Tuo Zhao

Lingkai Kong, Haoming Jiang, Yuchen Zhuang et al.

Fine-tuned pre-trained language models can suffer from severe miscalibration for both in-distribution and out-of-distribution (OOD) data due to over-parameterization. To mitigate this issue, we propose a regularized fine-tuning method. Our method introduces two types of regularization for better calibration: (1) On-manifold regularization, which generates pseudo on-manifold samples through interpolation within the data manifold. Augmented training with these pseudo samples imposes a smoothness regularization to improve in-distribution calibration. (2) Off-manifold regularization, which encourages the model to output uniform distributions for pseudo off-manifold samples to address the over-confidence issue for OOD data. Our experiments demonstrate that the proposed method outperforms existing calibration methods for text classification in terms of expectation calibration error, misclassification detection, and OOD detection on six datasets. Our code can be found at https://github.com/Lingkai-Kong/Calibrated-BERT-Fine-Tuning.

Ilgee Hong, Changlong Yu, Liang Qiu et al. · gatech

Reinforcement learning from human feedback (RLHF) has become a powerful post-training paradigm for aligning large language models with human preferences. A core challenge in RLHF is constructing accurate reward signals, where the conventional Bradley-Terry reward models (BT RMs) often suffer from sensitivity to data size and coverage, as well as vulnerability to reward hacking. Generative reward models (GenRMs) offer a more robust alternative by generating chain-of-thought (CoT) rationales followed by a final reward. However, existing GenRMs rely on shallow, vertically scaled reasoning, limiting their capacity to handle nuanced or complex (e.g., reasoning-intensive) tasks. Moreover, their pairwise preference outputs are incompatible with standard RLHF algorithms that require pointwise reward signals. In this work, we introduce Think-RM, a training framework that enables long-horizon reasoning in GenRMs by modeling an internal thinking process. Rather than producing structured, externally provided rationales, Think-RM generates flexible, self-guided reasoning traces that support advanced capabilities such as self-reflection, hypothetical reasoning, and divergent reasoning. To elicit these reasoning abilities, we first warm-up the models by supervised fine-tuning (SFT) over long CoT data. We then further improve the model's long-horizon abilities by rule-based reinforcement learning (RL). In addition, we propose a novel pairwise RLHF pipeline that directly optimizes policies using pairwise preference rewards, eliminating the need for pointwise reward conversion and enabling more effective use of Think-RM outputs. Experiments show that Think-RM achieves state-of-the-art results on RM-Bench, outperforming both BT RM and vertically scaled GenRM by 8%. When combined with our pairwise RLHF pipeline, it demonstrates superior end-policy performance compared to traditional approaches.

Hoang Huy Nguyen, Yan Li, Tuo Zhao

In modern decentralized applications, ensuring communication efficiency and privacy for the users are the key challenges. In order to train machine-learning models, the algorithm has to communicate to the data center and sample data for its gradient computation, thus exposing the data and increasing the communication cost. This gives rise to the need for a decentralized optimization algorithm that is communication-efficient and minimizes the number of gradient computations. To this end, we propose the primal-dual sliding with conditional gradient sliding framework, which is communication-efficient and achieves an $\varepsilon$-approximate solution with the optimal gradient complexity of $O(1/\sqrt{\varepsilon}+σ^2/{\varepsilon^2})$ and $O(\log(1/\varepsilon)+σ^2/\varepsilon)$ for the convex and strongly convex setting respectively and an LO (Linear Optimization) complexity of $O(1/\varepsilon^2)$ for both settings given a stochastic gradient oracle with variance $σ^2$. Compared with the prior work \cite{wai-fw-2017}, our framework relaxes the assumption of the optimal solution being a strict interior point of the feasible set and enjoys wider applicability for large-scale training using a stochastic gradient oracle. We also demonstrate the efficiency of our algorithms with various numerical experiments.

Liming Liu, Zixuan Zhang, Simon Du et al.

Recent advances in deep learning optimization have unveiled two intriguing phenomena under large learning rates: Edge of Stability (EoS) and Progressive Sharpening (PS), challenging classical Gradient Descent (GD) analyses. Current research approaches, using either generalist frameworks or minimalist examples, face significant limitations in explaining these phenomena. This paper advances the minimalist approach by introducing a two-layer network with a two-dimensional input, where one dimension is relevant to the response and the other is irrelevant. Through this model, we rigorously prove the existence of progressive sharpening and self-stabilization under large learning rates, and establish non-asymptotic analysis of the training dynamics and sharpness along the entire GD trajectory. Besides, we connect our minimalist example to existing works by reconciling the existence of a well-behaved ``stable set" between minimalist and generalist analyses, and extending the analysis of Gradient Flow Solution sharpness to our two-dimensional input scenario. These findings provide new insights into the EoS phenomenon from both parameter and input data distribution perspectives, potentially informing more effective optimization strategies in deep learning practice.

Yuqing Wang, Minshuo Chen, Tuo Zhao et al.

Recent empirical advances show that training deep models with large learning rate often improves generalization performance. However, theoretical justifications on the benefits of large learning rate are highly limited, due to challenges in analysis. In this paper, we consider using Gradient Descent (GD) with a large learning rate on a homogeneous matrix factorization problem, i.e., $\min_{X, Y} \|A - XY^\top\|_{\sf F}^2$. We prove a convergence theory for constant large learning rates well beyond $2/L$, where $L$ is the largest eigenvalue of Hessian at the initialization. Moreover, we rigorously establish an implicit bias of GD induced by such a large learning rate, termed 'balancing', meaning that magnitudes of $X$ and $Y$ at the limit of GD iterations will be close even if their initialization is significantly unbalanced. Numerical experiments are provided to support our theory.

