We present a method for Temporal Difference (TD) learning that addresses several challenges faced by robots learning to navigate in a marine environment. For improved data efficiency, our method reduces TD updates to Gaussian Process regression. To make predictions amenable to online settings, we introduce a sparse approximation with improved quality over current rejection-based sparse methods. We derive the predictive value function posterior and use the moments to obtain a new algorithm for model-free policy evaluation, SPGP-SARSA. With simple changes, we show SPGP-SARSA can be reduced to a model-based equivalent, SPGP-TD. We perform comprehensive simulation studies and also conduct physical learning trials with an underwater robot. Our results show SPGP-SARSA can outperform the state-of-the-art sparse method, replicate the prediction quality of its exact counterpart, and be applied to solve underwater navigation tasks.
When it comes to other approximate methods, the state-of-the-art uses a low-rank approximation to the full covariance matrix @cite_14 : @math , where @math is a projection matrix. Before adding new data to the active set, checks if new data points increase the conditional covariance by a desired error threshold, @math . In Section , we compare the approximation quality of and .
@cite_14
2103193999
This paper presents a new method for estimating high dimensional covariance matrices. The method, permuted rank-penalized least-squares (PRLS), is based on a Kronecker product series expansion of the true covariance matrix. Assuming an i.i.d. Gaussian random sample, we establish high dimensional rates of convergence to the true covariance as both the number of samples and the number of variables go to infinity. For covariance matrices of low separation rank, our results establish that PRLS has significantly faster convergence than the standard sample covariance matrix (SCM) estimator. The convergence rate captures a fundamental tradeoff between estimation error and approximation error, thus providing a scalable covariance estimation framework in terms of separation rank, similar to low rank approximation of covariance matrices . The MSE convergence rates generalize the high dimensional rates recently obtained for the ML Flip-flop algorithm , for Kronecker product covariance estimation. We show that a class of block Toeplitz covariance matrices is approximatable by low separation rank and give bounds on the minimal separation rank r that ensures a given level of bias. Simulations are presented to validate the theoretical bounds. As a real world application, we illustrate the utility of the proposed Kronecker covariance estimator for spatio-temporal linear least squares prediction of multivariate wind speed measurements.