Multivariate time series (MTS) analysis prevails in real-world applications such as finance, climate science and healthcare. The various self-attention mechanisms, the backbone of the state-of-the-art Transformer-based models, efficiently discover the temporal dependencies, yet cannot well capture the intricate cross-correlation between different features of MTS data, which inherently stems from complex dynamical systems in practice. To this end, we propose a novel correlated attention mechanism, which not only efficiently captures feature-wise dependencies, but can also be seamlessly integrated within the encoder blocks of existing well-known Transformers to gain efficiency improvement. In particular, correlated attention operates across feature channels to compute cross-covariance matrices between queries and keys with different lag values, and selectively aggregate representations at the sub-series level. This architecture automates representation learning of not only instantaneous but also lagged cross-correlations, while inherently capturing time series auto-correlation. When combined with prevalent Transformer baselines, correlated attention mechanism constitutes a better alternative for encoder-only architectures and achieves state-of-the-art results in imputation and classification.

Dynamic Time Warping (DTW) is a widely used metric for time series matching. However, when applied to few-shot action recognition (FSAR), DTW often encounters the "identical matching" issue, where multiple frames from one video are matched to a single frame from another. To address this, we introduce FTP-FSAR, a novel metric-based FSAR approach designed to mitigate this challenge. FTP-FSAR proposes an innovative alignment metric that incorporates temporal priors, guiding the matching process by encouraging the alignment of frames with similar temporal progression, thus improving the accuracy of frame matching. Additionally, FTP-FSAR integrates a dual framework, combining a foundation model with transductive learning to optimize feature extraction. Extensive experiments across multiple datasets demonstrate that FTP-FSAR outperforms existing methods, achieving the best results in 3 out of 4 benchmarks across 1-shot, 3-shot, and 5-shot settings, with performance improvements of up to 4.5%.

The state-of-the-art (SOTA) deep learning based time series models are inspired by convolutional neural networks (CNN), recurrent neural networks (RNN) or transformers which are successful architectures for domains like vision, text, etc. However, the gold standard architecture for time series modeling is not yet established. In this paper, we propose a new neural network structure that can be used as a strong baseline for time series problems, leveraging dilated kernels with fully convolutional networks (FCNs). The proposed model, called the dilated multi-kernel fully convolutional network (DM-FCN), is a composite model that leverages a vast receptive field and is designed to capture the long-distance interaction in multivariate time series data. We evaluate the performance of the DM-FCN model on a variety of time series benchmarks. Our results show that the baseline DM-FCN model outperforms state-of-the-art models on many of the benchmarks by a large margin. By integrating statistical insights, we also evaluated different variations of DM-FCN and deliberated on model selections across diverse time series data.

This paper studies the Partial Optimal Transport (POT) problem between two unbalanced measures with at most n supports and its applications in various AI tasks such as color transfer or domain adaptation. There is hence a need for fast approximations of POT with increasingly large problem sizes in arising applications. We first theoretically and experimentally investigate the infeasibility of the state-of-the-art Sinkhorn algorithm for POT, which consequently degrades its qualitative performance in real world applications like point-cloud registration. To this end, we propose a novel rounding algorithm for POT, and then provide a feasible Sinkhorn procedure with a revised computation complexity of O(n^2/epsilon^4). Our rounding algorithm also permits the development of two first-order methods to approximate the POT problem. The first algorithm, Adaptive Primal-Dual Accelerated Gradient Descent (APDAGD), finds an epsilon-approximate solution to the POT problem in O(n^2.5/epsilon). The second method, Dual Extrapolation, achieves the computation complexity of O(n^2/epsilon), thereby being the best in the literature. We further demonstrate the flexibility of POT compared to standard OT as well as the practicality of our algorithms on real applications where two marginal distributions are unbalanced.

Neural networks are powerful tools in various applications, and quantifying their uncertainty is crucial for reliable decision-making. In the deep learning field, the uncertainties are usually categorized into aleatoric (data) and epistemic (model) uncertainty. In this paper, we point out that the existing popular variance attenuation method highly overestimates aleatoric uncertainty. To address this issue, we proposed a new estimation method by actively de-noising the observed data. By conducting a broad range of experiments, we demonstrate that our proposed approach provides a much closer approximation to the actual data uncertainty than the standard method.

