We propose efficient numerical schemes for implementing the natural gradient descent (NGD) for a broad range of metric spaces with applications to PDE-based optimization problems. Our technique represents the natural gradient direction as a solution to a standard least-squares problem. Hence, instead of calculating, storing, or inverting the information matrix directly, we apply efficient methods from numerical linear algebra. We treat both scenarios where the Jacobian, i.e., the derivative of the state variable with respect to the parameter, is either explicitly known or implicitly given through constraints. We can thus reliably compute several natural NGDs for a large-scale parameter space. In particular, we are able to compute Wasserstein NGD in thousands of dimensions, which was believed to be out of reach. Finally, our numerical results shed light on the qualitative differences between the standard gradient descent and various NGD methods based on different metric spaces in nonconvex optimization problems.
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Stochastic Gradient Descent-Ascent (SGDA) is one of the most prominent algorithms for solving min-max optimization and variational inequalities problems (VIP) appearing in various machine learning tasks. The success of the method led to several advanced extensions of the classical SGDA, including variants with arbitrary sampling, variance reduction, coordinate randomization, and distributed variants with compression, which were extensively studied in the literature, especially during the last few years. In this paper, we propose a unified convergence analysis that covers a large variety of stochastic gradient descent-ascent methods, which so far have required different intuitions, have different applications and have been developed separately in various communities. A key to our unified framework is a parametric assumption on the stochastic estimates. Via our general theoretical framework, we either recover the sharpest known rates for the known special cases or tighten them. Moreover, to illustrate the flexibility of our approach we develop several new variants of SGDA such as a new variance-reduced method (L-SVRGDA), new distributed methods with compression (QSGDA, DIANA-SGDA, VR-DIANA-SGDA), and a new method with coordinate randomization (SEGA-SGDA). Although variants of the new methods are known for solving minimization problems, they were never considered or analyzed for solving min-max problems and VIPs. We also demonstrate the most important properties of the new methods through extensive numerical experiments.
In this paper, we make the key delineation on the roles of resolution and statistical uncertainty in hierarchical bandits-based black-box optimization algorithms, guiding a more general analysis and a more efficient algorithm design. We introduce the \textit{optimum-statistical collaboration}, an algorithm framework of managing the interaction between optimization error flux and statistical error flux evolving in the optimization process. We provide a general analysis of this framework without specifying the forms of statistical error and uncertainty quantifier. Our framework and its analysis, due to their generality, can be applied to a large family of functions and partitions that satisfy different local smoothness assumptions and have different numbers of local optimums, which is much richer than the class of functions studied in prior works. Our framework also inspires us to propose a better measure of the statistical uncertainty and consequently a variance-adaptive algorithm \texttt{VHCT}. In theory, we prove the algorithm enjoys rate-optimal regret bounds under different local smoothness assumptions; in experiments, we show the algorithm outperforms prior efforts in different settings.
Bilevel optimization is a popular two-level hierarchical optimization, which has been widely applied to many machine learning tasks such as hyperparameter learning, meta learning and continual learning. Although many bilevel optimization methods recently have been developed, the bilevel methods are not well studied when the lower-level problem is nonconvex. To fill this gap, in the paper, we study a class of nonconvex bilevel optimization problems, which both upper-level and lower-level problems are nonconvex, and the lower-level problem satisfies Polyak-Lojasiewicz (PL) condition. We propose an efficient momentum-based gradient bilevel method (MGBiO) to solve these deterministic problems. Meanwhile, we propose a class of efficient momentum-based stochastic gradient bilevel methods (MSGBiO and VR-MSGBiO) to solve these stochastic problems. Moreover, we provide a useful convergence analysis framework for our methods. Specifically, under some mild conditions, we prove that our MGBiO method has a sample (or gradient) complexity of $O(\epsilon^{-2})$ for finding an $\epsilon$-stationary solution of the deterministic bilevel problems (i.e., $\|\nabla F(x)\|\leq \epsilon$), which improves the existing best results by a factor of $O(\epsilon^{-1})$. Meanwhile, we prove that our MSGBiO and VR-MSGBiO methods have sample complexities of $\tilde{O}(\epsilon^{-4})$ and $\tilde{O}(\epsilon^{-3})$, respectively, in finding an $\epsilon$-stationary solution of the stochastic bilevel problems (i.e., $\mathbb{E}\|\nabla F(x)\|\leq \epsilon$), which improves the existing best results by a factor of $O(\epsilon^{-3})$. This manuscript commemorates the mathematician Boris Polyak (1935 -2023).
We present adapted Zhang Neural Networks (AZNN) in which the parameter settings for the exponential decay constant $\eta$ and the length of the start-up phase of basic ZNN are adapted to the problem at hand. Specifically we study experiments with AZNN for time-varying square matrix factorizations as a product of time-varying symmetric matrices and for the time-varying matrix square roots problem. Differing from generally used small $\eta$ values and minimal start-up length phases in ZNN, we adapt the basic ZNN method to work with large or even gigantic $\eta$ settings and arbitrary length start-ups using Euler's low accuracy finite difference formula. These adaptations improve the speed of AZNN's convergence and lower its solution error bounds for our chosen problems significantly to near machine constant or even lower levels. Parameter-varying AZNN also allows us to find full rank symmetrizers of static matrices reliably, for example for the Kahan and Frank matrices and for matrices with highly ill-conditioned eigenvalues and complicated Jordan structures of dimensions from $n = 2$ on up. This helps in cases where full rank static matrix symmetrizers have never been successfully computed before.
