Regularization is one of the most fundamental topics in optimization, statistics and machine learning. To get sparsity in estimating a parameter u Rd, an ℓq penalty term, u q, is usually added to the objective function. What is the probabilistic distribution corresponding to such ℓq penalty? What is the correct stochastic process corresponding to u q when we model functions u Lq? This is important for statistically modeling high-dimensional objects such as images, with penalty to preserve certain properties, e.g.

Spectral clustering has gained popularity for clustering non-convex data due to its simplicity and effectiveness. It is essential to construct a similarity graph using a high-quality affinity measure that models the local neighborhood relations among the data samples. However, incomplete data can lead to inaccurate affinity measures, resulting in degraded clustering performance. To address these issues, we propose an imputation-free framework with two novel approaches to improve spectral clustering on incomplete data. Firstly, we introduce a new kernel correction method that enhances the quality of the kernel matrix estimated on incomplete data with a theoretical guarantee, benefiting classical spectral clustering on pre-defined kernels. Secondly, we develop a series of affinity learning methods that equip the selfexpressive framework with ℓp-norm to construct an intrinsic affinity matrix with an adaptive extension. Our methods outperform existing data imputation and distance calibration techniques on benchmark datasets, offering a promising solution to spectral clustering on incomplete data in various real-world applications.

We focus on the task of approximating the optimal value function in deep reinforcement learning. This iterative process is comprised of solving a sequence of optimization problems where the loss function changes per iteration. The common approach to solving this sequence of problems is to employ modern variants of the stochastic gradient descent algorithm such as Adam. These optimizers maintain their own internal parameters such as estimates of the first-order and the secondorder moments of the gradient, and update them over time. Therefore, information obtained in previous iterations is used to solve the optimization problem in the current iteration. We demonstrate that this can contaminate the moment estimates because the optimization landscape can change arbitrarily from one iteration to the next one. To hedge against this negative effect, a simple idea is to reset the internal parameters of the optimizer when starting a new iteration. We empirically investigate this resetting idea by employing various optimizers in conjunction with the Rainbow algorithm. We demonstrate that this simple modification significantly improves the performance of deep RL on the Atari benchmark.

Reproducibility319 The backbone recommendation model, DLRM by Naumov et al. [2019], has an open-source PyTorch320 implementation available on Github which includes an implementation of CE. For CCE you need a321 fast library for K-means. We recommend the open-sourced implementation by Johnson et al. [2019]322 for better performance, but you can also use the implementation in Scikit-learn [Pedregosa et al.,323 2011]. The baseline result should be straightforward to reproduce as we closely follow the instructions324 provided by Naumov et al. [2019]. For the CE methods, we only need to change two functions in325 the code: create_emband apply_emb. We suggest using a class for each CE method; see Figure 3.326 For the random hash function, one could use a universal hash function or numpy.random.randint.327