We study the Riemannian gradient method for PCA on which a crucial fact is that despite the simplicity of the considered setting, i.e., deterministic version of Krasulina's method, the convergence rate has not been well-understood yet. In this work, we provide a general tight analysis for the gap-dependent rate at O(\frac{1}{\Delta}\log\frac{1}{\epsilon}) that holds for any real symmetric matrix. More importantly, when the gap \Delta is significantly smaller than the target accuracy \epsilon on the objective sub-optimality of the final solution, the rate of this type is actually not tight any more, which calls for a worst-case rate. We further give the first worst-case analysis that achieves a rate of convergence at O(\frac{1}{\epsilon}\log\frac{1}{\epsilon}) . Particularly, our gap-dependent analysis suggests a new promising learning rate for stochastic variance reduced PCA algorithms.

Stochastic Gradient Descent (SGD) has been the method of choice for learning large-scale non-convex models. While a general analysis of when SGD works has been elusive, there has been a lot of recent progress in understanding the convergence of Gradient Flow (GF) on the population loss, partly due to the simplicity that a continuous-time analysis buys us. An overarching theme of our paper is providing general conditions under which SGD converges, assuming that GF on the population loss converges. Our main tool to establish this connection is a general \textit{converse Lyapunov} like theorem, which implies the existence of a Lyapunov potential under mild assumptions on the rates of convergence of GF. In fact, using these potentials, we show a one-to-one correspondence between rates of convergence of GF and geometrical properties of the underlying objective.

Deep neural networks often suffer from poor generalization caused by complex and non-convex loss landscapes. One of the popular solutions is Sharpness-Aware Minimization (SAM), which smooths the loss landscape via minimizing the maximized change of training loss when adding a perturbation to the weight. However, we find the indiscriminate perturbation of SAM on all parameters is suboptimal, which also results in excessive computation, \emph{i.e.}, double the overhead of common optimizers like Stochastic Gradient Descent (SGD). In this paper, we propose an efficient and effective training scheme coined as Sparse SAM (SSAM), which achieves sparse perturbation by a binary mask. To obtain the sparse mask, we provide two solutions which are based on Fisher information and dynamic sparse training, respectively.

One common approach to this problem, exemplified by stochastic gradient descent, involves sampling one f_i term at every iteration to make progress. This approach crucially relies on a notion of uniformity across the f_i's, formally captured by their condition number. In this work, we give an algorithm that minimizes the above convex formulation to \epsilon -accuracy in \widetilde{O}(\sum_{i 1} n d_i \log (1 /\epsilon)) gradient computations, with no assumptions on the condition number. The previous best algorithm independent of the condition number is the standard cutting plane method, which requires O(nd \log (1/\epsilon)) gradient computations. As a corollary, we improve upon the evaluation oracle complexity for decomposable submodular minimization by [Axiotis, Karczmarz, Mukherjee, Sankowski and Vladu, ICML 2021].

Bilevel optimization has been shown to be a powerful framework for formulating multi-task machine learning problems, e.g., reinforcement learning (RL) and meta-learning, where the decision variables are coupled in both levels of the minimization problems. In practice, the learning tasks would be located at different computing resource environments, and thus there is a need for deploying a decentralized training framework to implement multi-agent and multi-task learning. We develop a stochastic linearized augmented Lagrangian method (SLAM) for solving general nonconvex bilevel optimization problems over a graph, where both upper and lower optimization variables are able to achieve a consensus. We also establish that the theoretical convergence rate of the proposed SLAM to the Karush-Kuhn-Tucker (KKT) points of this class of problems is on the same order as the one achieved by the classical distributed stochastic gradient descent for only single-level nonconvex minimization problems. Numerical results tested on multi-agent RL problems showcase the superiority of SLAM compared with the benchmarks.

