Yi Zhou

Yiwei Qiu, Jiatong Li, Yangjun Zeng et al.

An emerging approach for large-scale renewable hydrogen production is integrating multiple alkaline water electrolysis (AWE) stacks into one balance-of-plant (BoP) system, sharing gas-lye separation and lye circulation components. While this configuration, termed $N$-in-1, reduces cost and complexity, its dynamic performance under fluctuating power remains unclear compared with conventional 1-in-1 systems. This paper develops a state-space model of the multi-stack AWE system, capturing lye circulation, temperature, and hydrogen-to-oxygen (HTO) dynamics, calibrated via experiments on a 4,000 Nm$^3$/h-rated 4-in-1 system. A nonlinear model predictive controller (NMPC) is then designed to coordinate inter-stack current distribution, lye flow, and cooling for load tracking and operational stability. Simulations on the experimental-validated model show that a $4$-in-1 system can achieve very similar performance compared to four parallel 1-in-1 systems. Differences in load-tracking error, temperature stabilization, and specific energy consumption remain below 0.015 MW, 0.346 K, and 0.001 kWh/Nm$^3$ under wind power supply.

Xiaoshuai Hao, Mengchuan Wei, Yifan Yang et al.

Driving systems often rely on high-definition (HD) maps for precise environmental information, which is crucial for planning and navigation. While current HD map constructors perform well under ideal conditions, their resilience to real-world challenges, \eg, adverse weather and sensor failures, is not well understood, raising safety concerns. This work introduces MapBench, the first comprehensive benchmark designed to evaluate the robustness of HD map construction methods against various sensor corruptions. Our benchmark encompasses a total of 29 types of corruptions that occur from cameras and LiDAR sensors. Extensive evaluations across 31 HD map constructors reveal significant performance degradation of existing methods under adverse weather conditions and sensor failures, underscoring critical safety concerns. We identify effective strategies for enhancing robustness, including innovative approaches that leverage multi-modal fusion, advanced data augmentation, and architectural techniques. These insights provide a pathway for developing more reliable HD map construction methods, which are essential for the advancement of autonomous driving technology. The benchmark toolkit and affiliated code and model checkpoints have been made publicly accessible.

Cheng Chen, Jiaying Zhou, Jie Ding et al.

In the era of big data, many big organizations are integrating machine learning into their work pipelines to facilitate data analysis. However, the performance of their trained models is often restricted by limited and imbalanced data available to them. In this work, we develop an assisted learning framework for assisting organizations to improve their learning performance. The organizations have sufficient computation resources but are subject to stringent data-sharing and collaboration policies. Their limited imbalanced data often cause biased inference and sub-optimal decision-making. In assisted learning, an organizational learner purchases assistance service from an external service provider and aims to enhance its model performance within only a few assistance rounds. We develop effective stochastic training algorithms for both assisted deep learning and assisted reinforcement learning. Different from existing distributed algorithms that need to frequently transmit gradients or models, our framework allows the learner to only occasionally share information with the service provider, but still obtain a model that achieves near-oracle performance as if all the data were centralized.

Ziyi Chen, Yi Zhou, Rongrong Chen et al.

Actor-critic (AC) algorithms have been widely adopted in decentralized multi-agent systems to learn the optimal joint control policy. However, existing decentralized AC algorithms either do not preserve the privacy of agents or are not sample and communication-efficient. In this work, we develop two decentralized AC and natural AC (NAC) algorithms that are private, and sample and communication-efficient. In both algorithms, agents share noisy information to preserve privacy and adopt mini-batch updates to improve sample and communication efficiency. Particularly for decentralized NAC, we develop a decentralized Markovian SGD algorithm with an adaptive mini-batch size to efficiently compute the natural policy gradient. Under Markovian sampling and linear function approximation, we prove the proposed decentralized AC and NAC algorithms achieve the state-of-the-art sample complexities $\mathcal{O}\big(ε^{-2}\ln(ε^{-1})\big)$ and $\mathcal{O}\big(ε^{-3}\ln(ε^{-1})\big)$, respectively, and the same small communication complexity $\mathcal{O}\big(ε^{-1}\ln(ε^{-1})\big)$. Numerical experiments demonstrate that the proposed algorithms achieve lower sample and communication complexities than the existing decentralized AC algorithm.

