The observation that stochastic gradient descent (SGD) favors flat minima has played a fundamental role in understanding implicit regularization of SGD and guiding the tuning of hyperparameters. In this paper, we provide a quantitative explanation of this striking phenomenon by relating the particular noise structure of SGD to its \emph{linear stability} (Wu et al., 2018). Specifically, we consider training over-parameterized models with square loss. We prove that if a global minimum $\theta^*$ is linearly stable for SGD, then it must satisfy $\|H(\theta^*)\|_F\leq O(\sqrt{B}/\eta)$, where $\|H(\theta^*)\|_F, B,\eta$ denote the Frobenius norm of Hessian at $\theta^*$, batch size, and learning rate, respectively. Otherwise, SGD will escape from that minimum \emph{exponentially} fast. Hence, for minima accessible to SGD, the flatness -- as measured by the Frobenius norm of the Hessian -- is bounded independently of the model size and sample size. The key to obtaining these results is exploiting the particular geometry awareness of SGD noise: 1) the noise magnitude is proportional to loss value; 2) the noise directions concentrate in the sharp directions of local landscape. This property of SGD noise provably holds for linear networks and random feature models (RFMs) and is empirically verified for nonlinear networks. Moreover, the validity and practical relevance of our theoretical findings are justified by extensive numerical experiments.
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Consider the approximation of stochastic Allen-Cahn-type equations (i.e. $1+1$-dimensional space-time white noise-driven stochastic PDEs with polynomial nonlinearities $F$ such that $F(\pm \infty)=\mp \infty$) by a fully discrete space-time explicit finite difference scheme. The consensus in literature, supported by rigorous lower bounds, is that strong convergence rate $1/2$ with respect to the parabolic grid meshsize is expected to be optimal. We show that one can reach almost sure convergence rate $1$ (and no better) when measuring the error in appropriate negative Besov norms, by temporarily `pretending' that the SPDE is singular.
In this paper, we propose a new covering technique localized for the trajectories of SGD. This localization provides an algorithm-specific complexity measured by the covering number, which can have dimension-independent cardinality in contrast to standard uniform covering arguments that result in exponential dimension dependency. Based on this localized construction, we show that if the objective function is a finite perturbation of a piecewise strongly convex and smooth function with $P$ pieces, i.e. non-convex and non-smooth in general, the generalization error can be upper bounded by $O(\sqrt{(\log n\log(nP))/n})$, where $n$ is the number of data samples. In particular, this rate is independent of dimension and does not require early stopping and decaying step size. Finally, we employ these results in various contexts and derive generalization bounds for multi-index linear models, multi-class support vector machines, and $K$-means clustering for both hard and soft label setups, improving the known state-of-the-art rates.
The importance of human mobility analyses is growing in both research and practice, especially as applications for urban planning and mobility rely on them. Aggregate statistics and visualizations play an essential role as building blocks of data explorations and summary reports, the latter being increasingly released to third parties such as municipal administrations or in the context of citizen participation. However, such explorations already pose a threat to privacy as they reveal potentially sensitive location information, and thus should not be shared without further privacy measures. There is a substantial gap between state-of-the-art research on privacy methods and their utilization in practice. We thus conceptualize a standardized mobility report with differential privacy guarantees and implement it as open-source software to enable a privacy-preserving exploration of key aspects of mobility data in an easily accessible way. Moreover, we evaluate the benefits of limiting user contributions using three data sets relevant to research and practice. Our results show that even a strong limit on user contribution alters the original geospatial distribution only within a comparatively small range, while significantly reducing the error introduced by adding noise to achieve privacy guarantees.
The quality of generalized linear models (GLMs), frequently used by insurance companies, depends on the choice of interacting variables. The search for interactions is time-consuming, especially for data sets with a large number of variables, depends much on expert judgement of actuaries, and often relies on visual performance indicators. Therefore, we present an approach to automating the process of finding interactions that should be added to GLMs to improve their predictive power. Our approach relies on neural networks and a model-specific interaction detection method, which is computationally faster than the traditionally used methods like Friedman H-Statistic or SHAP values. In numerical studies, we provide the results of our approach on different data sets: open-source data, artificial data, and proprietary data.
Recently there is a large amount of work devoted to the study of Markov chain stochastic gradient methods (MC-SGMs) which mainly focus on their convergence analysis for solving minimization problems. In this paper, we provide a comprehensive generalization analysis of MC-SGMs for both minimization and minimax problems through the lens of algorithmic stability in the framework of statistical learning theory. For empirical risk minimization (ERM) problems, we establish the optimal excess population risk bounds for both smooth and non-smooth cases by introducing on-average argument stability. For minimax problems, we develop a quantitative connection between on-average argument stability and generalization error which extends the existing results for uniform stability \cite{lei2021stability}. We further develop the first nearly optimal convergence rates for convex-concave problems both in expectation and with high probability, which, combined with our stability results, show that the optimal generalization bounds can be attained for both smooth and non-smooth cases. To the best of our knowledge, this is the first generalization analysis of SGMs when the gradients are sampled from a Markov process.
