Many machine learning approaches for decision making, such as reinforcement learning, rely on simulators or predictive models to forecast the time-evolution of quantities of interest, e.g., the state of an agent or the reward of a policy. Forecasts of such complex phenomena are commonly described by highly nonlinear dynamical systems, making their use in optimization-based decision-making challenging. Koopman operator theory offers a beneficial paradigm for addressing this problem by characterizing forecasts via linear dynamical systems. This makes system analysis and long-term predictions simple -- involving only matrix multiplications. However, the transformation to a linear system is generally non-trivial and unknown, requiring learning-based approaches. While there exists a variety of approaches, they usually lack crucial learning-theoretic guarantees, such that the behavior of the obtained models with increasing data and dimensionality is often unclear. We address the aforementioned by deriving a novel reproducing kernel Hilbert space (RKHS) that solely spans transformations into linear dynamical systems. The resulting Koopman Kernel Regression (KKR) framework enables the use of statistical learning tools from function approximation for novel convergence results and generalization risk bounds under weaker assumptions than existing work. Our numerical experiments indicate advantages over state-of-the-art statistical learning approaches for Koopman-based predictors.
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Bayesian model comparison (BMC) offers a principled approach for assessing the relative merits of competing computational models and propagating uncertainty into model selection decisions. However, BMC is often intractable for the popular class of hierarchical models due to their high-dimensional nested parameter structure. To address this intractability, we propose a deep learning method for performing BMC on any set of hierarchical models which can be instantiated as probabilistic programs. Since our method enables amortized inference, it allows efficient re-estimation of posterior model probabilities and fast performance validation prior to any real-data application. In a series of extensive validation studies, we benchmark the performance of our method against the state-of-the-art bridge sampling method and demonstrate excellent amortized inference across all BMC settings. We then showcase our method by comparing four hierarchical evidence accumulation models that have previously been deemed intractable for BMC due to partly implicit likelihoods. In this application, we corroborate evidence for the recently proposed L\'evy flight model of decision-making and show how transfer learning can be leveraged to enhance training efficiency. We provide reproducible code for all analyses and an open-source implementation of our method.
Analysis of high-dimensional data, where the number of covariates is larger than the sample size, is a topic of current interest. In such settings, an important goal is to estimate the signal level $\tau^2$ and noise level $\sigma^2$, i.e., to quantify how much variation in the response variable can be explained by the covariates, versus how much of the variation is left unexplained. This thesis considers the estimation of these quantities in a semi-supervised setting, where for many observations only the vector of covariates $X$ is given with no responses $Y$. Our main research question is: how can one use the unlabeled data to better estimate $\tau^2$ and $\sigma^2$? We consider two frameworks: a linear regression model and a linear projection model in which linearity is not assumed. In the first framework, while linear regression is used, no sparsity assumptions on the coefficients are made. In the second framework, the linearity assumption is also relaxed and we aim to estimate the signal and noise levels defined by the linear projection. We first propose a naive estimator which is unbiased and consistent, under some assumptions, in both frameworks. We then show how the naive estimator can be improved by using zero-estimators, where a zero-estimator is a statistic arising from the unlabeled data, whose expected value is zero. In the first framework, we calculate the optimal zero-estimator improvement and discuss ways to approximate the optimal improvement. In the second framework, such optimality does no longer hold and we suggest two zero-estimators that improve the naive estimator although not necessarily optimally. Furthermore, we show that our approach reduces the variance for general initial estimators and we present an algorithm that potentially improves any initial estimator. Lastly, we consider four datasets and study the performance of our suggested methods.
Standard multiparameter eigenvalue problems (MEPs) are systems of $k\ge 2$ linear $k$-parameter square matrix pencils. Recently, a new form of multiparameter eigenvalue problems has emerged: a rectangular MEP (RMEP) with only one multivariate rectangular matrix pencil, where we are looking for combinations of the parameters for which the rank of the pencil is not full. Applications include finding the optimal least squares autoregressive moving average (ARMA) model and the optimal least squares realization of autonomous linear time-invariant (LTI) dynamical system. For linear and polynomial RMEPs, we give the number of solutions and show how these problems can be solved numerically by a transformation into a standard MEP. For the transformation we provide new linearizations for quadratic multivariate matrix polynomials with a specific structure of monomials and consider mixed systems of rectangular and square multivariate matrix polynomials. This numerical approach seems computationally considerably more attractive than the block Macaulay method, the only other currently available numerical method for polynomial RMEPs.
The accurate modeling of dynamics in interactive environments is critical for successful long-range prediction. Such a capability could advance Reinforcement Learning (RL) and Planning algorithms, but achieving it is challenging. Inaccuracies in model estimates can compound, resulting in increased errors over long horizons. We approach this problem from the lens of Koopman theory, where the nonlinear dynamics of the environment can be linearized in a high-dimensional latent space. This allows us to efficiently parallelize the sequential problem of long-range prediction using convolution, while accounting for the agent's action at every time step. Our approach also enables stability analysis and better control over gradients through time. Taken together, these advantages result in significant improvement over the existing approaches, both in the efficiency and the accuracy of modeling dynamics over extended horizons. We also report promising experimental results in dynamics modeling for the scenarios of both model-based planning and model-free RL.