Yue Wang, Shaofeng Zou, Yi Zhou

Temporal-difference learning with gradient correction (TDC) is a two time-scale algorithm for policy evaluation in reinforcement learning. This algorithm was initially proposed with linear function approximation, and was later extended to the one with general smooth function approximation. The asymptotic convergence for the on-policy setting with general smooth function approximation was established in [bhatnagar2009convergent], however, the finite-sample analysis remains unsolved due to challenges in the non-linear and two-time-scale update structure, non-convex objective function and the time-varying projection onto a tangent plane. In this paper, we develop novel techniques to explicitly characterize the finite-sample error bound for the general off-policy setting with i.i.d.\ or Markovian samples, and show that it converges as fast as $\mathcal O(1/\sqrt T)$ (up to a factor of $\mathcal O(\log T)$). Our approach can be applied to a wide range of value-based reinforcement learning algorithms with general smooth function approximation.

Shaocong Ma, Ziyi Chen, Yi Zhou et al.

Greedy-GQ is a value-based reinforcement learning (RL) algorithm for optimal control. Recently, the finite-time analysis of Greedy-GQ has been developed under linear function approximation and Markovian sampling, and the algorithm is shown to achieve an $ε$-stationary point with a sample complexity in the order of $\mathcal{O}(ε^{-3})$. Such a high sample complexity is due to the large variance induced by the Markovian samples. In this paper, we propose a variance-reduced Greedy-GQ (VR-Greedy-GQ) algorithm for off-policy optimal control. In particular, the algorithm applies the SVRG-based variance reduction scheme to reduce the stochastic variance of the two time-scale updates. We study the finite-time convergence of VR-Greedy-GQ under linear function approximation and Markovian sampling and show that the algorithm achieves a much smaller bias and variance error than the original Greedy-GQ. In particular, we prove that VR-Greedy-GQ achieves an improved sample complexity that is in the order of $\mathcal{O}(ε^{-2})$. We further compare the performance of VR-Greedy-GQ with that of Greedy-GQ in various RL experiments to corroborate our theoretical findings.

Cheng Chen, Bhavya Kailkhura, Ryan Goldhahn et al.

Federated learning is an emerging data-private distributed learning framework, which, however, is vulnerable to adversarial attacks. Although several heuristic defenses are proposed to enhance the robustness of federated learning, they do not provide certifiable robustness guarantees. In this paper, we incorporate randomized smoothing techniques into federated adversarial training to enable data-private distributed learning with certifiable robustness to test-time adversarial perturbations. Our experiments show that such an advanced federated adversarial learning framework can deliver models as robust as those trained by the centralized training. Further, this enables provably-robust classifiers to $\ell_2$-bounded adversarial perturbations in a distributed setup. We also show that one-point gradient estimation based training approach is $2-3\times$ faster than popular stochastic estimator based approach without any noticeable certified robustness differences.

Ziyi Chen, Yi Zhou, Rongrong Chen

The finite-time convergence of off-policy TD learning has been comprehensively studied recently. However, such a type of convergence has not been well established for off-policy TD learning in the multi-agent setting, which covers broader applications and is fundamentally more challenging. This work develops two decentralized TD with correction (TDC) algorithms for multi-agent off-policy TD learning under Markovian sampling. In particular, our algorithms preserve full privacy of the actions, policies and rewards of the agents, and adopt mini-batch sampling to reduce the sampling variance and communication frequency. Under Markovian sampling and linear function approximation, we proved that the finite-time sample complexity of both algorithms for achieving an $ε$-accurate solution is in the order of $\mathcal{O}(ε^{-1}\ln ε^{-1})$, matching the near-optimal sample complexity of centralized TD(0) and TDC. Importantly, the communication complexity of our algorithms is in the order of $\mathcal{O}(\ln ε^{-1})$, which is significantly lower than the communication complexity $\mathcal{O}(ε^{-1}\ln ε^{-1})$ of the existing decentralized TD(0). Experiments corroborate our theoretical findings.

Ziyi Chen, Yi Zhou, Tengyu Xu et al.

The gradient descent-ascent (GDA) algorithm has been widely applied to solve minimax optimization problems. In order to achieve convergent policy parameters for minimax optimization, it is important that GDA generates convergent variable sequences rather than convergent sequences of function values or gradient norms. However, the variable convergence of GDA has been proved only under convexity geometries, and there lacks understanding for general nonconvex minimax optimization. This paper fills such a gap by studying the convergence of a more general proximal-GDA for regularized nonconvex-strongly-concave minimax optimization. Specifically, we show that proximal-GDA admits a novel Lyapunov function, which monotonically decreases in the minimax optimization process and drives the variable sequence to a critical point. By leveraging this Lyapunov function and the KŁ geometry that parameterizes the local geometries of general nonconvex functions, we formally establish the variable convergence of proximal-GDA to a critical point $x^*$, i.e., $x_t\to x^*, y_t\to y^*(x^*)$. Furthermore, over the full spectrum of the KŁ-parameterized geometry, we show that proximal-GDA achieves different types of convergence rates ranging from sublinear convergence up to finite-step convergence, depending on the geometry associated with the KŁ parameter. This is the first theoretical result on the variable convergence for nonconvex minimax optimization.

