This paper introduces SparseOptimizer, a novel deep learning optimizer that exploits Moreau-Yosida regularization to naturally induce sparsity in large language models such as BERT, ALBERT and GPT. Key to the design of SparseOptimizer is an embedded shrinkage operator, which imparts sparsity directly within the optimization process. This operator, backed by a sound theoretical framework, includes an analytical solution, thereby reinforcing the optimizer's robustness and efficacy. Crucially, SparseOptimizer's plug-and-play functionality eradicates the need for code modifications, making it a universally adaptable tool for a wide array of large language models. Empirical evaluations on benchmark datasets such as GLUE, RACE, SQuAD1, and SQuAD2 confirm that SparseBERT and SparseALBERT, when sparsified using SparseOptimizer, achieve performance comparable to their dense counterparts, BERT and ALBERT, while significantly reducing their parameter count. Further, this work proposes an innovative optimizer-compiler co-design strategy, demonstrating the potential of inference acceleration (\textbf{3.37x}, \textbf{6.30x}, and \textbf{7.15x} in comparison with Pytorch, TensorFlow, and LLVM generic compile, respectively) in SparseBERT when paired with an appropriately designed compiler. This study represents a significant step forward in the evolution of efficient, scalable, and high-performing large language models, setting a precedent for future exploration and optimization in this domain. The SparseOptimizer code and SparseALBERT model will be publicly available upon paper acceptance.

We consider the tree structured group Lasso where the structure over the features can be represented as a tree with leaf nodes as features and internal nodes as clusters of the features. The structured regularization with a pre-defined tree structure is based on a group-Lasso penalty, where one group is defined for each node in the tree. Such a regularization can help uncover the structured sparsity, which is desirable for applications with some meaningful tree structures on the features. However, the tree structured group Lasso is challenging to solve due to the complex regularization. In this paper, we develop an efficient algorithm for the tree structured group Lasso. One of the key steps in the proposed algorithm is to solve the Moreau-Yosida regularization associated with the grouped tree structure.

We study the function approximation aspect of distributionally robust optimization (DRO) based on probability metrics, such as the Wasserstein and the maximum mean discrepancy. Our analysis leverages the insight that existing DRO paradigms hinge on function majorants such as the Moreau-Yosida regularization (supremal convolution). Deviating from those, this paper instead proposes robust learning algorithms based on smooth function approximation and interpolation. Our methods are simple in forms and apply to general loss functions without knowing functional norms a priori. Furthermore, we analyze the DRO risk bound decomposition by leveraging smooth function approximators and the convergence rate for empirical kernel mean embedding.

We consider a commonly studied supervised classification of a synthetic dataset whose labels are generated by feeding a one-layer neural network with random iid inputs. We study the generalization performances of standard classifiers in the high-dimensional regime where $\alpha=n/d$ is kept finite in the limit of a high dimension $d$ and number of samples $n$. Our contribution is three-fold: First, we prove a formula for the generalization error achieved by $\ell_2$ regularized classifiers that minimize a convex loss. This formula was first obtained by the heuristic replica method of statistical physics. Secondly, focussing on commonly used loss functions and optimizing the $\ell_2$ regularization strength, we observe that while ridge regression performance is poor, logistic and hinge regression are surprisingly able to approach the Bayes-optimal generalization error extremely closely. As $\alpha \to \infty$ they lead to Bayes-optimal rates, a fact that does not follow from predictions of margin-based generalization error bounds. Third, we design an optimal loss and regularizer that provably leads to Bayes-optimal generalization error.

We consider the tree structured group Lasso where the structure over the features can be represented as a tree with leaf nodes as features and internal nodes as clusters of the features. The structured regularization with a pre-defined tree structure is based on a group-Lasso penalty, where one group is defined for each node in the tree. Such a regularization can help uncover the structured sparsity, which is desirable for applications with some meaningful tree structures on the features. However, the tree structured group Lasso is challenging to solve due to the complex regularization. In this paper, we develop an efficient algorithm for the tree structured group Lasso.

We propose a generic approach to accelerate gradient-based optimization algorithms with quasi-Newton principles. The proposed scheme, called QuickeNing, can be applied to incremental first-order methods such as stochastic variance-reduced gradient (SVRG) or incremental surrogate optimization (MISO). It is also compatible with composite objectives, meaning that it has the ability to provide exactly sparse solutions when the objective involves a sparsity-inducing regularization. QuickeNing relies on limited-memory BFGS rules, making it appropriate for solving high-dimensional optimization problems. Besides, it enjoys a worst-case linear convergence rate for strongly convex problems. We present experimental results where QuickeNing gives significant improvements over competing methods for solving large-scale high-dimensional machine learning problems.

We consider the tree structured group Lasso where the structure over the features can be represented as a tree with leaf nodes as features and internal nodes as clusters of the features. The structured regularization with a pre-defined tree structure is based on a group-Lasso penalty, where one group is defined for each node in the tree. Such a regularization can help uncover the structured sparsity, which is desirable for applications with some meaningful tree structures on the features. However, the tree structured group Lasso is challenging to solve due to the complex regularization. In this paper, we develop an efficient algorithm for the tree structured group Lasso. One of the key steps in the proposed algorithm is to solve the Moreau-Yosida regularization associated with the grouped tree structure. The main technical contributions of this paper include (1) we show that the associated Moreau-Yosida regularization admits an analytical solution, and (2) we develop an efficient algorithm for determining the effective interval for the regularization parameter. Our experimental results on the AR and JAFFE face data sets demonstrate the efficiency and effectiveness of the proposed algorithm.