In the paper, we study a class of useful minimax problems on Riemanian manifolds and propose a class of effective Riemanian gradient-based methods to solve these minimax problems. Specifically, we propose an effective Riemannian gradient descent ascent (RGDA) algorithm for the deterministic minimax optimization. Moreover, we prove that our RGDA has a sample complexity of $O(\kappa^2\epsilon^{-2})$ for finding an $\epsilon$-stationary solution of the Geodesically-Nonconvex Strongly-Concave (GNSC) minimax problems, where $\kappa$ denotes the condition number. At the same time, we present an effective Riemannian stochastic gradient descent ascent (RSGDA) algorithm for the stochastic minimax optimization, which has a sample complexity of $O(\kappa^4\epsilon^{-4})$ for finding an $\epsilon$-stationary solution. To further reduce the sample complexity, we propose an accelerated Riemannian stochastic gradient descent ascent (Acc-RSGDA) algorithm based on the momentum-based variance-reduced technique. We prove that our Acc-RSGDA algorithm achieves a lower sample complexity of $\tilde{O}(\kappa^{4}\epsilon^{-3})$ in searching for an $\epsilon$-stationary solution of the GNSC minimax problems. Extensive experimental results on the robust distributional optimization and robust Deep Neural Networks (DNNs) training over Stiefel manifold demonstrate efficiency of our algorithms.

Deep learning models are known to put the privacy of their training data at risk, which poses challenges for their safe and ethical release to the public. Differentially private stochastic gradient descent is the de facto standard for training neural networks without leaking sensitive information about the training data. However, applying it to models for graph-structured data poses a novel challenge: unlike with i.i.d. data, sensitive information about a node in a graph cannot only leak through its gradients, but also through the gradients of all nodes within a larger neighborhood. In practice, this limits privacy-preserving deep learning on graphs to very shallow graph neural networks. We propose to solve this issue by training graph neural networks on disjoint subgraphs of a given training graph. We develop three random-walk-based methods for generating such disjoint subgraphs and perform a careful analysis of the data-generating distributions to provide strong privacy guarantees. Through extensive experiments, we show that our method greatly outperforms the state-of-the-art baseline on three large graphs, and matches or outperforms it on four smaller ones.

As the agent's choice is often influenced by additional covariates, also referred to as contexts, contextual bandit problems have gained renewed attention in the past decades (Woodroofe, 1979; Langford and Zhang, 2007, etc.). With the development of internet and data technology, contextual bandit algorithms play an important role in sequential decision-making applications, such as online advertisement (Li et al., 2010), precision medicine (Kim et al., 2011), e-commence (Qiang and Bayati, 2016; Chen et al., 2022), and public policy (Kasy and Sautmann, 2021). Such decisions are often referred to as recommendations, treatments, interventions, and public orders, while the rewards can be healthcare outcomes, welfare utility, revenue as well as any measure of satisfaction of decisions. Most contextual bandit algorithms are built with the goal of learning the best action under different contexts. In sequential settings, it is often formulated as minimizing the expected cumulative regret that the practitioner would have received if she knows the optimal action. While the importance of this regret minimization is undisputed, reliable uncertainty quantification of the learned decision rule is evidently important in many featured applications. For example, in a personalized medicine application where the intervention decision is to choose t''he best medical treatment to optimize some health outcome, the risk for the selected treatment plays a critical and even sometimes life-threatening role in decision-making. Such examples call for the crucial need for a valid and reliable statistical inference procedure accompanying the decision-making process to provide guidance on policy interventions. Inferential studies help not only prompt risk alerts in recommendations, but also gain scientific knowledge of questions such as the effectiveness of medicines.

This paper concerns the study of optimal (supremum and infimum) uncertainty bounds for systems where the input (or prior) probability measure is only partially/imperfectly known (e.g., with only statistical moments and/or on a coarse topology) rather than fully specified. Such partial knowledge provides constraints on the input probability measures. The theory of Optimal Uncertainty Quantification allows us to convert the task into a constraint optimization problem where one seeks to compute the least upper/greatest lower bound of the system's output uncertainties by finding the extremal probability measure of the input. Such optimization requires repeated evaluation of the system's performance indicator (input to performance map) and is high-dimensional and non-convex by nature. Therefore, it is difficult to find the optimal uncertainty bounds in practice. In this paper, we examine the use of machine learning, especially deep neural networks, to address the challenge. We achieve this by introducing a neural network classifier to approximate the performance indicator combined with the stochastic gradient descent method to solve the optimization problem. We demonstrate the learning based framework on the uncertainty quantification of the impact of magnesium alloys, which are promising light-weight structural and protective materials. Finally, we show that the approach can be used to construct maps for the performance certificate and safety design in engineering practice.