Foundational models (FMs), pretrained on extensive datasets using self-supervised techniques, are capable of learning generalized patterns from large amounts of data. This reduces the need for extensive labeled datasets for each new task, saving both time and resources by leveraging the broad knowledge base established during pretraining. Most research on FMs has primarily focused on unstructured data, such as text and images, or semi-structured data, like time-series. However, there has been limited attention to structured data, such as tabular data, which, despite its prevalence, remains under-studied due to a lack of clean datasets and insufficient research on the transferability of FMs for various tabular data tasks. In response to this gap, we introduce a framework called TabularFM1, which incorporates state-of-the-art methods for developing FMs specifically for tabular data. This includes variations of neural architectures such as GANs, VAEs, and Transformers. We have curated a thousand tabular datasets and released cleaned versions to facilitate the development of tabular FMs. We pretrained FMs on this curated data, benchmarked various learning methods on these datasets, and released the pretrained models along with leaderboards for future comparative studies. Our fully open-sourced system provides a comprehensive analysis of the transferability of tabular FMs.

In this work, we investigate the time series representation learning problem using self-supervised techniques. Contrastive learning is well-known in this area as it is a powerful method for extracting information from the series and generating task-appropriate representations. Despite its proficiency in capturing time series characteristics, these techniques often overlook a critical factor - the inherent noise in this type of data, a consideration usually emphasized in general time series analysis. Moreover, there is a notable absence of attention to developing efficient yet lightweight encoder architectures, with an undue focus on delivering contrastive losses. Our work address these gaps by proposing an innovative training strategy that promotes consistent representation learning, accounting for the presence of noise-prone signals in natural time series. Furthermore, we propose an encoder architecture that incorporates dilated convolution within the Inception block, resulting in a scalable and robust network with a wide receptive field. Experimental findings underscore the effectiveness of our method, consistently outperforming state-of-the-art approaches across various tasks, including forecasting, classification, and abnormality detection. Notably, our method attains the top rank in over two-thirds of the classification UCR datasets, utilizing only 40% of the parameters compared to the second-best approach.

We introduce a multiple target optimization framework for DP-SGD referred to as pro-active DP. In contrast to traditional DP accountants, which are used to track the expenditure of privacy budgets, the pro-active DP scheme allows one to *a-priori* select parameters of DP-SGD based on a fixed privacy budget (in terms of $\epsilon$ and $\delta$) in such a way to optimize the anticipated utility (test accuracy) the most. To achieve this objective, we first propose significant improvements to the moment account method, presenting a closed-form $(\epsilon,\delta)$-DP guarantee that connects all parameters in the DP-SGD setup. Generally, DP-SGD is $(\epsilon\leq 1/2,\delta=1/N)$-DP if $\sigma=\sqrt{2(\epsilon +\ln(1/\delta))/\epsilon}$ with $T$ at least $\approx 2k^2/\epsilon$ and $(2/e)^2k^2-1/2\geq \ln(N)$, where $T$ is the total number of rounds, and $K=kN$ is the total number of gradient computations where $k$ measures $K$ in number of epochs of size $N$ of the local data set. We prove that our expression is close to tight in that if $T$ is more than a constant factor $\approx 4$ smaller than the lower bound $\approx 2k^2/\epsilon$, then the $(\epsilon,\delta)$-DP guarantee is violated. Our enhanced DP theory allows us to create a utility graph and DP calculator. These tools link privacy and utility objectives and search for optimal experiment setups, efficiently taking into account both accuracy and privacy objectives, as well as implementation goals. We furnish a comprehensive implementation flow of our proactive DP, with rigorous experiments to showcase the proof-of-concept.

This paper aims at developing novel shuffling gradient-based methods for tackling two classes of minimax problems: nonconvex-linear and nonconvex-strongly concave settings. The first algorithm addresses the nonconvex-linear minimax model and achieves the state-of-the-art oracle complexity typically observed in nonconvex optimization. It also employs a new shuffling estimator for the ``hyper-gradient'', departing from standard shuffling techniques in optimization. The second method consists of two variants: semi-shuffling and full-shuffling schemes. These variants tackle the nonconvex-strongly concave minimax setting. We establish their oracle complexity bounds under standard assumptions, which, to our best knowledge, are the best-known for this specific setting. Numerical examples demonstrate the performance of our algorithms and compare them with two other methods. Our results show that the new methods achieve comparable performance with SGD, supporting the potential of incorporating shuffling strategies into minimax algorithms.

In time-series analysis, many recent works seek to provide a unified view and representation for time-series across multiple domains, leading to the development of foundation models for time-series data. Despite diverse modeling techniques, existing models are black boxes and fail to provide insights and explanations about their representations. In this paper, we present VQShape, a pre-trained, generalizable, and interpretable model for time-series representation learning and classification. By introducing a novel representation for time-series data, we forge a connection between the latent space of VQShape and shape-level features. Using vector quantization, we show that time-series from different domains can be described using a unified set of low-dimensional codes, where each code can be represented as an abstracted shape in the time domain. On classification tasks, we show that the representations of VQShape can be utilized to build interpretable classifiers, achieving comparable performance to specialist models. Additionally, in zero-shot learning, VQShape and its codebook can generalize to previously unseen datasets and domains that are not included in the pre-training process. The code and pre-trained weights are available at https://github.com/YunshiWen/VQShape.

