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. In addition, we theoretically prove that SSAM can converge at the same rate as SAM,~\emph{i.e.}, $O(\log T/\sqrt{T})$. Sparse SAM not only has the potential for training acceleration but also smooths the loss landscape effectively. Extensive experimental results on CIFAR10, CIFAR100, and ImageNet-1K confirm the superior efficiency of our method to SAM, and the performance is preserved or even better with a perturbation of merely 50\% sparsity.

Understanding when and why interpolating methods generalize well has recently been a topic of interest in statistical learning theory. However, systematically connecting interpolating methods to achievable notions of optimality has only received partial attention. In this paper, we ask the question of what is the optimal way to interpolate in linear regression using functions that are linear in the response variable (as the case for the Bayes optimal estimator in ridge regression) and depend on the data, the population covariance of the data, the signal-to-noise ratio and the covariance of the prior for the signal, but do not depend on the value of the signal itself nor the noise vector in the training data. We provide a closed-form expression for the interpolator that achieves this notion of optimality and show that it can be derived as the limit of preconditioned gradient descent with a specific initialization. We identify a regime where the minimum-norm interpolator provably generalizes arbitrarily worse than the optimal response-linear achievable interpolator that we introduce, and validate with numerical experiments that the notion of optimality we consider can be achieved by interpolating methods that only use the training data as input in the case of an isotropic prior. Finally, we extend the notion of optimal response-linear interpolation to random features regression under a linear data-generating model.

We show, both theoretically and empirically, that the algorithm naturally adapts to data low-rankness and converges exponentially fast to the ground-truth principal subspace. Notably, our result suggests that despite various recent efforts to accelerate the convergence of stochastic-gradient based methods by adding a O(n)-time variance reduction step, for the k-PCA problem, a truly online SGD variant suffices to achieve exponential convergence on intrinsically low-rank data.

This paper introduces a new approach for the scalable Tucker decomposition problem. Given a tensor X, the method proposed allows to infer the latent factors by processing one subtensor drawn from X at a time. The key principle of our approach is based on the recursive computations of gradient and on cyclic update of factors involving only one single step of gradient descent. We further improve the computational efficiency of this algorithm by proposing an inexact gradient version. These two algorithms are backed with theoretical guarantees of convergence and convergence rate under mild conditions. The scalabilty of the proposed approaches which can be easily extended to handle some common constraints encountered in tensor decomposition (e.g non-negativity), is proven via numerical experiments on both synthetic and real data sets.

Many fundamental problems in machine learning can be formulated by the convex program \[ \min_{\theta\in \mathbb{R}^d}\ \sum_{i=1}^{n}f_{i}(\theta), \]where each $f_i$ is a convex, Lipschitz function supported on a subset of $d_i$ coordinates of $\theta$. 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]. Our main technical contribution is an adaptive procedure to select an $f_i$ term at every iteration via a novel combination of cutting-plane and interior-point methods.

In deep neural nets, lower level embedding layers account for a large portion of the total number of parameters. Tikhonov regularization, graph-based regularization, and hard parameter sharing are approaches that introduce explicit biases into training in a hope to reduce statistical complexity. Alternatively, we propose stochastically shared embeddings (SSE), a data-driven approach to regularizing embedding layers, which stochastically transitions between embeddings during stochastic gradient descent (SGD). Because SSE integrates seamlessly with existing SGD algorithms, it can be used with only minor modifications when training large scale neural networks. We develop two versions of SSE: SSE-Graph using knowledge graphs of embeddings; SSE-SE using no prior information. We provide theoretical guarantees for our method and show its empirical effectiveness on 6 distinct tasks, from simple neural networks with one hidden layer in recommender systems, to the transformer and BERT in natural languages. We find that when used along with widely-used regularization methods such as weight decay and dropout, our proposed SSE can further reduce overfitting, which often leads to more favorable generalization results.

Hamiltonian Monte Carlo (HMC) is a popular Markov Chain Monte Carlo (MCMC) algorithm to sample from an unnormalized probability distribution. A leapfrog integrator is commonly used to implement HMC in practice, but its performance can be sensitive to the choice of mass matrix used therein. We develop a gradient-based algorithm that allows for the adaptation of the mass matrix by encouraging the leapfrog integrator to have high acceptance rates while also exploring all dimensions jointly. In contrast to previous work that adapt the hyperparameters of HMC using some form of expected squared jumping distance, the adaptation strategy suggested here aims to increase sampling efficiency by maximizing an approximation of the proposal entropy. We illustrate that using multiple gradients in the HMC proposal can be beneficial compared to a single gradient-step in Metropolis-adjusted Langevin proposals. Empirical evidence suggests that the adaptation method can outperform different versions of HMC schemes by adjusting the mass matrix to the geometry of the target distribution and by providing some control on the integration time.

Distributionally robust learning (DRL) is increasingly seen as a viable method to train machine learning models for improved model generalization. These min-max formulations, however, are more difficult to solve. We provide a new stochastic gradient descent algorithm to efficiently solve this DRL formulation. Our approach applies gradient descent to the outer minimization formulation and estimates the gradient of the inner maximization based on a sample average approximation. The latter uses a subset of the data sampled without replacement in each iteration, progressively increasing the subset size to ensure convergence. We rigorously establish convergence to a near-optimal solution under standard regularity assumptions and, for strongly convex losses, match the best known $O(\epsilon{ 1})$ rate of convergence up to a known threshold. Empirical results demonstrate the significant benefits of our approach over previous work in improving learning for model generalization.

Minimax optimal convergence rates for numerous classes of stochastic convex optimization problems are well characterized, where the majority of results utilize iterate averaged stochastic gradient descent (SGD) with polynomially decaying step sizes. In contrast, the behavior of SGD's final iterate has received much less attention despite the widespread use in practice. Motivated by this observation, this work provides a detailed study of the following question: what rate is achievable using the final iterate of SGD for the streaming least squares regression problem with and without strong convexity? First, this work shows that even if the time horizon T (i.e. the number of iterations that SGD is run for) is known in advance, the behavior of SGD's final iterate with any polynomially decaying learning rate scheme is highly sub-optimal compared to the statistical minimax rate (by a condition number factor in the strongly convex case and a factor of $\sqrt{T}$ in the non-strongly convex case). In contrast, this paper shows that Step Decay schedules, which cut the learning rate by a constant factor every constant number of epochs (i.e., the learning rate decays geometrically) offer significant improvements over any polynomially decaying step size schedule. In particular, the behavior of the final iterate with step decay schedules is off from the statistical minimax rate by only log factors (in the condition number for the strongly convex case, and in T in the non-strongly convex case). Finally, in stark contrast to the known horizon case, this paper shows that the anytime (i.e. the limiting) behavior of SGD's final iterate is poor (in that it queries iterates with highly sub-optimal function value infinitely often, i.e. in a limsup sense) irrespective of the step size scheme employed. These results demonstrate the subtlety in establishing optimal learning rate schedules (for the final iterate) for stochastic gradient procedures in fixed time horizon settings.

Using a low-dimensional parametrization of signals is a generic and powerful way to enhance performance in signal processing and statistical inference. A very popular and widely explored type of dimensionality reduction is sparsity; another type is generative modelling of signal distributions. Generative models based on neural networks, such as GANs or variational auto-encoders, are particularly performant and are gaining on applicability. In this paper we study spiked matrix models, where a low-rank matrix is observed through a noisy channel. This problem with sparse structure of the spikes has attracted broad attention in the past literature.