Cheng Chen, Junjie Yang, Yi Zhou

Modern deep neural network (DNN) trainings utilize various training techniques, e.g., nonlinear activation functions, batch normalization, skip-connections, etc. Despite their effectiveness, it is still mysterious how they help accelerate DNN trainings in practice. In this paper, we provide an empirical study of the regularization effect of these training techniques on DNN optimization. Specifically, we find that the optimization trajectories of successful DNN trainings consistently obey a certain regularity principle that regularizes the model update direction to be aligned with the trajectory direction. Theoretically, we show that such a regularity principle leads to a convergence guarantee in nonconvex optimization and the convergence rate depends on a regularization parameter. Empirically, we find that DNN trainings that apply the training techniques achieve a fast convergence and obey the regularity principle with a large regularization parameter, implying that the model updates are well aligned with the trajectory. On the other hand, DNN trainings without the training techniques have slow convergence and obey the regularity principle with a small regularization parameter, implying that the model updates are not well aligned with the trajectory. Therefore, different training techniques regularize the model update direction via the regularity principle to facilitate the convergence.

Shaocong Ma, Yi Zhou, Shaofeng Zou

Variance reduction techniques have been successfully applied to temporal-difference (TD) learning and help to improve the sample complexity in policy evaluation. However, the existing work applied variance reduction to either the less popular one time-scale TD algorithm or the two time-scale GTD algorithm but with a finite number of i.i.d.\ samples, and both algorithms apply to only the on-policy setting. In this work, we develop a variance reduction scheme for the two time-scale TDC algorithm in the off-policy setting and analyze its non-asymptotic convergence rate over both i.i.d.\ and Markovian samples. In the i.i.d.\ setting, our algorithm {matches the best-known lower bound $\tilde{O}(ε^{-1}$).} In the Markovian setting, our algorithm achieves the state-of-the-art sample complexity $O(ε^{-1} \log ε^{-1})$ that is near-optimal. Experiments demonstrate that the proposed variance-reduced TDC achieves a smaller asymptotic convergence error than both the conventional TDC and the variance-reduced TD.

Cheng Chen, Ziyi Chen, Yi Zhou et al.

We develop FedCluster--a novel federated learning framework with improved optimization efficiency, and investigate its theoretical convergence properties. The FedCluster groups the devices into multiple clusters that perform federated learning cyclically in each learning round. Therefore, each learning round of FedCluster consists of multiple cycles of meta-update that boost the overall convergence. In nonconvex optimization, we show that FedCluster with the devices implementing the local {stochastic gradient descent (SGD)} algorithm achieves a faster convergence rate than the conventional {federated averaging (FedAvg)} algorithm in the presence of device-level data heterogeneity. We conduct experiments on deep learning applications and demonstrate that FedCluster converges significantly faster than the conventional federated learning under diverse levels of device-level data heterogeneity for a variety of local optimizers.

Shaocong Ma, Yi Zhou

Although SGD with random reshuffle has been widely-used in machine learning applications, there is a limited understanding of how model characteristics affect the convergence of the algorithm. In this work, we introduce model incoherence to characterize the diversity of model characteristics and study its impact on convergence of SGD with random reshuffle under weak strong convexity. Specifically, minimizer incoherence measures the discrepancy between the global minimizers of a sample loss and those of the total loss and affects the convergence error of SGD with random reshuffle. In particular, we show that the variable sequence generated by SGD with random reshuffle converges to a certain global minimizer of the total loss under full minimizer coherence. The other curvature incoherence measures the quality of condition numbers of the sample losses and determines the convergence rate of SGD. With model incoherence, our results show that SGD has a faster convergence rate and smaller convergence error under random reshuffle than those under random sampling, and hence provide justifications to the superior practical performance of SGD with random reshuffle.