Fine-tuning pre-trained models is a popular approach in machine learning for solving complex tasks with moderate data. However, fine-tuning the entire pre-trained model is ineffective in federated data scenarios where local data distributions are diversely skewed. To address this, we explore integrating federated learning with a more effective prompt-tuning method, optimizing for a small set of input prefixes to reprogram the pre-trained model's behavior. Our approach transforms federated learning into a distributed set modeling task, aggregating diverse sets of prompts to globally fine-tune the pre-trained model. We benchmark various baselines based on direct adaptations of existing federated model aggregation techniques and introduce a new probabilistic prompt aggregation method that substantially outperforms these baselines. Our reported results on a variety of computer vision datasets confirm that the proposed method is most effective to combat extreme data heterogeneity in federated learning.

In numerous machine learning applications, there is a preference for classifiers characterized by a polyhedral description, as they are intended for utilization within optimization frameworks or for interpretability purposes. Here, we present a structured classifier designed to cater to downstream decision-making tasks. The classification method is achieved through the process of partitioning the feature domain into clusters and encompassing each cluster within a polytope. We employ a combined approach that integrates semi-supervised k-means with SVM. This unified optimization framework enables the simultaneous generation of multiple polytopes. The central concept involves using a k-means-based clustering method for the clustering step, followed by the utilization of SVM to construct hyperplanes between each pair of clusters. Notably, the clustering process for each class considers classification loss as well as information from other classes when allocating sample points to clusters. We propose an algorithm to solve the integer program. Our numerical experiments demonstrate the competitiveness of the proposed method across a wide spectrum of datasets, exhibiting its efficacy in comparison to existing hyperplane-based classifiers and nonlinear classifiers.

Federated learning (FL) is the distributed machine learning framework that enables collaborative training across multiple parties while ensuring data privacy. Practical adaptation of XGBoost, the state-of-the-art tree boosting framework, to FL remains nascent due to high cost incurred by conventional privacy-preserving methods. Such limitations can be attributed to the lack of formal analytical model to enable new privacy methods well customized to federated XGBoost. To this end, we propose a novel formulation, termed splitting matrix, in the context of federated XGBoost that mathematically characterizes the role of passive party (PP) having been neglected in the literature. This new formulation facilitates our novel adoption of secure matrix multiplication protocol into federated XGBoost to propose FedXGBoost as a framework for secure XGBoost in federated setting with lossless accuracy and negligible overhead. Extensive experiments on both synthetic and real datasets exhibit our algorithm’s empirical outperformance over known methods in the literature.

Randomized smoothing has recently attracted attentions in the field of adversarial robustness to provide provable robustness guarantees on smoothed neural network classifiers. However, existing works show that vanillarandomized smoothing usually does not provide good robustness performance and often requires (re)training techniques on the base classifier in order to boost the robustness of the resulting smoothed classifier. In this work, we propose two cost-effective approaches to boost the robustness of randomized smoothing while preserving its clean performance. The first approach introduces a new robust training method AdvMacer which combines adversarial training and robustness certification maximization for randomized smoothing. We show that AdvMacer can improve the robustness performance of randomized smoothing classifiers compared to SOTA baselines, while being 3x faster to train than MACER baseline. The second approach introduces a post-processing method EsbRS which greatly improves the robustness certificate based on building model ensembles. Extensive experiments verify the superior performance of our methods on various datasets. Our code is available at https://github.com/Trustworthy-ML-Lab/AdvMacer_and_EsbRS.

In recent years, a proliferation of methods were developed for cooperative multi-agent reinforcement learning (c-MARL). However, the robustness of c-MARL agents against adversarial attacks has been rarely explored. In this paper, we propose to evaluate the robustness of c-MARL agents via a model-based approach, named c-MBA. Our proposed formulation can craft much stronger adversarial state perturbations of c-MARL agents to lower total team rewards than existing model-free approaches. In addition, we propose the first victim-agent selection strategy and the first data-driven approach to define targeted failure states where each of them allows us to develop even stronger adversarial attack without the expert knowledge to the underlying environment. Our numerical experiments on two representative MARL benchmarks illustrate the advantage of our approach over other baselines: our model-based attack consistently outperforms other baselines in all tested environments.