The Gaussian kernel and its traditional normalizations (e.g., row-stochastic) are popular approaches for assessing similarities between data points, commonly used for manifold learning and clustering, as well as supervised and semi-supervised learning on graphs. In many practical situations, the data can be corrupted by noise that prohibits traditional affinity matrices from correctly assessing similarities, especially if the noise magnitudes vary considerably across the data, e.g., under heteroskedasticity or outliers. An alternative approach that provides a more stable behavior under noise is the doubly stochastic normalization of the Gaussian kernel. In this work, we investigate this normalization in a setting where points are sampled from an unknown density on a low-dimensional manifold embedded in high-dimensional space and corrupted by possibly strong, non-identically distributed, sub-Gaussian noise. We establish the pointwise concentration of the doubly stochastic affinity matrix and its scaling factors around certain population forms. We then utilize these results to develop several tools for robust inference. First, we derive a robust density estimator that can substantially outperform the standard kernel density estimator under high-dimensional noise. Second, we provide estimators for the pointwise noise magnitudes, the pointwise signal magnitudes, and the pairwise Euclidean distances between clean data points. Lastly, we derive robust graph Laplacian normalizations that approximate popular manifold Laplacians, including the Laplace Beltrami operator, showing that the local geometry of the manifold can be recovered under high-dimensional noise. We exemplify our results in simulations and on real single-cell RNA-sequencing data. In the latter, we show that our proposed normalizations are robust to technical variability associated with different cell types.
Prophet inequalities for rewards maximization are fundamental results from optimal stopping theory with several applications to mechanism design and online optimization. We study the cost minimization counterpart of the classical prophet inequality, where one is facing a sequence of costs $X_1, X_2, \dots, X_n$ in an online manner and must ''stop'' at some point and take the last cost seen. Given that the $X_i$'s are independent, drawn from known distributions, the goal is to devise a stopping strategy $S$ (online algorithm) that minimizes the expected cost. We first observe that if the $X_i$'s are not identically distributed, then no strategy can achieve a bounded approximation, no matter if the arrival order is adversarial or random. This leads us to consider the case where the $X_i$'s are I.I.D.. For the I.I.D. case, we give a complete characterization of the optimal stopping strategy. We show that it achieves a (distribution-dependent) constant-factor approximation to the prophet's cost for almost all distributions and that this constant is tight. In particular, for distributions for which the integral of the hazard rate is a polynomial $H(x) = \sum_{i=1}^k a_i x^{d_i}$, where $d_1 < \dots < d_k$, the approximation factor is $\lambda(d_1)$, a decreasing function of $d_1$. Furthermore, for MHR distributions, we show that this constant is at most $2$, and this is again tight. We also analyze single-threshold strategies for the cost prophet inequality problem. We design a threshold that achieves a $\operatorname{O}(\operatorname{polylog}n)$-factor approximation, where the exponent in the logarithmic factor is a distribution-dependent constant, and we show a matching lower bound. We believe that our results are of independent interest for analyzing approximately optimal (posted price-style) mechanisms for procuring items.
We consider the stochastic gradient descent (SGD) algorithm driven by a general stochastic sequence, including i.i.d noise and random walk on an arbitrary graph, among others; and analyze it in the asymptotic sense. Specifically, we employ the notion of `efficiency ordering', a well-analyzed tool for comparing the performance of Markov Chain Monte Carlo (MCMC) samplers, for SGD algorithms in the form of Loewner ordering of covariance matrices associated with the scaled iterate errors in the long term. Using this ordering, we show that input sequences that are more efficient for MCMC sampling also lead to smaller covariance of the errors for SGD algorithms in the limit. This also suggests that an arbitrarily weighted MSE of SGD iterates in the limit becomes smaller when driven by more efficient chains. Our finding is of particular interest in applications such as decentralized optimization and swarm learning, where SGD is implemented in a random walk fashion on the underlying communication graph for cost issues and/or data privacy. We demonstrate how certain non-Markovian processes, for which typical mixing-time based non-asymptotic bounds are intractable, can outperform their Markovian counterparts in the sense of efficiency ordering for SGD. We show the utility of our method by applying it to gradient descent with shuffling and mini-batch gradient descent, reaffirming key results from existing literature under a unified framework. Empirically, we also observe efficiency ordering for variants of SGD such as accelerated SGD and Adam, open up the possibility of extending our notion of efficiency ordering to a broader family of stochastic optimization algorithms.
Designing learning systems which are invariant to certain data transformations is critical in machine learning. Practitioners can typically enforce a desired invariance on the trained model through the choice of a network architecture, e.g. using convolutions for translations, or using data augmentation. Yet, enforcing true invariance in the network can be difficult, and data invariances are not always known a piori. State-of-the-art methods for learning data augmentation policies require held-out data and are based on bilevel optimization problems, which are complex to solve and often computationally demanding. In this work we investigate new ways of learning invariances only from the training data. Using learnable augmentation layers built directly in the network, we demonstrate that our method is very versatile. It can incorporate any type of differentiable augmentation and be applied to a broad class of learning problems beyond computer vision. We provide empirical evidence showing that our approach is easier and faster to train than modern automatic data augmentation techniques based on bilevel optimization, while achieving comparable results. Experiments show that while the invariances transferred to a model through automatic data augmentation are limited by the model expressivity, the invariance yielded by our approach is insensitive to it by design.
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.
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