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The Convolutional Sparse Coding (CSC) model has recently gained considerable traction in the signal and image processing communities. By providing a global, yet tractable, model that operates on the whole image, the CSC was shown to overcome several limitations of the patch-based sparse model while achieving superior performance in various applications. Contemporary methods for pursuit and learning the CSC dictionary often rely on the Alternating Direction Method of Multipliers (ADMM) in the Fourier domain for the computational convenience of convolutions, while ignoring the local characterizations of the image. A recent work by Papyan et al. [1] suggested the SBDL algorithm for the CSC, while operating locally on image patches. SBDL demonstrates better performance compared to the Fourier-based methods, albeit still relying on the ADMM.In this work we maintain the localized strategy of the SBDL, while proposing a new and much simpler approach based on the Block Coordinate Descent algorithm -this method is termed Local Block Coordinate Descent (LoBCoD). Furthermore, we introduce a novel stochastic gradient descent version of LoBCoD for training the convolutional filters. The Stochastic-LoBCoD leverages the benefits of online learning, while being applicable to a single training image. We demonstrate the advantages of the proposed algorithms for image inpainting and multi-focus image fusion, achieving state-of-the-art results.

In the Bayesian reinforcement learning (RL) setting, a prior distribution over the unknown problem parameters -- the rewards and transitions -- is assumed, and a policy that optimizes the (posterior) expected return is sought. A common approximation, which has been recently popularized as meta-RL, is to train the agent on a sample of N problem instances from the prior, with the hope that for large enough N, good generalization behavior to an unseen test instance will be obtained. In this work, we study generalization in Bayesian RL under the probably approximately correct (PAC) framework, using the method of algorithmic stability. Our main contribution is showing that by adding regularization, the optimal policy becomes uniformly stable in an appropriate sense. Most stability results in the literature build on strong convexity of the regularized loss -- an approach that is not suitable for RL as Markov decision processes (MDPs) are not convex. Instead, building on recent results of fast convergence rates for mirror descent in regularized MDPs, we show that regularized MDPs satisfy a certain quadratic growth criterion, which is sufficient to establish stability. This result, which may be of independent interest, allows us to study the effect of regularization on generalization in the Bayesian RL setting.

In the Bayesian reinforcement learning (RL) setting, a prior distribution over the unknown problem parameters -the rewards and transitions -is assumed, and a policy that optimizes the (posterior) expected return is sought. A common approximation, which has been recently popularized as meta-RL, is to train the agent on a sample of N problem instances from the prior, with the hope that for large enough N , good generalization behavior to an unseen test instance will be obtained. In this work, we study generalization in Bayesian RL under the probably approximately correct (PAC) framework, using the method of algorithmic stability. Our main contribution is showing that by adding regularization, the optimal policy becomes stable in an appropriate sense. Most stability results in the literature build on strong convexity of the regularized loss -an approach that is not suitable for RL as Markov decision processes (MDPs) are not convex. Instead, building on recent results of fast convergence rates for mirror descent in regularized MDPs, we show that regularized MDPs satisfy a certain quadratic growth criterion, which is sufficient to establish stability. This result, which may be of independent interest, allows us to study the effect of regularization on generalization in the Bayesian RL setting.