Previous works on stochastic gradient descent (SGD) often focus on its success. In this work, we construct worst-case optimization problems illustrating that, when not in the regimes that the previous works often assume, SGD can exhibit many strange and potentially undesirable behaviors. Specifically, we construct landscapes and data distributions such that (1) SGD converges to local maxima, (2) SGD escapes saddle points arbitrarily slowly, (3) SGD prefers sharp minima over flat ones, and (4) AMSGrad converges to local maxima. We also realize results in a minimal neural network-like example. Our results highlight the importance of simultaneously analyzing the minibatch sampling, discrete-time updates rules, and realistic landscapes to understand the role of SGD in deep learning.
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We formulate standard and multilevel Monte Carlo methods for the $k$th moment $\mathbb{M}^k_\varepsilon[\xi]$ of a Banach space valued random variable $\xi\colon\Omega\to E$, interpreted as an element of the $k$-fold injective tensor product space $\otimes^k_\varepsilon E$. For the standard Monte Carlo estimator of $\mathbb{M}^k_\varepsilon[\xi]$, we prove the $k$-independent convergence rate $1-\frac{1}{p}$ in the $L_q(\Omega;\otimes^k_\varepsilon E)$-norm, provided that (i) $\xi\in L_{kq}(\Omega;E)$ and (ii) $q\in[p,\infty)$, where $p\in[1,2]$ is the Rademacher type of $E$. By using the fact that Rademacher averages are dominated by Gaussian sums combined with a version of Slepian's inequality for Gaussian processes due to Fernique, we moreover derive corresponding results for multilevel Monte Carlo methods, including a rigorous error estimate in the $L_q(\Omega;\otimes^k_\varepsilon E)$-norm and the optimization of the computational cost for a given accuracy. Whenever the type of the Banach space $E$ is $p=2$, our findings coincide with known results for Hilbert space valued random variables. We illustrate the abstract results by three model problems: second-order elliptic PDEs with random forcing or random coefficient, and stochastic evolution equations. In these cases, the solution processes naturally take values in non-Hilbertian Banach spaces. Further applications, where physical modeling constraints impose a setting in Banach spaces of type $p<2$, are indicated.
Computer vision methods for depth estimation usually use simple camera models with idealized optics. For modern machine learning approaches, this creates an issue when attempting to train deep networks with simulated data, especially for focus-sensitive tasks like Depth-from-Focus. In this work, we investigate the domain gap caused by off-axis aberrations that will affect the decision of the best-focused frame in a focal stack. We then explore bridging this domain gap through aberration-aware training (AAT). Our approach involves a lightweight network that models lens aberrations at different positions and focus distances, which is then integrated into the conventional network training pipeline. We evaluate the generality of pretrained models on both synthetic and real-world data. Our experimental results demonstrate that the proposed AAT scheme can improve depth estimation accuracy without fine-tuning the model or modifying the network architecture.
Decentralized learning (DL) has recently employed local updates to reduce the communication cost for general non-convex optimization problems. Specifically, local updates require each node to perform multiple update steps on the parameters of the local model before communicating with others. However, most existing methods could be highly sensitive to data heterogeneity (i.e., non-iid data distribution) and adversely affected by the stochastic gradient noise. In this paper, we propose DSE-MVR to address these problems.Specifically, DSE-MVR introduces a dual-slow estimation strategy that utilizes the gradient tracking technique to estimate the global accumulated update direction for handling the data heterogeneity problem; also for stochastic noise, the method uses the mini-batch momentum-based variance-reduction technique.We theoretically prove that DSE-MVR can achieve optimal convergence results for general non-convex optimization in both iid and non-iid data distribution settings. In particular, the leading terms in the convergence rates derived by DSE-MVR are independent of the stochastic noise for large-batches or large partial average intervals (i.e., the number of local update steps). Further, we put forward DSE-SGD and theoretically justify the importance of the dual-slow estimation strategy in the data heterogeneity setting. Finally, we conduct extensive experiments to show the superiority of DSE-MVR against other state-of-the-art approaches.
Location determination finds wide applications in daily life. Instead of existing efforts devoted to localizing tourist photos captured by perspective cameras, in this article, we focus on devising person positioning solutions using overhead fisheye cameras. Such solutions are advantageous in large field of view (FOV), low cost, anti-occlusion, and unaggressive work mode (without the necessity of cameras carried by persons). However, related studies are quite scarce, due to the paucity of data. To stimulate research in this exciting area, we present LOAF, the first large-scale overhead fisheye dataset for person detection and localization. LOAF is built with many essential features, e.g., i) the data cover abundant diversities in scenes, human pose, density, and location; ii) it contains currently the largest number of annotated pedestrian, i.e., 457K bounding boxes with groundtruth location information; iii) the body-boxes are labeled as radius-aligned so as to fully address the positioning challenge. To approach localization, we build a fisheye person detection network, which exploits the fisheye distortions by a rotation-equivariant training strategy and predict radius-aligned human boxes end-to-end. Then, the actual locations of the detected persons are calculated by a numerical solution on the fisheye model and camera altitude data. Extensive experiments on LOAF validate the superiority of our fisheye detector w.r.t. previous methods, and show that our whole fisheye positioning solution is able to locate all persons in FOV with an accuracy of 0.5 m, within 0.1 s.