Deep neural networks (DNN) have shown great capacity of modeling a dynamical system; nevertheless, they usually do not obey physics constraints such as conservation laws. This paper proposes a new learning framework named $\textbf{ConCerNet}$ to improve the trustworthiness of the DNN based dynamics modeling to endow the invariant properties. $\textbf{ConCerNet}$ consists of two steps: (i) a contrastive learning method to automatically capture the system invariants (i.e. conservation properties) along the trajectory observations; (ii) a neural projection layer to guarantee that the learned dynamics models preserve the learned invariants. We theoretically prove the functional relationship between the learned latent representation and the unknown system invariant function. Experiments show that our method consistently outperforms the baseline neural networks in both coordinate error and conservation metrics by a large margin. With neural network based parameterization and no dependence on prior knowledge, our method can be extended to complex and large-scale dynamics by leveraging an autoencoder.

Deep neural networks have been increasingly used in real-world applications, making it critical to ensure their ability to adapt to new, unseen data. In this paper, we study the generalization capability of neural networks trained with (stochastic) gradient flow. We establish a new connection between the loss dynamics of gradient flow and general kernel machines by proposing a new kernel, called loss path kernel. This kernel measures the similarity between two data points by evaluating the agreement between loss gradients along the path determined by the gradient flow. Based on this connection, we derive a new generalization upper bound that applies to general neural network architectures. This new bound is tight and strongly correlated with the true generalization error. We apply our results to guide the design of neural architecture search (NAS) and demonstrate favorable performance compared with state-of-the-art NAS algorithms through numerical experiments.

Stochastic gradient descent (SGD) algorithm is the method of choice in many machine learning tasks thanks to its scalability and efficiency in dealing with large-scale problems. In this paper, we focus on the shuffling version of SGD which matches the mainstream practical heuristics. We show the convergence to a global solution of shuffling SGD for a class of non-convex functions under over-parameterized settings. Our analysis employs more relaxed non-convex assumptions than previous literature. Nevertheless, we maintain the desired computational complexity as shuffling SGD has achieved in the general convex setting.

We study the Unbalanced Optimal Transport (UOT) between two measures of possibly different masses with at most $n$ components, where the marginal constraints of standard Optimal Transport (OT) are relaxed via Kullback-Leibler divergence with regularization factor $\tau$. Although only Sinkhorn-based UOT solvers have been analyzed in the literature with the iteration complexity of ${O}\big(\tfrac{\tau \log(n)}{\varepsilon} \log\big(\tfrac{\log(n)}{{\varepsilon}}\big)\big)$ and per-iteration cost of $O(n^2)$ for achieving the desired error $\varepsilon$, their positively dense output transportation plans strongly hinder the practicality. On the other hand, while being vastly used as heuristics for computing UOT in modern deep learning applications and having shown success in sparse OT problem, gradient methods applied to UOT have not been formally studied. In this paper, we propose a novel algorithm based on Gradient Extrapolation Method (GEM-UOT) to find an $\varepsilon$-approximate solution to the UOT problem in $O\big( \kappa \log\big(\frac{\tau n}{\varepsilon}\big) \big)$ iterations with $\widetilde{O}(n^2)$ per-iteration cost, where $\kappa$ is the condition number depending on only the two input measures. Our proof technique is based on a novel dual formulation of the squared $\ell_2$-norm UOT objective, which fills the lack of sparse UOT literature and also leads to a new characterization of approximation error between UOT and OT. To this end, we further present a novel approach of OT retrieval from UOT, which is based on GEM-UOT with fine tuned $\tau$ and a post-process projection step. Extensive experiments on synthetic and real datasets validate our theories and demonstrate the favorable performance of our methods in practice. We showcase GEM-UOT on the task of color transfer in terms of both the quality of the transfer image and the sparsity of the transportation plan.

Clustering is a popular unsupervised learning tool often used to discover groups within a larger population such as customer segments, or patient subtypes. However, despite its use as a tool for subgroup discovery and description few state-of-the-art algorithms provide any rationale or description behind the clusters found. We propose a novel approach for interpretable clustering that both clusters data points and constructs polytopes around the discovered clusters to explain them. Our framework allows for additional constraints on the polytopes including ensuring that the hyperplanes constructing the polytope are axis-parallel or sparse with integer coefficients. We formulate the problem of constructing clusters via polytopes as a Mixed-Integer Non-Linear Program (MINLP). To solve our formulation we propose a two phase approach where we first initialize clusters and polytopes using alternating minimization, and then use coordinate descent to boost clustering performance. We benchmark our approach on a suite of synthetic and real world clustering problems, where our algorithm outperforms state of the art interpretable and non-interpretable clustering algorithms.