Ziyi Chen, Yi Zhou

Composition optimization is widely-applied in nonconvex machine learning. Various advanced stochastic algorithms that adopt momentum and variance reduction techniques have been developed for composition optimization. However, these algorithms do not fully exploit both techniques to accelerate the convergence and are lack of convergence guarantee in nonconvex optimization. This paper complements the existing literature by developing various momentum schemes with SPIDER-based variance reduction for non-convex composition optimization. In particular, our momentum design requires less number of proximal mapping evaluations per-iteration than that required by the existing Katyusha momentum. Furthermore, our algorithm achieves near-optimal sample complexity results in both non-convex finite-sum and online composition optimization and achieves a linear convergence rate under the gradient dominant condition. Numerical experiments demonstrate that our algorithm converges significantly faster than existing algorithms in nonconvex composition optimization.

Yiren Li, Zheng Huang, Junchi Yan et al.

Tabular data is a crucial form of information expression, which can organize data in a standard structure for easy information retrieval and comparison. However, in financial industry and many other fields tables are often disclosed in unstructured digital files, e.g. Portable Document Format (PDF) and images, which are difficult to be extracted directly. In this paper, to facilitate deep learning based table extraction from unstructured digital files, we publish a standard Chinese dataset named FinTab, which contains more than 1,600 financial tables of diverse kinds and their corresponding structure representation in JSON. In addition, we propose a novel graph-based convolutional neural network model named GFTE as a baseline for future comparison. GFTE integrates image feature, position feature and textual feature together for precise edge prediction and reaches overall good results.

Yi Zhou, Zhe Wang, Kaiyi Ji et al.

Various types of parameter restart schemes have been proposed for accelerated gradient algorithms to facilitate their practical convergence in convex optimization. However, the convergence properties of accelerated gradient algorithms under parameter restart remain obscure in nonconvex optimization. In this paper, we propose a novel accelerated proximal gradient algorithm with parameter restart (named APG-restart) for solving nonconvex and nonsmooth problems. Our APG-restart is designed to 1) allow for adopting flexible parameter restart schemes that cover many existing ones; 2) have a global sub-linear convergence rate in nonconvex and nonsmooth optimization; and 3) have guaranteed convergence to a critical point and have various types of asymptotic convergence rates depending on the parameterization of local geometry in nonconvex and nonsmooth optimization. Numerical experiments demonstrate the effectiveness of our proposed algorithm.

Tengyu Xu, Zhe Wang, Yi Zhou et al.

Temporal difference (TD) learning is a popular algorithm for policy evaluation in reinforcement learning, but the vanilla TD can substantially suffer from the inherent optimization variance. A variance reduced TD (VRTD) algorithm was proposed by Korda and La (2015), which applies the variance reduction technique directly to the online TD learning with Markovian samples. In this work, we first point out the technical errors in the analysis of VRTD in Korda and La (2015), and then provide a mathematically solid analysis of the non-asymptotic convergence of VRTD and its variance reduction performance. We show that VRTD is guaranteed to converge to a neighborhood of the fixed-point solution of TD at a linear convergence rate. Furthermore, the variance error (for both i.i.d.\ and Markovian sampling) and the bias error (for Markovian sampling) of VRTD are significantly reduced by the batch size of variance reduction in comparison to those of vanilla TD. As a result, the overall computational complexity of VRTD to attain a given accurate solution outperforms that of TD under Markov sampling and outperforms that of TD under i.i.d.\ sampling for a sufficiently small conditional number.

Kaiyi Ji, Zhe Wang, Yi Zhou et al.

Two types of zeroth-order stochastic algorithms have recently been designed for nonconvex optimization respectively based on the first-order techniques SVRG and SARAH/SPIDER. This paper addresses several important issues that are still open in these methods. First, all existing SVRG-type zeroth-order algorithms suffer from worse function query complexities than either zeroth-order gradient descent (ZO-GD) or stochastic gradient descent (ZO-SGD). In this paper, we propose a new algorithm ZO-SVRG-Coord-Rand and develop a new analysis for an existing ZO-SVRG-Coord algorithm proposed in Liu et al. 2018b, and show that both ZO-SVRG-Coord-Rand and ZO-SVRG-Coord (under our new analysis) outperform other exiting SVRG-type zeroth-order methods as well as ZO-GD and ZO-SGD. Second, the existing SPIDER-type algorithm SPIDER-SZO (Fang et al. 2018) has superior theoretical performance, but suffers from the generation of a large number of Gaussian random variables as well as a $\sqrtε$-level stepsize in practice. In this paper, we develop a new algorithm ZO-SPIDER-Coord, which is free from Gaussian variable generation and allows a large constant stepsize while maintaining the same convergence rate and query complexity, and we further show that ZO-SPIDER-Coord automatically achieves a linear convergence rate as the iterate enters into a local PL region without restart and algorithmic modification.