Traditional network embedding primarily focuses on learning a continuous vector representation for each node, preserving network structure and/or node content information, such that off-the-shelf machine learning algorithms can be easily applied to the vector-format node representations for network analysis. However, the learned continuous vector representations are inefficient for large-scale similarity search, which often involves finding nearest neighbors measured by distance or similarity in a continuous vector space. In this paper, we propose a search efficient binary network embedding algorithm called BinaryNE to learn a binary code for each node, by simultaneously modeling node context relations and node attribute relations through a three-layer neural network. BinaryNE learns binary node representations through a stochastic gradient descent based online learning algorithm. The learned binary encoding not only reduces memory usage to represent each node, but also allows fast bit-wise comparisons to support faster node similarity search than using Euclidean distance or other distance measures. Extensive experiments and comparisons demonstrate that BinaryNE not only delivers more than 25 times faster search speed, but also provides comparable or better search quality than traditional continuous vector based network embedding methods. The binary codes learned by BinaryNE also render competitive performance on node classification and node clustering tasks. The source code of this paper is available at https://github.com/daokunzhang/BinaryNE.

In the recent years, various gradient descent algorithms including the methods of gradient descent, gradient descent with momentum, adaptive gradient (AdaGrad), root-mean-square propagation (RMSProp) and adaptive moment estimation (Adam) have been applied to the parameter optimization of several deep learning models with higher accuracies or lower errors. These optimization algorithms may need to set the values of several hyperparameters which include a learning rate, momentum coefficients, etc. Furthermore, the convergence speed and solution accuracy may be influenced by the values of hyperparameters. Therefore, this study proposes an analytical framework to use mathematical models for analyzing the mean error of each objective function based on various gradient descent algorithms. Moreover, the suitable value of each hyperparameter could be determined by minimizing the mean error. The principles of hyperparameter value setting have been generalized based on analysis results for model optimization. The experimental results show that higher efficiency convergences and lower errors can be obtained by the proposed method.

CS 221 ― Artificial Intelligence My twin brother Afshine and I created this set of illustrated Artificial Intelligence cheatsheets covering the content of the CS 221 class, which I TA-ed in Spring 2019 at Stanford. They can (hopefully!) be useful to all future students of this course as well as to anyone else interested in Artificial Intelligence. You can help us translating them on GitHub!

To lower the communication complexity of federated min-max learning, a natural approach is to utilize the idea of infrequent communications (through multiple local updates) same as in conventional federated learning. However, due to the more complicated inter-outer problem structure in federated min-max learning, theoretical understandings of communication complexity for federated min-max learning with infrequent communications remain very limited in the literature. This is particularly true for settings with non-i.i.d. datasets and partial client participation. To address this challenge, in this paper, we propose a new algorithmic framework called stochastic sampling averaging gradient descent ascent (SAGDA), which i) assembles stochastic gradient estimators from randomly sampled clients as control variates and ii) leverages two learning rates on both server and client sides. We show that SAGDA achieves a linear speedup in terms of both the number of clients and local update steps, which yields an $\mathcal{O}(\epsilon^{-2})$ communication complexity that is orders of magnitude lower than the state of the art. Interestingly, by noting that the standard federated stochastic gradient descent ascent (FSGDA) is in fact a control-variate-free special version of SAGDA, we immediately arrive at an $\mathcal{O}(\epsilon^{-2})$ communication complexity result for FSGDA. Therefore, through the lens of SAGDA, we also advance the current understanding on communication complexity of the standard FSGDA method for federated min-max learning.

An oft-cited open problem of federated learning is the existence of data heterogeneity at the clients. One pathway to understanding the drastic accuracy drop in federated learning is by scrutinizing the behavior of the clients' deep models on data with different levels of "difficulty", which has been left unaddressed. In this paper, we investigate a different and rarely studied dimension of FL: ordered learning. Specifically, we aim to investigate how ordered learning principles can contribute to alleviating the heterogeneity effects in FL. We present theoretical analysis and conduct extensive empirical studies on the efficacy of orderings spanning three kinds of learning: curriculum, anti-curriculum, and random curriculum. We find that curriculum learning largely alleviates non-IIDness. Interestingly, the more disparate the data distributions across clients the more they benefit from ordered learning. We provide analysis explaining this phenomenon, specifically indicating how curriculum training appears to make the objective landscape progressively less convex, suggesting fast converging iterations at the beginning of the training procedure. We derive quantitative results of convergence for both convex and nonconvex objectives by modeling the curriculum training on federated devices as local SGD with locally biased stochastic gradients. Also, inspired by ordered learning, we propose a novel client selection technique that benefits from the real-world disparity in the clients. Our proposed approach to client selection has a synergic effect when applied together with ordered learning in FL.

The Information Bottleneck theory provides a theoretical and computational framework for finding approximate minimum sufficient statistics. Analysis of the Stochastic Gradient Descent (SGD) training of a neural network on a toy problem has shown the existence of two phases, fitting and compression. In this work, we analyze the SGD training process of a Deep Neural Network on MNIST classification and confirm the existence of two phases of SGD training. We also propose a setup for estimating the mutual information for a Deep Neural Network through Variational Inference.