In this paper, we propose Nesterov Accelerated Shuffling Gradient (NASG), a new algorithm for the convex finite-sum minimization problems. Our method integrates the traditional Nesterov’s acceleration momentum with different shuffling sampling schemes. We show that our algorithm has an improved rate of $\Ocal(1/T)$ using unified shuffling schemes, where $T$ is the number of epochs. This rate is better than that of any other shuffling gradient methods in convex regime. Our convergence analysis does not require an assumption on bounded domain or a bounded gradient condition. For randomized shuffling schemes, we improve the convergence bound further. When employing some initial condition, we show that our method converges faster near the small neighborhood of the solution. Numerical simulations demonstrate the efficiency of our algorithm.

In this paper, we propose the StepDIRECT algorithm for derivative-free optimization (DFO), in which the black-box objective function has a stepwise landscape. Our framework is based on the well-known DIRECT algorithm. By incorporating the local variability to explore the flatness, we provide a new criterion to select the potentially optimal hyper-rectangles. In addition, we introduce a stochastic local search algorithm performing on potentially optimal hyper-rectangles to improve the solution quality and convergence speed. Global convergence of the StepDIRECT algorithm is provided. Numerical experiments on optimization for random forest models and hyper-parameter tuning are presented to support the efficacy of our algorithm. The proposed StepDIRECT algorithm shows competitive performance results compared with other state-of-the-art baseline DFO methods including the original DIRECT algorithm.

Hogwild! implements asynchronous Stochastic Gradient Descent (SGD) where multiple threads in parallel access a common repository containing training data, perform SGD iterations and update shared state that represents a jointly learned (global) model. We consider big data analysis where training data is distributed among local data sets in a heterogeneous way – and we wish to move SGD computations to local compute nodes where local data resides. The results of these local SGD computations are aggregated by a central “aggregator” which mimics Hogwild!. We show how local compute nodes can start choosing small mini-batch sizes which increase to larger ones in order to reduce communication cost (round interaction with the aggregator). We improve state-of-the-art literature and show O(K^{0.5}) communication rounds for heterogeneous data for strongly convex problems, where K is the total number of gradient computations across all local compute nodes. For our scheme, we prove a tight and novel non-trivial convergence analysis for strongly convex problems for heterogeneous data which does not use the bounded gradient assumption as seen in many existing publications. The tightness is a consequence of our proofs for lower and upper bounds of the convergence rate, which show a constant factor difference. We show experimental results for plain convex and non-convex problems for biased (i.e., heterogeneous) and unbiased local data sets.

We combine two advanced ideas widely used in optimization for machine learning: \textit{shuffling} strategy and \textit{momentum} technique to develop a novel shuffling gradient-based method with momentum, coined \textbf{S}huffling \textbf{M}omentum \textbf{G}radient (SMG), for non-convex finite-sum optimization problems. While our method is inspired by momentum techniques, its update is fundamentally different from existing momentum-based methods. We establish state-of-the-art convergence rates of SMG for any shuffling strategy using either constant or diminishing learning rate under standard assumptions (i.e. \textit{$L$-smoothness} and \textit{bounded variance}). When the shuffling strategy is fixed, we develop another new algorithm that is similar to existing momentum methods, and prove the same convergence rates for this algorithm under the $L$-smoothness and bounded gradient assumptions. We demonstrate our algorithms via numerical simulations on standard datasets and compare them with existing shuffling methods. Our tests have shown encouraging performance of the new algorithms.

Recent research shows that the dynamics of an infinitely wide neural network (NN) trained by gradient descent can be characterized by Neural Tangent Kernel (NTK) \citep{jacot2018neural}. Under the squared loss, the infinite-width NN trained by gradient descent with an infinitely small learning rate is equivalent to kernel regression with NTK \citep{arora2019exact}. However, the equivalence is only known for ridge regression currently \citep{arora2019harnessing}, while the equivalence between NN and other kernel machines (KMs), e.g. support vector machine (SVM), remains unknown. Therefore, in this work, we propose to establish the equivalence between NN and SVM, and specifically, the infinitely wide NN trained by soft margin loss and the standard soft margin SVM with NTK trained by subgradient descent. Our main theoretical results include establishing the equivalence between NN and a broad family of $\ell_2$ regularized KMs with finite-width bounds, which cannot be handled by prior work, and showing that every finite-width NN trained by such regularized loss functions is approximately a KM. Furthermore, we demonstrate our theory can enable three practical applications, including (i) \textit{non-vacuous} generalization bound of NN via the corresponding KM; (ii) \textit{nontrivial} robustness certificate for the infinite-width NN (while existing robustness verification methods would provide vacuous bounds); (iii) intrinsically more robust infinite-width NNs than those from previous kernel regression.