Deep reinforcement learning (RL) has shown immense potential for learning to control systems through data alone. However, one challenge deep RL faces is that the full state of the system is often not observable. When this is the case, the policy needs to leverage the history of observations to infer the current state. At the same time, differences between the training and testing environments makes it critical for the policy not to overfit to the sequence of observations it sees at training time. As such, there is an important balancing act between having the history encoder be flexible enough to extract relevant information, yet be robust to changes in the environment. To strike this balance, we look to the PID controller for inspiration. We assert the PID controller's success shows that only summing and differencing are needed to accumulate information over time for many control tasks. Following this principle, we propose two architectures for encoding history: one that directly uses PID features and another that extends these core ideas and can be used in arbitrary control tasks. When compared with prior approaches, our encoders produce policies that are often more robust and achieve better performance on a variety of tracking tasks. Going beyond tracking tasks, our policies achieve 1.7x better performance on average over previous state-of-the-art methods on a suite of high dimensional control tasks.
The accurate modeling and control of nonlinear dynamical effects are crucial for numerous robotic systems. The Koopman formalism emerges as a valuable tool for linear control design in nonlinear systems within unknown environments. However, it still remains a challenging task to learn the Koopman operator with control from data, and in particular, the simultaneous identification of the Koopman linear dynamics and the mapping between the state and Koopman spaces. Conventional approaches, based on single-level unconstrained optimization, may lack model robustness, training efficiency, and long-term predictive accuracy. This paper presents a bi-level optimization framework that jointly learns the Koopman embedding mapping and Koopman dynamics with explicit multi-step dynamical constraints, eliminating the need for heuristically-tuned loss terms. Leveraging implicit differentiation, our formulation allows back-propagation in standard learning framework and the use of state-of-the-art optimizers, yielding more stable and robust system performance over various applications compared to conventional methods.
Vector autoregressions (VARs) have an associated order $p$; conditional on observations at the preceding $p$ time points, the variable at time $t$ is conditionally independent of all the earlier history. Learning the order of the model is therefore vital for its characterisation and subsequent use in forecasting. It is common to assume that a VAR is stationary. This prevents the predictive variance of the process from increasing without bound as the forecast horizon increases and facilitates interpretation of the relationships between variables. A VAR is stable if and only if the roots of its characteristic equation lie outside the unit circle, constraining the autoregressive coefficient matrices to lie in the stationary region. Unfortunately, the geometry of the stationary region is very complicated which impedes specification of a prior. In this work, the autoregressive coefficients are mapped to a set of transformed partial autocorrelation matrices which are unconstrained, allowing for straightforward prior specification, routine computational inference, and meaningful interpretation of the magnitude of the elements in the matrix. The multiplicative gamma process is used to build a prior for the unconstrained matrices, which encourages increasing shrinkage of the partial autocorrelation parameters as the lag increases. Identifying the lag beyond which the partial autocorrelations become equal to zero then determines the order of the process. Posterior inference is performed using Hamiltonian Monte Carlo via Stan. A truncation criterion is used to determine whether a partial autocorrelation matrix has been effectively shrunk to zero. The value of the truncation threshold is motivated by classical theory on the sampling distribution of the partial autocorrelation function. The work is applied to neural activity data in order to investigate ultradian rhythms in the brain.
Cellwise contamination remains a challenging problem for data scientists, particularly in research fields that require the selection of sparse features. Traditional robust methods may not be feasible nor efficient in dealing with such contaminated datasets. We propose CR-Lasso, a robust Lasso-type cellwise regularization procedure that performs feature selection in the presence of cellwise outliers by minimising a regression loss and cell deviation measure simultaneously. To evaluate the approach, we conduct empirical studies comparing its selection and prediction performance with several sparse regression methods. We show that CR-Lasso is competitive under the settings considered. We illustrate the effectiveness of the proposed method on real data through an analysis of a bone mineral density dataset.
We present a simple linear regression based approach for learning the weights and biases of a neural network, as an alternative to standard gradient based backpropagation. The present work is exploratory in nature, and we restrict the description and experiments to (i) simple feedforward neural networks, (ii) scalar (single output) regression problems, and (iii) invertible activation functions. However, the approach is intended to be extensible to larger, more complex architectures. The key idea is the observation that the input to every neuron in a neural network is a linear combination of the activations of neurons in the previous layer, as well as the parameters (weights and biases) of the layer. If we are able to compute the ideal total input values to every neuron by working backwards from the output, we can formulate the learning problem as a linear least squares problem which iterates between updating the parameters and the activation values. We present an explicit algorithm that implements this idea, and we show that (at least for simple problems) the approach is more stable and faster than gradient-based backpropagation.
This paper presents a decentralized algorithm for solving distributed convex optimization problems in dynamic networks with time-varying objectives. The unique feature of the algorithm lies in its ability to accommodate a wide range of communication systems, including previously unsupported ones, by abstractly modeling the information exchange in the network. Specifically, it supports a novel communication protocol based on the "over-the-air" function computation (OTA-C) technology, that is designed for an efficient and truly decentralized implementation of the consensus step of the algorithm. Unlike existing OTA-C protocols, the proposed protocol does not require the knowledge of network graph structure or channel state information, making it particularly suitable for decentralized implementation over ultra-dense wireless networks with time-varying topologies and fading channels. Furthermore, the proposed algorithm synergizes with the "superiorization" methodology, allowing the development of new distributed algorithms with enhanced performance for the intended applications. The theoretical analysis establishes sufficient conditions for almost sure convergence of the algorithm to a common time-invariant solution for all agents, assuming such a solution exists. Our algorithm is applied to a real-world distributed random field estimation problem, showcasing its efficacy in terms of convergence speed, scalability, and spectral efficiency. Furthermore, we present a superiorized version of our algorithm that achieves faster convergence with significantly reduced energy consumption compared to the unsuperiorized algorithm.
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