We establish an equivalence between the $\ell_2$-regularized solution path for a convex loss function, and the solution of an ordinary differentiable equation (ODE). Importantly, this equivalence reveals that the solution path can be viewed as the flow of a hybrid of gradient descent and Newton method applying to the empirical loss, which is similar to a widely used optimization technique called trust region method. This provides an interesting algorithmic view of $\ell_2$ regularization, and is in contrast to the conventional view that the $\ell_2$ regularization solution path is similar to the gradient flow of the empirical loss.New path-following algorithms based on homotopy methods and numerical ODE solvers are proposed to numerically approximate the solution path. In particular, we consider respectively Newton method and gradient descent method as the basis algorithm for the homotopy method, and establish their approximation error rates over the solution path. Importantly, our theory suggests novel schemes to choose grid points that guarantee an arbitrarily small suboptimality for the solution path. In terms of computational cost, we prove that in order to achieve an $\epsilon$-suboptimality for the entire solution path, the number of Newton steps required for the Newton method is $\mathcal O(\epsilon^{-1/2})$, while the number of gradient steps required for the gradient descent method is $\mathcal O\left(\epsilon^{-1} \ln(\epsilon^{-1})\right)$. Finally, we use $\ell_2$-regularized logistic regression as an illustrating example to demonstrate the effectiveness of the proposed path-following algorithms.

Network data are ubiquitous in modern machine learning, with tasks of interest including node classification, node clustering and link prediction. A frequent approach begins by learning an Euclidean embedding of the network, to which algorithms developed for vector-valued data are applied. For large networks, embeddings are learned using stochastic gradient methods where the sub-sampling scheme can be freely chosen. Despite the strong empirical performance of such methods, they are not well understood theoretically. Our work encapsulates representation methods using a subsampling approach, such as node2vec, into a single unifying framework. We prove, under the assumption that the graph is exchangeable, that the distribution of the learned embedding vectors asymptotically decouples. Moreover, we characterize the asymptotic distribution and provided rates of convergence, in terms of the latent parameters, which includes the choice of loss function and the embedding dimension. This provides a theoretical foundation to understand what the embedding vectors represent and how well these methods perform on downstream tasks. Notably, we observe that typically used loss functions may lead to shortcomings, such as a lack of Fisher consistency.

Despite the empirical success of deep learning, it still lacks theoretical understandings to explain why randomly initialized neural network trained by first-order optimization methods is able to achieve zero training loss, even though its landscape is non-convex and non-smooth. Recently, there are some works to demystifies this phenomenon under over-parameterized regime. In this work, we make further progress on this area by considering a commonly used momentum optimization algorithm: Nesterov accelerated method (NAG). We analyze the convergence of NAG for two-layer fully connected neural network with ReLU activation. Specifically, we prove that the error of NAG converges to zero at a linear convergence rate $1-\Theta(1/\sqrt{\kappa})$, where $\kappa > 1$ is determined by the initialization and the architecture of neural network. Comparing to the rate $1-\Theta(1/\kappa)$ of gradient descent, NAG achieves an acceleration. Besides, it also validates NAG and Heavy-ball method can achieve a similar convergence rate.

Transfer-based adversarial attacks can effectively evaluate model robustness in the black-box setting. Though several methods have demonstrated impressive transferability of untargeted adversarial examples, targeted adversarial transferability is still challenging. The existing methods either have low targeted transferability or sacrifice computational efficiency. In this paper, we develop a simple yet practical framework to efficiently craft targeted transfer-based adversarial examples. Specifically, we propose a conditional generative attacking model, which can generate the adversarial examples targeted at different classes by simply altering the class embedding and share a single backbone. Extensive experiments demonstrate that our method improves the success rates of targeted black-box attacks by a significant margin over the existing methods -- it reaches an average success rate of 29.6\% against six diverse models based only on one substitute white-box model in the standard testing of NeurIPS 2017 competition, which outperforms the state-of-the-art gradient-based attack methods (with an average success rate of $<$2\%) by a large margin. Moreover, the proposed method is also more efficient beyond an order of magnitude than gradient-based methods.

Successful applications of InfoNCE and its variants have popularized the use of contrastive variational mutual information (MI) estimators in machine learning. While featuring superior stability, these estimators crucially depend on costly large-batch training, and they sacrifice bound tightness for variance reduction. To overcome these limitations, we revisit the mathematics of popular variational MI bounds from the lens of unnormalized statistical modeling and convex optimization. Our investigation not only yields a new unified theoretical framework encompassing popular variational MI bounds but also leads to a novel, simple, and powerful contrastive MI estimator named as FLO. Theoretically, we show that the FLO estimator is tight, and it provably converges under stochastic gradient descent. Empirically, our FLO estimator overcomes the limitations of its predecessors and learns more efficiently. The utility of FLO is verified using an extensive set of benchmarks, which also reveals the trade-offs in practical MI estimation.