In many machine learning tasks, models are trained to predict structure data such as graphs. For example, in natural language processing, it is very common to parse texts into dependency trees or abstract meaning representation (AMR) graphs. On the other hand, ensemble methods combine predictions from multiple models to create a new one that is more robust and accurate than individual predictions. In the literature, there are many ensembling techniques proposed for classification or regression problems, however, ensemble graph prediction has not been studied thoroughly. In this work, we formalize this problem as mining the largest graph that is the most supported by a collection of graph predictions. As the problem is NP-Hard, we propose an efficient heuristic algorithm to approximate the optimal solution. To validate our approach, we carried out experiments in AMR parsing problems. The experimental results demonstrate that the proposed approach can combine the strength of state-of-the-art AMR parsers to create new predictions that are more accurate than any individual models in five standard benchmark datasets.

We develop two new algorithms, called, FedDR and asyncFedDR, for solving a fundamental nonconvex composite optimization problem in federated learning. Our algorithms rely on a novel combination between a nonconvex Douglas-Rachford splitting method, randomized block-coordinate strategies, and asynchronous im- plementation. They can also handle convex regularizers. Unlike recent methods in the literature, e.g., FedSplit and FedPD, our algorithms update only a subset of users at each communication round, and possibly in an asynchronous manner, making them more practical. These new algorithms can handle statistical and sys- tem heterogeneity, which are the two main challenges in federated learning, while achieving the best known communication complexity. In fact, our new algorithms match the communication complexity lower bound up to a constant factor under standard assumptions. Our numerical experiments illustrate the advantages of our methods over existing algorithms on synthetic and real datasets.

In this paper, we propose a unified convergence analysis for a class of generic shuffling-type gradient methods for solving finite-sum optimization problems. Our analysis works with any sampling without replacement strategy and covers many known variants such as randomized reshuffling, deterministic or randomized single permutation, and cyclic and incremental gradient schemes. We focus on two different settings: strongly convex and nonconvex problems, but also discuss the non-strongly convex case. Our main contribution consists of new non-asymptotic and asymptotic convergence rates for a wide class of shuffling-type gradient methods in both nonconvex and convex settings. We also study uniformly randomized shuffling variants with different learning rates and model assumptions. While our rate in the nonconvex case is new and significantly improved over existing works under standard assumptions, the rate on the strongly convex one matches the existing best-known rates prior to this paper up to a constant factor without imposing a bounded gradient condition. Finally, we empirically illustrate our theoretical results via two numerical examples: nonconvex logistic regression and neural network training examples. As byproducts, our results suggest some appropriate choices for diminishing learning rates in certain shuffling variants.

We propose a novel hybrid stochastic policy gradient estimator by combining an unbiased policy gradient estimator, the REINFORCE estimator, with another biased one, an adapted SARAH estimator for policy optimization. The hybrid policy gradient estimator is shown to be biased, but has variance reduced property. Using this estimator, we develop a new Proximal Hybrid Stochastic Policy Gradient Algorithm (ProxHSPGA) to solve a composite policy optimization problem that allows us to handle constraints or regularizers on the policy parameters. We first propose a single-looped algorithm then introduce a more practical restarting variant. We prove that both algorithms can achieve the best-known trajectory complexity to attain a first-order stationary point for the composite problem which is better than existing REINFORCE/GPOMDP and SVRPG in the non-composite setting. We evaluate the performance of our algorithm on several well-known examples in reinforcement learning. Numerical results show that our algorithm outperforms two existing methods on these examples. Moreover, the composite settings indeed have some advantages compared to the non-composite ones on certain problems.

Deep neural networks (DNNs) give state-of-the-art accuracy in many tasks, but they can require large amounts of memory storage, energy consumption, and long inference times. Modern DNNs can have hundreds of million parameters, which make it difficult for DNNs to be deployed in some applications with low-resource environments. Pruning redundant connections without sacrificing accuracy is one of popular approaches to overcome these limitations. We propose two l0-constrained optimization models for pruning deep neural networks layer-by-layer. The first model is devoted to a general activation function, while the second one is specifically for a ReLU. We introduce an efficient cutting plane algorithm to solve the latter to optimality. Our experiments show that the proposed approach achieves competitive compression rates over several state-of-the-art baseline methods.