Kaiyi Ji, Zhe Wang, Bowen Weng et al.

Variance-reduced algorithms, although achieve great theoretical performance, can run slowly in practice due to the periodic gradient estimation with a large batch of data. Batch-size adaptation thus arises as a promising approach to accelerate such algorithms. However, existing schemes either apply prescribed batch-size adaption rule or exploit the information along optimization path via additional backtracking and condition verification steps. In this paper, we propose a novel scheme, which eliminates backtracking line search but still exploits the information along optimization path by adapting the batch size via history stochastic gradients. We further theoretically show that such a scheme substantially reduces the overall complexity for popular variance-reduced algorithms SVRG and SARAH/SPIDER for both conventional nonconvex optimization and reinforcement learning problems. To this end, we develop a new convergence analysis framework to handle the dependence of the batch size on history stochastic gradients. Extensive experiments validate the effectiveness of the proposed batch-size adaptation scheme.

Cat P. Le, Yi Zhou, Jie Ding et al.

Classical supervised classification tasks search for a nonlinear mapping that maps each encoded feature directly to a probability mass over the labels. Such a learning framework typically lacks the intuition that encoded features from the same class tend to be similar and thus has little interpretability for the learned features. In this paper, we propose a novel supervised learning model named Supervised-Encoding Quantizer (SEQ). The SEQ applies a quantizer to cluster and classify the encoded features. We found that the quantizer provides an interpretable graph where each cluster in the graph represents a class of data samples that have a particular style. We also trained a decoder that can decode convex combinations of the encoded features from similar and different clusters and provide guidance on style transfer between sub-classes.

Jayanth Regatti, Gaurav Tendolkar, Yi Zhou et al.

The performance of fully synchronized distributed systems has faced a bottleneck due to the big data trend, under which asynchronous distributed systems are becoming a major popularity due to their powerful scalability. In this paper, we study the generalization performance of stochastic gradient descent (SGD) on a distributed asynchronous system. The system consists of multiple worker machines that compute stochastic gradients which are further sent to and aggregated on a common parameter server to update the variables, and the communication in the system suffers from possible delays. Under the algorithm stability framework, we prove that distributed asynchronous SGD generalizes well given enough data samples in the training optimization. In particular, our results suggest to reduce the learning rate as we allow more asynchrony in the distributed system. Such adaptive learning rate strategy improves the stability of the distributed algorithm and reduces the corresponding generalization error. Then, we confirm our theoretical findings via numerical experiments.

Guanghui Lan, Zhize Li, Yi Zhou

We propose a novel randomized incremental gradient algorithm, namely, VAriance-Reduced Accelerated Gradient (Varag), for finite-sum optimization. Equipped with a unified step-size policy that adjusts itself to the value of the condition number, Varag exhibits the unified optimal rates of convergence for solving smooth convex finite-sum problems directly regardless of their strong convexity. Moreover, Varag is the first accelerated randomized incremental gradient method that benefits from the strong convexity of the data-fidelity term to achieve the optimal linear convergence. It also establishes an optimal linear rate of convergence for solving a wide class of problems only satisfying a certain error bound condition rather than strong convexity. Varag can also be extended to solve stochastic finite-sum problems.

Yi Zhou, Zhe Wang, Kaiyi Ji et al.

Two new stochastic variance-reduced algorithms named SARAH and SPIDER have been recently proposed, and SPIDER has been shown to achieve a near-optimal gradient oracle complexity for nonconvex optimization. However, the theoretical advantage of SPIDER does not lead to substantial improvement of practical performance over SVRG. To address this issue, momentum technique can be a good candidate to improve the performance of SPIDER. However, existing momentum schemes used in variance-reduced algorithms are designed specifically for convex optimization, and are not applicable to nonconvex scenarios. In this paper, we develop novel momentum schemes with flexible coefficient settings to accelerate SPIDER for nonconvex and nonsmooth composite optimization, and show that the resulting algorithms achieve the near-optimal gradient oracle complexity for achieving a generalized first-order stationary condition. Furthermore, we generalize our algorithm to online nonconvex and nonsmooth optimization, and establish an oracle complexity result that matches the state-of-the-art. Our extensive experiments demonstrate the superior performance of our proposed algorithm over other stochastic variance-reduced algorithms.

Yi Zhou, Junjie Yang, Huishuai Zhang et al.