With the advent of big data acquisition technology, it has become increasingly important to integrate information across multiple datasets collected on a common set of subjects. Canonical correlation analysis (CCA), first proposed by Hotelling [20], is a widely used statistical tool to integrate information from two datasets: It seeks linear combinations of variables within each dataset such that their correlation is maximized. However, recent advances in fields such as multi-omics and multimodal brain imaging have presented us with new challenges, since scientists are often able to collect more than two datasets on the same set of subjects nowadays. To tackle these challenges, we turn to a useful generalization of CCA called generalized correlation analysis (GCA) [23] which aims to explore linear relationships across multiple data sources. Kettenring [23] proposed five different techniques for generalized correlation analysis of multiple datasets, where different methods correspond to maximization of different objective functions of covariances and correlations, subject to certain normalization constraints.

Two of the most prominent algorithms for solving unconstrained smooth games are the classical stochastic gradient descent-ascent (SGDA) and the recently introduced stochastic consensus optimization (SCO) (Mescheder et al., 2017). SGDA is known to converge to a stationary point for specific classes of games, but current convergence analyses require a bounded variance assumption. SCO is used successfully for solving large-scale adversarial problems, but its convergence guarantees are limited to its deterministic variant. In this work, we introduce the expected co-coercivity condition, explain its benefits, and provide the first last-iterate convergence guarantees of SGDA and SCO under this condition for solving a class of stochastic variational inequality problems that are potentially non-monotone. We prove linear convergence of both methods to a neighborhood of the solution when they use constant step-size, and we propose insightful stepsize-switching rules to guarantee convergence to the exact solution. In addition, our convergence guarantees hold under the arbitrary sampling paradigm, and as such, we give insights into the complexity of minibatching.

We study online convex optimization in the random order model, recently proposed by \citet{garber2020online}, where the loss functions may be chosen by an adversary, but are then presented to the online algorithm in a uniformly random order. Focusing on the scenario where the cumulative loss function is (strongly) convex, yet individual loss functions are smooth but might be non-convex, we give algorithms that achieve the optimal bounds and significantly outperform the results of \citet{garber2020online}, completely removing the dimension dependence and improving their scaling with respect to the strong convexity parameter. Our analysis relies on novel connections between algorithmic stability and generalization for sampling without-replacement analogous to those studied in the with-replacement i.i.d.~setting, as well as on a refined average stability analysis of stochastic gradient descent.

We propose a method that meta-learns a knowledge on matrix factorization from various matrices, and uses the knowledge for factorizing unseen matrices. The proposed method uses a neural network that takes a matrix as input, and generates prior distributions of factorized matrices of the given matrix. The neural network is meta-learned such that the expected imputation error is minimized when the factorized matrices are adapted to each matrix by a maximum a posteriori (MAP) estimation. We use a gradient descent method for the MAP estimation, which enables us to backpropagate the expected imputation error through the gradient descent steps for updating neural network parameters since each gradient descent step is written in a closed form and is differentiable. The proposed method can meta-learn from matrices even when their rows and columns are not shared, and their sizes are different from each other. In our experiments with three user-item rating datasets, we demonstrate that our proposed method can impute the missing values from a limited number of observations in unseen matrices after being trained with different matrices.

Recently there has been significant theoretical progress on understanding the convergence and generalization of gradient-based methods on nonconvex losses with overparameterized models. Nevertheless, many aspects of optimization and generalization and in particular the critical role of small random initialization are not fully understood. In this paper, we take a step towards demystifying this role by proving that small random initialization followed by a few iterations of gradient descent behaves akin to popular spectral methods. We also show that this implicit spectral bias from small random initialization, which is provably more prominent for overparameterized models, also puts the gradient descent iterations on a particular trajectory towards solutions that are not only globally optimal but also generalize well. Concretely, we focus on the problem of reconstructing a low-rank matrix from a few measurements via a natural nonconvex formulation. In this setting, we show that the trajectory of the gradient descent iterations from small random initialization can be approximately decomposed into three phases: (I) a spectral or alignment phase where we show that that the iterates have an implicit spectral bias akin to spectral initialization allowing us to show that at the end of this phase the column space of the iterates and the underlying low-rank matrix are sufficiently aligned, (II) a saddle avoidance/refinement phase where we show that the trajectory of the gradient iterates moves away from certain degenerate saddle points, and (III) a local refinement phase where we show that after avoiding the saddles the iterates converge quickly to the underlying low-rank matrix. Underlying our analysis are insights for the analysis of overparameterized nonconvex optimization schemes that may have implications for computational problems beyond low-rank reconstruction.