We develop two new stochastic Gauss-Newton algorithms for solving a class of non-convex stochastic compositional optimization problems frequently arising in practice. We consider both the expectation and finite-sum settings under standard assumptions, and use both classical stochastic and SARAH estimators for approximating function values and Jacobians. In the expectation case, we establish $\BigO{\varepsilon^{-2}}$ iteration-complexity to achieve a stationary point in expectation and estimate the total number of stochastic oracle calls for both function value and its Jacobian, where $\varepsilon$ is a desired accuracy. In the finite sum case, we also estimate $\BigO{\varepsilon^{-2}}$ iteration-complexity and the total oracle calls with high probability. To our best knowledge, this is the first time such global stochastic oracle complexity is established for stochastic Gauss-Newton methods. Finally, we illustrate our theoretical results via two numerical examples on both synthetic and real datasets.

We develop a novel and single-loop variance-reduced algorithm to solve a class of stochastic nonconvex-convex minimax problems involving a nonconvex-linear objective function, which has various applications in different fields such as ma- chine learning and robust optimization. This problem class has several compu- tational challenges due to its nonsmoothness, nonconvexity, nonlinearity, and non-separability of the objective functions. Our approach relies on a new combi- nation of recent ideas, including smoothing and hybrid biased variance-reduced techniques. Our algorithm and its variants can achieve $\mathcal{O}(T^{-2/3})$-convergence rate and the best-known oracle complexity under standard assumptions, where T is the iteration counter. They have several computational advantages compared to exist- ing methods such as simple to implement and less parameter tuning requirements. They can also work with both single sample or mini-batch on derivative estimators, and with constant or diminishing step-sizes. We demonstrate the benefits of our algorithms over existing methods through two numerical examples, including a nonsmooth and nonconvex-non-strongly concave minimax model.

Several recent publications report advances in training optimal decision trees (ODTs) using mixed-integer programs (MIPs), due to algorithmic advances in integer programming and a growing interest in addressing the inherent suboptimality of heuristic approaches such as CART. In this paper, we propose a novel MIP formulation, based on 1-norm support vector machine model, to train a binary oblique ODT for classification problems. We further present techniques, such as cutting planes, to tighten its linear relaxation, to improve run times to reach optimality. Using 36 datasets from the University of California Irvine Machine Learning Repository, we demonstrate that our training approach outperforms its counterparts from literature in terms of out-of-sample performance (around 10% improvement in mean out-of-sample testing accuracy). Towards our goal of developing a scalable framework to train multivariate ODT on large datasets, we propose a new linear programming based data selection method to choose a subset of the data, and use it to train a decision tree through our proposed MIP model. We conclude this paper with extensive numerical testing results, that showcase the generalization performance of our new MIP formulation, and the improvement in mean out-of-sample accuracy on large datasets.

We propose a new stochastic first-order algorithmic framework to solve stochastic composite nonconvex optimization problems that covers both finite-sum and expectation settings. Our algorithms rely on the SARAH estimator and consist of two steps: a proximal gradient and an averaging step making them different from existing nonconvex proximal-type algorithms. The algorithms only require an average smoothness assumption of the nonconvex objective term and additional bounded variance assumption if applied to expectation problems. They work with both constant and dynamic step-sizes, while allowing single sample and mini-batches. In all these cases, we prove that our algorithms can achieve the best-known complexity bounds in terms of stochastic first-order oracle. One key step of our methods is the new constant and dynamic step-sizes resulting in the desired complexity bounds while improving practical performance. Our constant step-size is much larger than existing methods including proximal SVRG scheme in the single sample case. We also specify our framework to the non-composite case that covers existing state-of-the-arts in terms of oracle complexity bounds. Our update also allows one to trade-off between step-sizes and mini-batch sizes to improve performance. We test the proposed algorithms on two composite nonconvex problems and neural networks using several well-known data sets.

We study Stochastic Gradient Descent (SGD) with diminishing step sizes for convex objective functions. We introduce a definitional framework and theory that defines and characterizes a core property, called curvature, of convex objective functions. In terms of curvature we can derive a new inequality that can be used to compute an optimal sequence of diminishing step sizes by solving a differential equation. Our exact solutions confirm known results in literature and allows us to fully characterize a new regularizer with its corresponding expected convergence rates.

We propose a novel framework PROVEN to \textbf{PRO}babilistically \textbf{VE}rify \textbf{N}eural network’s robustness with statistical guarantees. PROVEN provides probability certificates of neural network robustness when the input perturbation follow distributional characterization. Notably, PROVEN is derived from current state-of-the-art worst-case neural network robustness verification frameworks, and therefore it can provide probability certificates with little computational overhead on top of existing methods such as Fast-Lin, CROWN and CNN-Cert. Experiments on small and large MNIST and CIFAR neural network models demonstrate our probabilistic approach can tighten up robustness certificate to around $1.8 \times$ and $3.5 \times$ with at least a $99.99%$ confidence compared with the worst-case robustness certificate by CROWN and CNN-Cert.