Stochastic gradient descent (SGD) has been found to be surprisingly effective in training a variety of deep neural networks. However, there is still a lack of understanding on how and why SGD can train these complex networks towards a global minimum. In this study, we establish the convergence of SGD to a global minimum for nonconvex optimization problems that are commonly encountered in neural network training. Our argument exploits the following two important properties: 1) the training loss can achieve zero value (approximately), which has been widely observed in deep learning; 2) SGD follows a star-convex path, which is verified by various experiments in this paper. In such a context, our analysis shows that SGD, although has long been considered as a randomized algorithm, converges in an intrinsically deterministic manner to a global minimum.

Zhe Wang, Kaiyi Ji, Yi Zhou et al.

SARAH and SPIDER are two recently developed stochastic variance-reduced algorithms, and SPIDER has been shown to achieve a near-optimal first-order oracle complexity in smooth nonconvex optimization. However, SPIDER uses an accuracy-dependent stepsize that slows down the convergence in practice, and cannot handle objective functions that involve nonsmooth regularizers. In this paper, we propose SpiderBoost as an improved scheme, which allows to use a much larger constant-level stepsize while maintaining the same near-optimal oracle complexity, and can be extended with proximal mapping to handle composite optimization (which is nonsmooth and nonconvex) with provable convergence guarantee. In particular, we show that proximal SpiderBoost achieves an oracle complexity of $\mathcal{O}(\min\{n^{1/2}ε^{-2},ε^{-3}\})$ in composite nonconvex optimization, improving the state-of-the-art result by a factor of $\mathcal{O}(\min\{n^{1/6},ε^{-1/3}\})$. We further develop a novel momentum scheme to accelerate SpiderBoost for composite optimization, which achieves the near-optimal oracle complexity in theory and substantial improvement in experiments.

Zhe Wang, Yi Zhou, Yingbin Liang et al.

Momentum is a popular technique to accelerate the convergence in practical training, and its impact on convergence guarantee has been well-studied for first-order algorithms. However, such a successful acceleration technique has not yet been proposed for second-order algorithms in nonconvex optimization.In this paper, we apply the momentum scheme to cubic regularized (CR) Newton's method and explore the potential for acceleration. Our numerical experiments on various nonconvex optimization problems demonstrate that the momentum scheme can substantially facilitate the convergence of cubic regularization, and perform even better than the Nesterov's acceleration scheme for CR. Theoretically, we prove that CR under momentum achieves the best possible convergence rate to a second-order stationary point for nonconvex optimization. Moreover, we study the proposed algorithm for solving problems satisfying an error bound condition and establish a local quadratic convergence rate. Then, particularly for finite-sum problems, we show that the proposed algorithm can allow computational inexactness that reduces the overall sample complexity without degrading the convergence rate.

Zhe Wang, Yi Zhou, Yingbin Liang et al.

This note considers the inexact cubic-regularized Newton's method (CR), which has been shown in \cite{Cartis2011a} to achieve the same order-level convergence rate to a secondary stationary point as the exact CR \citep{Nesterov2006}. However, the inexactness condition in \cite{Cartis2011a} is not implementable due to its dependence on future iterates variable. This note fixes such an issue by proving the same convergence rate for nonconvex optimization under an inexact adaptive condition that depends on only the current iterate. Our proof controls the sufficient decrease of the function value over the total iterations rather than each iteration as used in the previous studies, which can be of independent interest in other contexts.

Yi Zhou, Zhe Wang, Yingbin Liang

Cubic-regularized Newton's method (CR) is a popular algorithm that guarantees to produce a second-order stationary solution for solving nonconvex optimization problems. However, existing understandings of the convergence rate of CR are conditioned on special types of geometrical properties of the objective function. In this paper, we explore the asymptotic convergence rate of CR by exploiting the ubiquitous Kurdyka-Lojasiewicz (KL) property of nonconvex objective functions. In specific, we characterize the asymptotic convergence rate of various types of optimality measures for CR including function value gap, variable distance gap, gradient norm and least eigenvalue of the Hessian matrix. Our results fully characterize the diverse convergence behaviors of these optimality measures in the full parameter regime of the KL property. Moreover, we show that the obtained asymptotic convergence rates of CR are order-wise faster than those of first-order gradient descent algorithms under the KL property.

Tengyu Xu, Yi Zhou, Kaiyi Ji et al.