The Cox model is an indispensable tool for time-to-event analysis, particularly in biomedical research. However, medicine is undergoing a profound transformation, generating data at an unprecedented scale, which opens new frontiers to study and understand diseases. With the wealth of data collected, new challenges for statistical inference arise, as datasets are often high dimensional, exhibit an increasing number of measurements at irregularly spaced time points, and are simply too large to fit in memory. Many current implementations for time-to-event analysis are ill-suited for these problems as inference is computationally demanding and requires access to the full data at once. Here we propose a Bayesian version for the counting process representation of Cox's partial likelihood for efficient inference on large-scale datasets with millions of data points and thousands of time-dependent covariates. Through the combination of stochastic variational inference and a reweighting of the log-likelihood, we obtain an approximation for the posterior distribution that factorizes over subsamples of the data, enabling the analysis in big data settings. Crucially, the method produces viable uncertainty estimates for large-scale and high-dimensional datasets. We show the utility of our method through a simulation study and an application to myocardial infarction in the UK Biobank.
Existing neural radiance fields (NeRF) methods for large-scale scene modeling require days of training using multiple GPUs, hindering their applications in scenarios with limited computing resources. Despite fast optimization NeRF variants have been proposed based on the explicit dense or hash grid features, their effectivenesses are mainly demonstrated in object-scale scene representation. In this paper, we point out that the low feature resolution in explicit representation is the bottleneck for large-scale unbounded scene representation. To address this problem, we introduce a new and efficient hybrid feature representation for NeRF that fuses the 3D hash-grids and high-resolution 2D dense plane features. Compared with the dense-grid representation, the resolution of a dense 2D plane can be scaled up more efficiently. Based on this hybrid representation, we propose a fast optimization NeRF variant, called GP-NeRF, that achieves better rendering results while maintaining a compact model size. Extensive experiments on multiple large-scale unbounded scene datasets show that our model can converge in 1.5 hours using a single GPU while achieving results comparable to or even better than the existing method that requires about one day's training with 8 GPUs.
This paper studies deep neural networks for solving extremely large linear systems arising from highdimensional problems. Because of the curse of dimensionality, it is expensive to store both the solution and right-hand side vector in such extremely large linear systems. Our idea is to employ a neural network to characterize the solution with much fewer parameters than the size of the solution under a matrix-free setting. We present an error analysis of the proposed method, indicating that the solution error is bounded by the condition number of the matrix and the neural network approximation error. Several numerical examples from partial differential equations, queueing problems, and probabilistic Boolean networks are presented to demonstrate that the solutions of linear systems can be learned quite accurately.
By learning the mappings between infinite function spaces using carefully designed neural networks, the operator learning methodology has exhibited significantly more efficiency than traditional methods in solving complex problems such as differential equations, but faces concerns about their accuracy and reliability. To overcomes these limitations, combined with the structures of the spectral numerical method, a general neural architecture named spectral operator learning (SOL) is introduced, and one variant called the orthogonal polynomial neural operator (OPNO), developed for PDEs with Dirichlet, Neumann and Robin boundary conditions (BCs), is proposed later. The strict BC satisfaction properties and the universal approximation capacity of the OPNO are theoretically proven. A variety of numerical experiments with physical backgrounds show that the OPNO outperforms other existing deep learning methodologies, as well as the traditional 2nd-order finite difference method (FDM) with a considerably fine mesh (with the relative errors reaching the order of 1e-6), and is up to almost 5 magnitudes faster than the traditional method.
Classic algorithms and machine learning systems like neural networks are both abundant in everyday life. While classic computer science algorithms are suitable for precise execution of exactly defined tasks such as finding the shortest path in a large graph, neural networks allow learning from data to predict the most likely answer in more complex tasks such as image classification, which cannot be reduced to an exact algorithm. To get the best of both worlds, this thesis explores combining both concepts leading to more robust, better performing, more interpretable, more computationally efficient, and more data efficient architectures. The thesis formalizes the idea of algorithmic supervision, which allows a neural network to learn from or in conjunction with an algorithm. When integrating an algorithm into a neural architecture, it is important that the algorithm is differentiable such that the architecture can be trained end-to-end and gradients can be propagated back through the algorithm in a meaningful way. To make algorithms differentiable, this thesis proposes a general method for continuously relaxing algorithms by perturbing variables and approximating the expectation value in closed form, i.e., without sampling. In addition, this thesis proposes differentiable algorithms, such as differentiable sorting networks, differentiable renderers, and differentiable logic gate networks. Finally, this thesis presents alternative training strategies for learning with algorithms.
A community reveals the features and connections of its members that are different from those in other communities in a network. Detecting communities is of great significance in network analysis. Despite the classical spectral clustering and statistical inference methods, we notice a significant development of deep learning techniques for community detection in recent years with their advantages in handling high dimensional network data. Hence, a comprehensive overview of community detection's latest progress through deep learning is timely to both academics and practitioners. This survey devises and proposes a new taxonomy covering different categories of the state-of-the-art methods, including deep learning-based models upon deep neural networks, deep nonnegative matrix factorization and deep sparse filtering. The main category, i.e., deep neural networks, is further divided into convolutional networks, graph attention networks, generative adversarial networks and autoencoders. The survey also summarizes the popular benchmark data sets, model evaluation metrics, and open-source implementations to address experimentation settings. We then discuss the practical applications of community detection in various domains and point to implementation scenarios. Finally, we outline future directions by suggesting challenging topics in this fast-growing deep learning field.
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