terms of the number of subsamples, (ii) a constrained acquisition function optimization over sets, which uses symmetry of the feasible region that defines a set input. Finally, we present several numerical experiments which demonstrate that our method outperforms other methods.
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