We study the implicit bias of gradient descent methods in solving a binary classification problem over a linearly separable dataset. The classifier is described by a nonlinear ReLU model and the objective function adopts the exponential loss function. We first characterize the landscape of the loss function and show that there can exist spurious asymptotic local minima besides asymptotic global minima. We then show that gradient descent (GD) can converge to either a global or a local max-margin direction, or may diverge from the desired max-margin direction in a general context. For stochastic gradient descent (SGD), we show that it converges in expectation to either the global or the local max-margin direction if SGD converges. We further explore the implicit bias of these algorithms in learning a multi-neuron network under certain stationary conditions, and show that the learned classifier maximizes the margins of each sample pattern partition under the ReLU activation.

Zhe Wang, Yi Zhou, Yingbin Liang et al.

Cubic regularization (CR) is an optimization method with emerging popularity due to its capability to escape saddle points and converge to second-order stationary solutions for nonconvex optimization. However, CR encounters a high sample complexity issue for finite-sum problems with a large data size. %Various inexact variants of CR have been proposed to improve the sample complexity. In this paper, we propose a stochastic variance-reduced cubic-regularization (SVRC) method under random sampling, and study its convergence guarantee as well as sample complexity. We show that the iteration complexity of SVRC for achieving a second-order stationary solution within $ε$ accuracy is $O(ε^{-3/2})$, which matches the state-of-art result on CR types of methods. Moreover, our proposed variance reduction scheme significantly reduces the per-iteration sample complexity. The resulting total Hessian sample complexity of our SVRC is ${\Oc}(N^{2/3} ε^{-3/2})$, which outperforms the state-of-art result by a factor of $O(N^{2/15})$. We also study our SVRC under random sampling without replacement scheme, which yields a lower per-iteration sample complexity, and hence justifies its practical applicability.

Yi Zhou, Yingbin Liang, Huishuai Zhang

The success of deep learning has led to a rising interest in the generalization property of the stochastic gradient descent (SGD) method, and stability is one popular approach to study it. Existing works based on stability have studied nonconvex loss functions, but only considered the generalization error of the SGD in expectation. In this paper, we establish various generalization error bounds with probabilistic guarantee for the SGD. Specifically, for both general nonconvex loss functions and gradient dominant loss functions, we characterize the on-average stability of the iterates generated by SGD in terms of the on-average variance of the stochastic gradients. Such characterization leads to improved bounds for the generalization error for SGD. We then study the regularized risk minimization problem with strongly convex regularizers, and obtain improved generalization error bounds for proximal SGD. With strongly convex regularizers, we further establish the generalization error bounds for nonconvex loss functions under proximal SGD with high-probability guarantee, i.e., exponential concentration in probability.

Guanghui Lan, Yi Zhou

In this paper, we consider a class of finite-sum convex optimization problems defined over a distributed multiagent network with $m$ agents connected to a central server. In particular, the objective function consists of the average of $m$ ($\ge 1$) smooth components associated with each network agent together with a strongly convex term. Our major contribution is to develop a new randomized incremental gradient algorithm, namely random gradient extrapolation method (RGEM), which does not require any exact gradient evaluation even for the initial point, but can achieve the optimal ${\cal O}(\log(1/ε))$ complexity bound in terms of the total number of gradient evaluations of component functions to solve the finite-sum problems. Furthermore, we demonstrate that for stochastic finite-sum optimization problems, RGEM maintains the optimal ${\cal O}(1/ε)$ complexity (up to a certain logarithmic factor) in terms of the number of stochastic gradient computations, but attains an ${\cal O}(\log(1/ε))$ complexity in terms of communication rounds (each round involves only one agent). It is worth noting that the former bound is independent of the number of agents $m$, while the latter one only linearly depends on $m$ or even $\sqrt m$ for ill-conditioned problems. To the best of our knowledge, this is the first time that these complexity bounds have been obtained for distributed and stochastic optimization problems. Moreover, our algorithms were developed based on a novel dual perspective of Nesterov's accelerated gradient method.

Yi Zhou, Yingbin Liang

Due to the success of deep learning to solving a variety of challenging machine learning tasks, there is a rising interest in understanding loss functions for training neural networks from a theoretical aspect. Particularly, the properties of critical points and the landscape around them are of importance to determine the convergence performance of optimization algorithms. In this paper, we provide full (necessary and sufficient) characterization of the analytical forms for the critical points (as well as global minimizers) of the square loss functions for various neural networks. We show that the analytical forms of the critical points characterize the values of the corresponding loss functions as well as the necessary and sufficient conditions to achieve global minimum. Furthermore, we exploit the analytical forms of the critical points to characterize the landscape properties for the loss functions of these neural networks. One particular conclusion is that: The loss function of linear networks has no spurious local minimum, while the loss function of one-hidden-layer nonlinear networks with ReLU activation function does have local minimum that is not global minimum.