We study the bias of Stochastic Gradient Descent (SGD) to learn low-rank weight matrices when training deep ReLU neural networks. Our results show that training neural networks with mini-batch SGD and weight decay causes a bias towards rank minimization over the weight matrices. Specifically, we show, both theoretically and empirically, that this bias is more pronounced when using smaller batch sizes, higher learning rates, or increased weight decay. Additionally, we predict and observe empirically that weight decay is necessary to achieve this bias. In addition, we show that in the presence of intermediate neural collapse, the learned weights are particularly low-rank. Unlike previous literature, our analysis does not rely on assumptions about the data, convergence, or optimality of the weight matrices. Furthermore, it applies to a wide range of neural network architectures of any width or depth. Finally, we empirically investigate the connection between this bias and generalization, finding that it has a marginal effect on generalization.
We introduce a physics-driven deep latent variable model (PDDLVM) to learn simultaneously parameter-to-solution (forward) and solution-to-parameter (inverse) maps of parametric partial differential equations (PDEs). Our formulation leverages conventional PDE discretization techniques, deep neural networks, probabilistic modelling, and variational inference to assemble a fully probabilistic coherent framework. In the posited probabilistic model, both the forward and inverse maps are approximated as Gaussian distributions with a mean and covariance parameterized by deep neural networks. The PDE residual is assumed to be an observed random vector of value zero, hence we model it as a random vector with a zero mean and a user-prescribed covariance. The model is trained by maximizing the probability, that is the evidence or marginal likelihood, of observing a residual of zero by maximizing the evidence lower bound (ELBO). Consequently, the proposed methodology does not require any independent PDE solves and is physics-informed at training time, allowing the real-time solution of PDE forward and inverse problems after training. The proposed framework can be easily extended to seamlessly integrate observed data to solve inverse problems and to build generative models. We demonstrate the efficiency and robustness of our method on finite element discretized parametric PDE problems such as linear and nonlinear Poisson problems, elastic shells with complex 3D geometries, and time-dependent nonlinear and inhomogeneous PDEs using a physics-informed neural network (PINN) discretization. We achieve up to three orders of magnitude speed-up after training compared to traditional finite element method (FEM), while outputting coherent uncertainty estimates.
Free space optical (FSO) transmission has emerged as a key candidate technology for 6G to expand new spectrum and improve network capacity due to its advantages of large bandwidth, low electromagnetic interference, and high energy efficiency. Resonant beam operating in the infrared band utilizes spatially separated laser cavities to enable safe and mobile high-power energy and high-rate information transmission but is limited by line-of-sight (LOS) channel. In this paper, we propose a reconfigurable intelligent surface (RIS) assisted resonant beam simultaneous wireless information and power transfer (SWIPT) system and establish an optical field propagation model to analyze the channel state information (CSI), in which LOS obstruction can be detected sensitively and non-line-of-sight (NLOS) transmission can be realized by changing the phased of resonant beam in RIS. Numerical results demonstrate that, apart from the transmission distance, the NLOS performance depends on both the horizontal and vertical positions of RIS. The maximum NLOS energy efficiency can achieve 55% within a transfer distance of 10m, a translation distance of $\pm$4mm, and rotation angle of $\pm$50{\deg}.
Choice Modeling is at the core of many economics, operations, and marketing problems. In this paper, we propose a fundamental characterization of choice functions that encompasses a wide variety of extant choice models. We demonstrate how nonparametric estimators like neural nets can easily approximate such functionals and overcome the curse of dimensionality that is inherent in the non-parametric estimation of choice functions. We demonstrate through extensive simulations that our proposed functionals can flexibly capture underlying consumer behavior in a completely data-driven fashion and outperform traditional parametric models. As demand settings often exhibit endogenous features, we extend our framework to incorporate estimation under endogenous features. Further, we also describe a formal inference procedure to construct valid confidence intervals on objects of interest like price elasticity. Finally, to assess the practical applicability of our estimator, we utilize a real-world dataset from S. Berry, Levinsohn, and Pakes (1995). Our empirical analysis confirms that the estimator generates realistic and comparable own- and cross-price elasticities that are consistent with the observations reported in the existing literature.
Stochastic Gradient Descent (SGD) is one of the simplest and most popular algorithms in modern statistical and machine learning due to its computational and memory efficiency. Various averaging schemes have been proposed to accelerate the convergence of SGD in different settings. In this paper, we explore a general averaging scheme for SGD. Specifically, we establish the asymptotic normality of a broad range of weighted averaged SGD solutions and provide asymptotically valid online inference approaches. Furthermore, we propose an adaptive averaging scheme that exhibits both optimal statistical rate and favorable non-asymptotic convergence, drawing insights from the optimal weight for the linear model in terms of non-asymptotic mean squared error (MSE).
Interpretability methods are developed to understand the working mechanisms of black-box models, which is crucial to their responsible deployment. Fulfilling this goal requires both that the explanations generated by these methods are correct and that people can easily and reliably understand them. While the former has been addressed in prior work, the latter is often overlooked, resulting in informal model understanding derived from a handful of local explanations. In this paper, we introduce explanation summary (ExSum), a mathematical framework for quantifying model understanding, and propose metrics for its quality assessment. On two domains, ExSum highlights various limitations in the current practice, helps develop accurate model understanding, and reveals easily overlooked properties of the model. We also connect understandability to other properties of explanations such as human alignment, robustness, and counterfactual minimality and plausibility.
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