A fundamental open problem in deep learning theory is how to define and understand the stability of stochastic gradient descent (SGD) close to a fixed point. Conventional literature relies on the convergence of statistical moments, esp., the variance, of the parameters to quantify the stability. We revisit the definition of stability for SGD and use the \textit{convergence in probability} condition to define the \textit{probabilistic stability} of SGD. The proposed stability directly answers a fundamental question in deep learning theory: how SGD selects a meaningful solution for a neural network from an enormous number of solutions that may overfit badly. To achieve this, we show that only under the lens of probabilistic stability does SGD exhibit rich and practically relevant phases of learning, such as the phases of the complete loss of stability, incorrect learning, convergence to low-rank saddles, and correct learning. When applied to a neural network, these phase diagrams imply that SGD prefers low-rank saddles when the underlying gradient is noisy, thereby improving the learning performance. This result is in sharp contrast to the conventional wisdom that SGD prefers flatter minima to sharp ones, which we find insufficient to explain the experimental data. We also prove that the probabilistic stability of SGD can be quantified by the Lyapunov exponents of the SGD dynamics, which can easily be measured in practice. Our work potentially opens a new venue for addressing the fundamental question of how the learning algorithm affects the learning outcome in deep learning.
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We propose an online learning algorithm for a class of machine learning models under a separable stochastic approximation framework. The essence of our idea lies in the observation that certain parameters in the models are easier to optimize than others. In this paper, we focus on models where some parameters have a linear nature, which is common in machine learning. In one routine of the proposed algorithm, the linear parameters are updated by the recursive least squares (RLS) algorithm, which is equivalent to a stochastic Newton method; then, based on the updated linear parameters, the nonlinear parameters are updated by the stochastic gradient method (SGD). The proposed algorithm can be understood as a stochastic approximation version of block coordinate gradient descent approach in which one part of the parameters is updated by a second-order SGD method while the other part is updated by a first-order SGD. Global convergence of the proposed online algorithm for non-convex cases is established in terms of the expected violation of a first-order optimality condition. Numerical experiments have shown that the proposed method accelerates convergence significantly and produces more robust training and test performance when compared to other popular learning algorithms. Moreover, our algorithm is less sensitive to the learning rate and outperforms the recently proposed slimTrain algorithm. The code has been uploaded to GitHub for validation.
Random smoothing data augmentation is a unique form of regularization that can prevent overfitting by introducing noise to the input data, encouraging the model to learn more generalized features. Despite its success in various applications, there has been a lack of systematic study on the regularization ability of random smoothing. In this paper, we aim to bridge this gap by presenting a framework for random smoothing regularization that can adaptively and effectively learn a wide range of ground truth functions belonging to the classical Sobolev spaces. Specifically, we investigate two underlying function spaces: the Sobolev space of low intrinsic dimension, which includes the Sobolev space in $D$-dimensional Euclidean space or low-dimensional sub-manifolds as special cases, and the mixed smooth Sobolev space with a tensor structure. By using random smoothing regularization as novel convolution-based smoothing kernels, we can attain optimal convergence rates in these cases using a kernel gradient descent algorithm, either with early stopping or weight decay. It is noteworthy that our estimator can adapt to the structural assumptions of the underlying data and avoid the curse of dimensionality. This is achieved through various choices of injected noise distributions such as Gaussian, Laplace, or general polynomial noises, allowing for broad adaptation to the aforementioned structural assumptions of the underlying data. The convergence rate depends only on the effective dimension, which may be significantly smaller than the actual data dimension. We conduct numerical experiments on simulated data to validate our theoretical results.
We demonstrate the relevance of an algorithm called generalized iterative scaling (GIS) or simultaneous multiplicative algebraic reconstruction technique (SMART) and its rescaled block-iterative version (RBI-SMART) in the field of optimal transport (OT). Many OT problems can be tackled through the use of entropic regularization by solving the Schr\"odinger problem, which is an information projection problem, that is, with respect to the Kullback--Leibler divergence. Here we consider problems that have several affine constraints. It is well-known that cyclic information projections onto the individual affine sets converge to the solution. In practice, however, even these individual projections are not explicitly available in general. In this paper, we exchange them for one GIS iteration. If this is done for every affine set, we obtain RBI-SMART. We provide a convergence proof using an interpretation of these iterations as two-step affine projections in an equivalent problem. This is done in a slightly more general setting than RBI-SMART, since we use a mix of explicitly known information projections and GIS iterations. We proceed to specialize this algorithm to several OT applications. First, we find the measure that minimizes the regularized OT divergence to a given measure under moment constraints. Second and third, the proposed framework yields an algorithm for solving a regularized martingale OT problem, as well as a relaxed version of the barycentric weak OT problem. Finally, we show an approach from the literature for unbalanced OT problems.