Motivated by the recent successes of neural networks that have the ability to fit the data perfectly \emph{and} generalize well, we study the noiseless model in the fundamental least-squares setup. We assume that an optimum predictor perfectly fits the inputs and outputs \langle \theta_*, \phi(X) \rangle Y, where \phi(X) stands for a possibly infinite dimensional non-linear feature map. To solve this problem, we consider the estimator given by the last iterate of stochastic gradient descent (SGD) with constant step-size. In this context, our contribution is two fold: (i) \emph{from a (stochastic) optimization perspective}, we exhibit an archetypal problem where we can show explicitly the convergence of SGD final iterate for a non-strongly convex problem with constant step-size whereas usual results use some form of average and (ii) \emph{from a statistical perspective}, we give explicit non-asymptotic convergence rates in the over-parameterized setting and leverage a \emph{fine-grained} parameterization of the problem to exhibit polynomial rates that can be faster than O(1/T) . The link with reproducing kernel Hilbert spaces is established.

The implicit bias towards solutions with favorable properties is believed to be a key reason why neural networks trained by gradient-based optimization can generalize well. While the implicit bias of gradient flow has been widely studied for homogeneous neural networks (including ReLU and leaky ReLU networks), the implicit bias of gradient descent is currently only understood for smooth neural networks. Therefore, implicit bias in non-smooth neural networks trained by gradient descent remains an open question. In this paper, we aim to answer this question by studying the implicit bias of gradient descent for training two-layer fully connected (leaky) ReLU neural networks. We showed that when the training data are nearly-orthogonal, for leaky ReLU activation function, gradient descent will find a network with a stable rank that converges to 1, whereas for ReLU activation function, gradient descent will find a neural network with a stable rank that is upper bounded by a constant.

Recently, there is a growing interest in studying pairwise learning since it includes many important machine learning tasks as specific examples, e.g., metric learning, AUC maximization and ranking. While stochastic gradient descent (SGD) is an efficient method, there is a lacking study on its generalization behavior for pairwise learning. In this paper, we present a systematic study on the generalization analysis of SGD for pairwise learning to understand the balance between generalization and optimization. We develop a novel high-probability generalization bound for uniformly-stable algorithms to incorporate the variance information for better generalization, based on which we establish the first nonsmooth learning algorithm to achieve almost optimal high-probability and dimension-independent generalization bounds in linear time. We consider both convex and nonconvex pairwise learning problems.

We give sharper bounds for uniformly stable randomized algorithms in a PAC-Bayesian framework, which improve the existing results by up to a factor of \sqrt{n} (ignoring a log factor), where n is the sample size. The key idea is to bound the moment generating function of the generalization gap using concentration of weakly dependent random variables due to Bousquet et al (2020). We introduce an assumption of sub-exponential stability parameter, which allows a general treatment that we instantiate in two applications: stochastic gradient descent and randomized coordinate descent. Our results eliminate the requirement of strong convexity from previous results, and hold for non-smooth convex problems.

Many modern machine learning applications - from online principal component analysis to covariance matrix identification and dictionary learning - can be formulated as minimization problems on Riemannian manifolds, typically solved with a Riemannian stochastic gradient method (or some variant thereof). However, in many cases of interest, the resulting minimization problem is _not_ geodesically convex, so the convergence of the chosen solver to a desirable solution - i.e., a local minimizer - is by no means guaranteed. In this paper, we study precisely this question, that is, whether stochastic Riemannian optimization algorithms are guaranteed to avoid saddle points with probability 1 . For generality, we study a family of retraction-based methods which, in addition to having a potentially much lower per-iteration cost relative to Riemannian gradient descent, include other widely used algorithms, such as natural policy gradient methods and mirror descent in ordinary convex spaces. In this general setting, we show that, under mild assumptions for the ambient manifold and the oracle providing gradient information, the policies under study avoid strict saddle points / submanifolds with probability 1, from any initial condition. This result provides an important sanity check for the use of gradient methods on manifolds as it shows that, almost always, the end state of a stochastic Riemannian algorithm can only be a local minimizer.