In seeking for sparse and efficient neural network models, many previous works investigated on enforcing L1 or L0 regularizers to encourage weight sparsity during training. The L0 regularizer measures the parameter sparsity directly and is invariant to the scaling of parameter values, but it cannot provide useful gradients, and therefore requires complex optimization techniques. The L1 regularizer is almost everywhere differentiable and can be easily optimized with gradient descent. Yet it is not scale-invariant, causing the same shrinking rate to all parameters, which is inefficient in increasing sparsity. Inspired by the Hoyer measure (the ratio between L1 and L2 norms) used in traditional compressed sensing problems, we present DeepHoyer, a set of sparsity-inducing regularizers that are both differentiable almost everywhere and scale-invariant. Our experiments show that enforcing DeepHoyer regularizers can produce even sparser neural network models than previous works, under the same accuracy level. We also show that DeepHoyer can be applied to both element-wise and structural pruning.
Dai, Xin-Yu (State Key Laboratory for Novel Software Technology, Nanjing University) | Zhang, Jian-Bing (State Key Laboratory for Novel Software Technology, Nanjing University) | Huang, Shu-Jian (State Key Laboratory for Novel Software Technology, Nanjing University) | Chen, Jia-Jun (State Key Laboratory for Novel Software Technology, Nanjing University) | Zhou, Zhi-Hua (State Key Laboratory for Novel Software Technology, Nanjing University)
In many learning tasks with structural properties, structural sparsity methods help induce sparse models, usually leading to better interpretability and higher generalization performance. One popular approach is to use group sparsity regularization that enforces sparsity on the clustered groups of features, while another popular approach is to adopt graph sparsity regularization that considers sparsity on the link structure of graph embedded features. Both the group and graph structural properties co-exist in many applications. However, group sparsity and graph sparsity have not been considered simultaneously yet. In this paper, we propose a g 2 -regularization that takes group and graph sparsity into joint consideration, and present an effective approach for its optimization. Experiments on both synthetic and real data show that, enforcing group-graph sparsity lead to better performance than using group sparsity or graph sparsity only.
In this paper, the estimation problem for sparse reduced rank regression (SRRR) model is considered. The SRRR model is widely used for dimension reduction and variable selection with applications in signal processing, econometrics, etc. The problem is formulated to minimize the least squares loss with a sparsity-inducing penalty considering an orthogonality constraint. Convex sparsity-inducing functions have been used for SRRR in literature. In this work, a nonconvex function is proposed for better sparsity inducing. An efficient algorithm is developed based on the alternating minimization (or projection) method to solve the nonconvex optimization problem. Numerical simulations show that the proposed algorithm is much more efficient compared to the benchmark methods and the nonconvex function can result in a better estimation accuracy.
Various sparse regularizers have been applied to machine learning problems, among which structured sparsity has been proposed for a better adaption to structured data. In this paper, motivated by effectively classifying linked data (e.g. Web pages, tweets, articles with references, and biological network data) where a group structure exists over the whole dataset and links exist between specific samples, we propose a joint sparse representation model that combines group sparsity and graph sparsity, to select a small number of connected components from the graph of linked samples, meanwhile promoting the sparsity of edges that link samples from different groups in each connected component. Consequently, linked samples are selected from a few sparsely-connected groups. Both theoretical analysis and experimental results on four benchmark datasets show that the joint sparsity model outperforms traditional group sparsity model and graph sparsity model, as well as the latest group-graph sparsity model.
We consider regularized risk minimization in a large dictionary of Reproducing kernel Hilbert Spaces (RKHSs) over which the target function has a sparse representation. This setting, commonly referred to as Sparse Multiple Kernel Learning (MKL), may be viewed as the non-parametric extension of group sparsity in linear models. While the two dominant algorithmic strands of sparse learning, namely convex relaxations using l1 norm (e.g., Lasso) and greedy methods (e.g., OMP), have both been rigorously extended for group sparsity, the sparse MKL literature has so farmainly adopted the former withmild empirical success. In this paper, we close this gap by proposing a Group-OMP based framework for sparse multiple kernel learning. Unlike l1-MKL, our approach decouples the sparsity regularizer (via a direct l0 constraint) from the smoothness regularizer (via RKHS norms) which leads to better empirical performance as well as a simpler optimization procedure that only requires a black-box single-kernel solver.