Yi Zhou, Yingbin Liang

The past decade has witnessed a successful application of deep learning to solving many challenging problems in machine learning and artificial intelligence. However, the loss functions of deep neural networks (especially nonlinear networks) are still far from being well understood from a theoretical aspect. In this paper, we enrich the current understanding of the landscape of the square loss functions for three types of neural networks. Specifically, when the parameter matrices are square, we provide an explicit characterization of the global minimizers for linear networks, linear residual networks, and nonlinear networks with one hidden layer. Then, we establish two quadratic types of landscape properties for the square loss of these neural networks, i.e., the gradient dominance condition within the neighborhood of their full rank global minimizers, and the regularity condition along certain directions and within the neighborhood of their global minimizers. These two landscape properties are desirable for the optimization around the global minimizers of the loss function for these neural networks.

Qunwei Li, Yi Zhou, Yingbin Liang et al.

In many modern machine learning applications, structures of underlying mathematical models often yield nonconvex optimization problems. Due to the intractability of nonconvexity, there is a rising need to develop efficient methods for solving general nonconvex problems with certain performance guarantee. In this work, we investigate the accelerated proximal gradient method for nonconvex programming (APGnc). The method compares between a usual proximal gradient step and a linear extrapolation step, and accepts the one that has a lower function value to achieve a monotonic decrease. In specific, under a general nonsmooth and nonconvex setting, we provide a rigorous argument to show that the limit points of the sequence generated by APGnc are critical points of the objective function. Then, by exploiting the Kurdyka-Łojasiewicz (\KL) property for a broad class of functions, we establish the linear and sub-linear convergence rates of the function value sequence generated by APGnc. We further propose a stochastic variance reduced APGnc (SVRG-APGnc), and establish its linear convergence under a special case of the \KL property. We also extend the analysis to the inexact version of these methods and develop an adaptive momentum strategy that improves the numerical performance.

Guanghui Lan, Sebastian Pokutta, Yi Zhou et al.

In this work we introduce a conditional accelerated lazy stochastic gradient descent algorithm with optimal number of calls to a stochastic first-order oracle and convergence rate $O\left(\frac{1}{\varepsilon^2}\right)$ improving over the projection-free, Online Frank-Wolfe based stochastic gradient descent of Hazan and Kale [2012] with convergence rate $O\left(\frac{1}{\varepsilon^4}\right)$.

Guanghui Lan, Yi Zhou

In this paper, we consider a class of finite-sum convex optimization problems whose objective function is given by the summation of $m$ ($\ge 1$) smooth components together with some other relatively simple terms. We first introduce a deterministic primal-dual gradient (PDG) method that can achieve the optimal black-box iteration complexity for solving these composite optimization problems using a primal-dual termination criterion. Our major contribution is to develop a randomized primal-dual gradient (RPDG) method, which needs to compute the gradient of only one randomly selected smooth component at each iteration, but can possibly achieve better complexity than PDG in terms of the total number of gradient evaluations. More specifically, we show that the total number of gradient evaluations performed by RPDG can be ${\cal O} (\sqrt{m})$ times smaller, both in expectation and with high probability, than those performed by deterministic optimal first-order methods under favorable situations. We also show that the complexity of the RPDG method is not improvable by developing a new lower complexity bound for a general class of randomized methods for solving large-scale finite-sum convex optimization problems. Moreover, through the development of PDG and RPDG, we introduce a novel game-theoretic interpretation for these optimal methods for convex optimization.

Pengtao Xie, Jin Kyu Kim, Yi Zhou et al.

Matrix-parametrized models, including multiclass logistic regression and sparse coding, are used in machine learning (ML) applications ranging from computer vision to computational biology. When these models are applied to large-scale ML problems starting at millions of samples and tens of thousands of classes, their parameter matrix can grow at an unexpected rate, resulting in high parameter synchronization costs that greatly slow down distributed learning. To address this issue, we propose a Sufficient Factor Broadcasting (SFB) computation model for efficient distributed learning of a large family of matrix-parameterized models, which share the following property: the parameter update computed on each data sample is a rank-1 matrix, i.e., the outer product of two "sufficient factors" (SFs). By broadcasting the SFs among worker machines and reconstructing the update matrices locally at each worker, SFB improves communication efficiency --- communication costs are linear in the parameter matrix's dimensions, rather than quadratic --- without affecting computational correctness. We present a theoretical convergence analysis of SFB, and empirically corroborate its efficiency on four different matrix-parametrized ML models.