We consider alternating gradient descent (AGD) with fixed step size $\eta > 0$, applied to the asymmetric matrix factorization objective. We show that, for a rank-$r$ matrix $\mathbf{A} \in \mathbb{R}^{m \times n}$, $T = \left( \left(\frac{\sigma_1(\mathbf{A})}{\sigma_r(\mathbf{A})}\right)^2 \log(1/\epsilon)\right)$ iterations of alternating gradient descent suffice to reach an $\epsilon$-optimal factorization $\| \mathbf{A} - \mathbf{X}_T^{\vphantom{\intercal}} \mathbf{Y}_T^{\intercal} \|_{\rm F}^2 \leq \epsilon \| \mathbf{A} \|_{\rm F}^2$ with high probability starting from an atypical random initialization. The factors have rank $d>r$ so that $\mathbf{X}_T\in\mathbb{R}^{m \times d}$ and $\mathbf{Y}_T \in\mathbb{R}^{n \times d}$. Experiments suggest that our proposed initialization is not merely of theoretical benefit, but rather significantly improves convergence of gradient descent in practice. Our proof is conceptually simple: a uniform PL-inequality and uniform Lipschitz smoothness constant are guaranteed for a sufficient number of iterations, starting from our random initialization. Our proof method should be useful for extending and simplifying convergence analyses for a broader class of nonconvex low-rank factorization problems.
This paper provides a selective review of the statistical network analysis literature focused on clustering and inference problems for stochastic blockmodels and their variants. We survey asymptotic normality results for stochastic blockmodels as a means of thematically linking classical statistical concepts to contemporary research in network data analysis. Of note, multiple different forms of asymptotically Gaussian behavior arise in stochastic blockmodels and are useful for different purposes, pertaining to estimation and testing, the characterization of cluster structure in community detection, and understanding latent space geometry. This paper concludes with a discussion of open problems and ongoing research activities addressing asymptotic normality and its implications for statistical network modeling.
In this paper I propose a concept of a correct loss function in a generative model of supervised learning for an input space $\mathcal{X}$ and a label space $\mathcal{Y}$, which are measurable spaces. A correct loss function in a generative model of supervised learning must correctly measure the discrepancy between elements of a hypothesis space $\mathcal{H}$ of possible predictors and the supervisor operator, which may not belong to $\mathcal{H}$. To define correct loss functions, I propose a characterization of a regular conditional probability measure $\mu_{\mathcal{Y}|\mathcal{X}}$ for a probability measure $\mu$ on $\mathcal{X} \times \mathcal{Y}$ relative to the projection $\Pi_{\mathcal{X}}: \mathcal{X}\times\mathcal{Y}\to \mathcal{X}$ as a solution of a linear operator equation. If $\mathcal{Y}$ is a separable metrizable topological space with the Borel $\sigma$-algebra $ \mathcal{B} (\mathcal{Y})$, I propose another characterization of a regular conditional probability measure $\mu_{\mathcal{Y}|\mathcal{X}}$ as a minimizer of a mean square error on the space of Markov kernels, called probabilistic morphisms, from $\mathcal{X}$ to $\mathcal{Y}$, using kernel mean embedding. Using these results and using inner measure to quantify generalizability of a learning algorithm, I give a generalization of a result due to Cucker-Smale, which concerns the learnability of a regression model, to a setting of a conditional probability estimation problem. I also give a variant of Vapnik's method of solving stochastic ill-posed problem, using inner measure and discuss its applications.
