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.
There are several key assumptions underpinning our results. To guarantee the likelihood can be factored, we need to assume that transitions are uncorrelated. Had we modeled serial correlation in the noise process, @math would be distributed with a tridiagonal covariance @cite_14 , which cannot be factored directly. It is possible to obtain a factorable model by applying a whitening transform. However, this changes the observation process to Monte Carlo samples, which are known to be noisy @cite_16 . Under our simpler set of assumptions, we obtain an efficiently-computable, sparse representation of the value posterior that is amenable to smooth evidence maximization. Section details how to select the hyperparameters and pseudo inputs with gradient-based optimization.
@cite_14 @cite_16
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Given a weighted graph with N vertices, consider a real-valued regression problem in a semisupervised setting, where one observes n labeled vertices, and the task is to label the remaining ones. We present a theoretical study of p-based Laplacian regularization under a d-dimensional geometric random graph model. We provide a variational characterization of the performance of this regularized learner as N grows to infinity while n stays constant; the associated optimality conditions lead to a partial differential equation that must be satisfied by the associated function estimate f . From this formulation we derive several predictions on the limiting behavior the function f , including (a) a phase transition in its smoothness at the threshold p = d + 1; and (b) a tradeoff between smoothness and sensitivity to the underlying unlabeled data distribution P . Thus, over the range p ≤ d, the function estimate f is degenerate and “spiky,” whereas for p ≥ d + 1, the function estimate f is smooth. We show that the effect of the underlying density vanishes monotonically with p, such that in the limit p = ∞, corresponding to the so-called Absolutely Minimal Lipschitz Extension, the estimate f is independent of the distribution P . Under the assumption of semi-supervised smoothness, ignoring P can lead to poor statistical performance; in particular, we construct a specific example for d = 1 to demonstrate that p = 2 has lower risk than p =∞ due to the former penalty adapting to P and the latter ignoring it. We also provide simulations that verify the accuracy of our predictions for finite sample sizes. Together, these properties show that p = d + 1 is an optimal choice, yielding a function estimate f that is both smooth and non-degenerate, while remaining maximally sensitive to P . Most existing image denoising approaches assumed the noise to be homogeneous white Gaussian distributed with known intensity. However, in real noisy images, the noise models are usually unknown beforehand and can be much more complex. This paper addresses this problem and proposes a novel blind image denoising algorithm to recover the clean image from noisy one with the unknown noise model. To model the empirical noise of an image, our method introduces the mixture of Gaussian distribution, which is flexible enough to approximate different continuous distributions. The problem of blind image denoising is reformulated as a learning problem. The procedure is to first build a two-layer structural model for noisy patches and consider the clean ones as latent variable. To control the complexity of the noisy patch model, this work proposes a novel Bayesian nonparametric prior called “Dependent Dirichlet Process Tree” to build the model. Then, this study derives a variational inference algorithm to estimate model parameters and recover clean patches. We apply our method on synthesis and real noisy images with different noise models. Comparing with previous approaches, ours achieves better performance. The experimental results indicate the efficiency of the proposed algorithm to cope with practical image denoising tasks.