Pros: This paper is well written and clear. The authors do a good job analyzing their method from a theoretical standpoint. I like that this paper has good theory. I like the kinds of experiments the authors chose, and how they are presented. All in all I think this paper is good, and is a solid contribution to the literature on approximate Newton methods.

We consider the regime where n d.

The initial motivation seems to be the work of Hoffman et al on the use of clustering to speedup stochastic methods for ERM. Their method was not proved to converge to the optimal due to the use of biased stochastic gradients. Also, that work seemed to work only for small clusters due to the approach chosen. This papers goes a long way to develop the basic idea into a satisfying theoretical framework which also gives rise to efficient implementations. This paper is truly a pleasure to read – a very fine example of academic exposition.

However, this paper is not carefully written. For example, the references are missing on page 6, line 192 and page 7, line 206. The legend of red lines are missing for Figure 2c,d. The paper states only the necessary information but not sufficient for the readers to follow easily. I think the clarity of this paper could be greatly improved especially the authors did not use the full 8 pages.

Recent work on scaling up Gaussian process regression (GPR) to large datasets has primarily focused on sparse GPR, which leverages a small set of basis functions to approximate the full Gaussian process during inference. However, the majority of these approaches are batch methods that operate on the entire training dataset at once, precluding the use of datasets that are streaming or too large to fit into memory. Although previous work has considered incrementally solving variational sparse GPR, most algorithms fail to update the basis functions and therefore perform suboptimally. We propose a novel incremental learning algorithm for variational sparse GPR based on stochastic mirror ascent of probability densities in reproducing kernel Hilbert space. This new formulation allows our algorithm to update basis functions online in accordance with the manifold structure of probability densities for fast convergence. We conduct several experiments and show that our proposed approach achieves better empirical performance in terms of prediction error than the recent state-of-the-art incremental solutions to variational sparse GPR.

The paper studies stochastic convex optimization in a distributed master/workers framework, where on each round each machine out of m produces a stochastic gradient and sends it to the master, which aggregates these into a mini-batch. In this paper the authors allow a fraction of alpha of the machines to be Byzantine, i.e., they do not need to report valid stochastic gradients but may produce arbitrary vectors, even in an adversarial manner. The goal is to aggregate the reports of the machines and to converge to an optimal solution of the convex objective despite the malicious Byzantine machines. The authors present a novel variant of minibatch-SGD which tackles the difficulty the dealing with Byzantine machines. They prove upper-bounds on the convergence and nearly optimal matching lower-bounds on any algorithm working in such framework, and in this sense the results are quite satisfactory.

This paper describes a new approach for computing nonnegative matrix factorizations (NMFs) with linear programming. The key idea is a data-driven model for the factorization, in which the most salient features in the data are used to express the remaining features. More precisely, given a data matrix X, the algorithm identifies a matrix C that satisfies X CX and some linear constraints. The matrix C selects features, which are then used to compute a low-rank NMF of X. A theoretical analysis demonstrates that this approach has the same type of guarantees as the recent NMF algorithm of Arora et al. (2012).

Differentially private (stochastic) gradient descent is the workhorse of DP private machine learning in both the convex and non-convex settings. In this work, we investigate the prospect of using the second-order information from the loss function to accelerate DP convex optimization. We first develop a private variant of the regularized cubic Newton method of Nesterov and Polyak, and show that for the class of strongly convex loss functions, our algorithm has quadratic convergence and achieves the optimal excess loss. We theoretically and empirically study the performance of our algorithm. Empirical results show our algorithm consistently achieves the best excess loss compared to other baselines and is 10-40x faster than DP-GD/DP-SGD for challenging datasets.

We propose a framework which makes it feasible to directly train deep neural networks with respect to popular families of task-specific non-decomposable performance measures such as AUC, multi-class AUC, F -measure and others. A common feature of the optimization model that emerges from these tasks is that it involves solving a Linear Programs (LP) during training where representations learned by upstream layers characterize the constraints or the feasible set. The constraint matrix is not only large but the constraints are also modified at each iteration. We show how adopting a set of ingenious ideas proposed by Mangasarian for 1-norm SVMs -- which advocates for solving LPs with a generalized Newton method -- provides a simple and effective solution that can be run on the GPU. In particular, this strategy needs little unrolling, which makes it more efficient during backward pass.

Moreover, CoLA provides memory efficient automatic differentiation, low precision computation, and GPU acceleration in both JAX and PyTorch, while also accommodating new objects, operations, and rules in downstream packages via multiple dispatch. CoLA can accelerate many algebraic operations, while making it easy to prototype matrix structures and algorithms, providing an appealing drop-in tool for virtually any computational effort that requires linear algebra. We showcase its efficacy across a broad range of applications, including partial differential equations, Gaussian processes, equivariant model construction, and unsupervised learning.