Deep neural networks (DNNs) are increasingly being deployed to perform safety-critical tasks. The opacity of DNNs, which prevents humans from reasoning about them, presents new safety and security challenges. To address these challenges, the verification community has begun developing techniques for rigorously analyzing DNNs, with numerous verification algorithms proposed in recent years. While a significant amount of work has gone into developing these verification algorithms, little work has been devoted to rigorously studying the computability and complexity of the underlying theoretical problems. Here, we seek to contribute to the bridging of this gap. We focus on two kinds of DNNs: those that employ piecewise-linear activation functions (e.g., ReLU), and those that employ piecewise-smooth activation functions (e.g., Sigmoids). We prove the two following theorems: 1) The decidability of verifying DNNs with piecewise-smooth activation functions is equivalent to a well-known, open problem formulated by Tarski; and 2) The DNN verification problem for any quantifier-free linear arithmetic specification can be reduced to the DNN reachability problem, whose approximation is NP-complete. These results answer two fundamental questions about the computability and complexity of DNN verification, and the ways it is affected by the network's activation functions and error tolerance; and could help guide future efforts in developing DNN verification tools.
In this paper, we present a novel stochastic normal map-based algorithm ($\mathsf{norM}\text{-}\mathsf{SGD}$) for nonconvex composite-type optimization problems and discuss its convergence properties. Using a time window-based strategy, we first analyze the global convergence behavior of $\mathsf{norM}\text{-}\mathsf{SGD}$ and it is shown that every accumulation point of the generated sequence of iterates $\{\boldsymbol{x}^k\}_k$ corresponds to a stationary point almost surely and in an expectation sense. The obtained results hold under standard assumptions and extend the more limited convergence guarantees of the basic proximal stochastic gradient method. In addition, based on the well-known Kurdyka-{\L}ojasiewicz (KL) analysis framework, we provide novel point-wise convergence results for the iterates $\{\boldsymbol{x}^k\}_k$ and derive convergence rates that depend on the underlying KL exponent $\boldsymbol{\theta}$ and the step size dynamics $\{\alpha_k\}_k$. Specifically, for the popular step size scheme $\alpha_k=\mathcal{O}(1/k^\gamma)$, $\gamma \in (\frac23,1]$, (almost sure) rates of the form $\|\boldsymbol{x}^k-\boldsymbol{x}^*\| = \mathcal{O}(1/k^p)$, $p \in (0,\frac12)$, can be established. The obtained rates are faster than related and existing convergence rates for $\mathsf{SGD}$ and improve on the non-asymptotic complexity bounds for $\mathsf{norM}\text{-}\mathsf{SGD}$.
Sampling methods (e.g., node-wise, layer-wise, or subgraph) has become an indispensable strategy to speed up training large-scale Graph Neural Networks (GNNs). However, existing sampling methods are mostly based on the graph structural information and ignore the dynamicity of optimization, which leads to high variance in estimating the stochastic gradients. The high variance issue can be very pronounced in extremely large graphs, where it results in slow convergence and poor generalization. In this paper, we theoretically analyze the variance of sampling methods and show that, due to the composite structure of empirical risk, the variance of any sampling method can be decomposed into \textit{embedding approximation variance} in the forward stage and \textit{stochastic gradient variance} in the backward stage that necessities mitigating both types of variance to obtain faster convergence rate. We propose a decoupled variance reduction strategy that employs (approximate) gradient information to adaptively sample nodes with minimal variance, and explicitly reduces the variance introduced by embedding approximation. We show theoretically and empirically that the proposed method, even with smaller mini-batch sizes, enjoys a faster convergence rate and entails a better generalization compared to the existing methods.
Over the past few years, we have seen fundamental breakthroughs in core problems in machine learning, largely driven by advances in deep neural networks. At the same time, the amount of data collected in a wide array of scientific domains is dramatically increasing in both size and complexity. Taken together, this suggests many exciting opportunities for deep learning applications in scientific settings. But a significant challenge to this is simply knowing where to start. The sheer breadth and diversity of different deep learning techniques makes it difficult to determine what scientific problems might be most amenable to these methods, or which specific combination of methods might offer the most promising first approach. In this survey, we focus on addressing this central issue, providing an overview of many widely used deep learning models, spanning visual, sequential and graph structured data, associated tasks and different training methods, along with techniques to use deep learning with less data and better interpret these complex models --- two central considerations for many scientific use cases. We also include overviews of the full design process, implementation tips, and links to a plethora of tutorials, research summaries and open-sourced deep learning pipelines and pretrained models, developed by the community. We hope that this survey will help accelerate the use of deep learning across different scientific domains.
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