We study the convergence of Stochastic Gradient Descent (SGD) for strongly convex objective functions. We prove for all $t$ a lower bound on the expected convergence rate after the $t$-th SGD iteration; the lower bound is over all possible sequences of diminishing step sizes. It implies that recently proposed sequences of step sizes at ICML 2018 and ICML 2019 are {\em universally} close to optimal in that the expected convergence rate after {\em each} iteration is within a factor $32$ of our lower bound. This factor is independent of dimension $d$. We offer a framework for comparing with lower bounds in state-of-the-art literature and when applied to SGD for strongly convex objective functions our lower bound is a significant factor $775\cdot d$ larger compared to existing work.

The classical convergence analysis of SGD is carried out under the assumption that the norm of the stochastic gradient is uniformly bounded. While this might hold for some loss functions, it is violated for cases where the objective function is strongly convex. In Bottou et al. (2018), a new analysis of convergence of SGD is performed under the assumption that stochastic gradients are bounded with respect to the true gradient norm. We show that for stochastic problems arising in machine learning such bound always holds; and we also propose an alternative convergence analysis of SGD with diminishing learning rate regime. We then move on to the asynchronous parallel setting, and prove convergence of Hogwild! algorithm in the same regime in the case of diminished learning rate. It is well-known that SGD converges if a sequence of learning rates $\{\eta_t\}$ satisfies $\sum_{t=0}^\infty \eta_t \rightarrow \infty$ and $\sum_{t=0}^\infty \eta^2_t < \infty$. We show the convergence of SGD for strongly convex objective function without using bounded gradient assumption when $\{\eta_t\}$ is a diminishing sequence and $\sum_{t=0}^\infty \eta_t \rightarrow \infty$. In other words, we extend the current state-of-the-art class of learning rates satisfying the convergence of SGD.

Discovering the underlying dynamics leading up to an industrial asset failure is an important problem to be solved for successful development of Predictive Maintenance techniques. Existing work has largely focused on building complex ML/AI models for developing Predictive Maintenance solution patterns, but has largely avoided developing methods to explain the underlying failure dynamics. In this paper, we use an old but significantly improved change-pattern based technique to analyze IoT sensor data and failure information to generate useful and interpretable failure-centric insight. We discuss a solution pattern that we call ChieF, which when applied on multi-variate time series datasets, discover the leading failure indicators, generate associative patterns among multiple features, and output temporal dynamics of changes. Experimental analysis of ChieF on four datasets uncovers insights that may be valuable for predictive maintenance.

Stochastic gradient descent (SGD) is the optimization algorithm of choice in many machine learning applications such as regularized empirical risk minimization and training deep neural networks. The classical convergence analysis of SGD is carried out under the assumption that the norm of the stochastic gradient is uniformly bounded. While this might hold for some loss functions, it is always violated for cases where the objective function is strongly convex. In (Bottou et al.,2016), a new analysis of convergence of SGD is performed under the assumption that stochastic gradients are bounded with respect to the true gradient norm. Here we show that for stochastic problems arising in machine learning such bound always holds; and we also propose an alternative convergence analysis of SGD with diminishing learning rate regime, which results in more relaxed conditions than those in (Bottou et al.,2016). We then move on the asynchronous parallel setting, and prove convergence of Hogwild! algorithm in the same regime, obtaining the first convergence results for this method in the case of diminished learning rate.

In this paper, we propose a StochAstic Recursive grAdient algoritHm (SARAH), as well as its practical variant SARAH+, as a novel approach to the finite-sum minimization problems. Different from the vanilla SGD and other modern stochastic methods such as SVRG, S2GD, SAG and SAGA, SARAH admits a simple recursive framework for updating stochastic gradient estimates; when comparing to SAG/SAGA, SARAH does not require a storage of past gradients. The linear convergence rate of SARAH is proven under strong convexity assumption. We also prove a linear convergence rate (in the strongly convex case) for an inner loop of SARAH, the property that SVRG does not possess. Numerical experiments demonstrate the efficiency of our algorithm.

We consider a service system where agents (or, servers) are invited on-demand. Customers arrive as a Poisson process and join a customer queue. Customer service times are i.i.d. exponential. Agents' behavior is random in two respects. First, they can be invited into the system exogenously, and join the agent queue after a random time. Second, with some probability they rejoin the agent queue after a service completion, and otherwise leave the system. The objective is to design a real-time adaptive agent invitation scheme that keeps both customer and agent queues/waiting-times small. We study an adaptive scheme, which controls the number of pending agent invitations, based on queue-state feedback.