Hao Liu, Minshuo Chen, Tuo Zhao et al.

Most of existing statistical theories on deep neural networks have sample complexities cursed by the data dimension and therefore cannot well explain the empirical success of deep learning on high-dimensional data. To bridge this gap, we propose to exploit low-dimensional geometric structures of the real world data sets. We establish theoretical guarantees of convolutional residual networks (ConvResNet) in terms of function approximation and statistical estimation for binary classification. Specifically, given the data lying on a $d$-dimensional manifold isometrically embedded in $\mathbb{R}^D$, we prove that if the network architecture is properly chosen, ConvResNets can (1) approximate Besov functions on manifolds with arbitrary accuracy, and (2) learn a classifier by minimizing the empirical logistic risk, which gives an excess risk in the order of $n^{-\frac{s}{2s+2(s\vee d)}}$, where $s$ is a smoothness parameter. This implies that the sample complexity depends on the intrinsic dimension $d$, instead of the data dimension $D$. Our results demonstrate that ConvResNets are adaptive to low-dimensional structures of data sets.

Danqing Zhang, Zheng Li, Tianyu Cao et al.

We study the problem of query attribute value extraction, which aims to identify named entities from user queries as diverse surface form attribute values and afterward transform them into formally canonical forms. Such a problem consists of two phases: {named entity recognition (NER)} and {attribute value normalization (AVN)}. However, existing works only focus on the NER phase but neglect equally important AVN. To bridge this gap, this paper proposes a unified query attribute value extraction system in e-commerce search named QUEACO, which involves both two phases. Moreover, by leveraging large-scale weakly-labeled behavior data, we further improve the extraction performance with less supervision cost. Specifically, for the NER phase, QUEACO adopts a novel teacher-student network, where a teacher network that is trained on the strongly-labeled data generates pseudo-labels to refine the weakly-labeled data for training a student network. Meanwhile, the teacher network can be dynamically adapted by the feedback of the student's performance on strongly-labeled data to maximally denoise the noisy supervisions from the weak labels. For the AVN phase, we also leverage the weakly-labeled query-to-attribute behavior data to normalize surface form attribute values from queries into canonical forms from products. Extensive experiments on a real-world large-scale E-commerce dataset demonstrate the effectiveness of QUEACO.

Yan Li, Caleb Ju, Ethan X. Fang et al.

Bregman proximal point algorithm (BPPA) has witnessed emerging machine learning applications, yet its theoretical understanding has been largely unexplored. We study the computational properties of BPPA through learning linear classifiers with separable data, and demonstrate provable algorithmic regularization of BPPA. For any BPPA instantiated with a fixed Bregman divergence, we provide a lower bound of the margin obtained by BPPA with respect to an arbitrarily chosen norm. The obtained margin lower bound differs from the maximal margin by a multiplicative factor, which inversely depends on the condition number of the distance-generating function measured in the dual norm. We show that the dependence on the condition number is tight, thus demonstrating the importance of divergence in affecting the quality of the learned classifiers. We then extend our findings to mirror descent, for which we establish similar connections between the margin and Bregman divergence, together with a non-asymptotic analysis. Numerical experiments on both synthetic and real-world datasets are provided to support our theoretical findings. To the best of our knowledge, the aforementioned findings appear to be new in the literature of algorithmic regularization.

Haoming Jiang, Danqing Zhang, Tianyu Cao et al.

Weak supervision has shown promising results in many natural language processing tasks, such as Named Entity Recognition (NER). Existing work mainly focuses on learning deep NER models only with weak supervision, i.e., without any human annotation, and shows that by merely using weakly labeled data, one can achieve good performance, though still underperforms fully supervised NER with manually/strongly labeled data. In this paper, we consider a more practical scenario, where we have both a small amount of strongly labeled data and a large amount of weakly labeled data. Unfortunately, we observe that weakly labeled data does not necessarily improve, or even deteriorate the model performance (due to the extensive noise in the weak labels) when we train deep NER models over a simple or weighted combination of the strongly labeled and weakly labeled data. To address this issue, we propose a new multi-stage computational framework -- NEEDLE with three essential ingredients: (1) weak label completion, (2) noise-aware loss function, and (3) final fine-tuning over the strongly labeled data. Through experiments on E-commerce query NER and Biomedical NER, we demonstrate that NEEDLE can effectively suppress the noise of the weak labels and outperforms existing methods. In particular, we achieve new SOTA F1-scores on 3 Biomedical NER datasets: BC5CDR-chem 93.74, BC5CDR-disease 90.69, NCBI-disease 92.28.

Chen Liang, Simiao Zuo, Minshuo Chen et al.

The Lottery Ticket Hypothesis suggests that an over-parametrized network consists of ``lottery tickets'', and training a certain collection of them (i.e., a subnetwork) can match the performance of the full model. In this paper, we study such a collection of tickets, which is referred to as ``winning tickets'', in extremely over-parametrized models, e.g., pre-trained language models. We observe that at certain compression ratios, the generalization performance of the winning tickets can not only match but also exceed that of the full model. In particular, we observe a phase transition phenomenon: As the compression ratio increases, generalization performance of the winning tickets first improves then deteriorates after a certain threshold. We refer to the tickets on the threshold as ``super tickets''. We further show that the phase transition is task and model dependent -- as the model size becomes larger and the training data set becomes smaller, the transition becomes more pronounced. Our experiments on the GLUE benchmark show that the super tickets improve single task fine-tuning by $0.9$ points on BERT-base and $1.0$ points on BERT-large, in terms of task-average score. We also demonstrate that adaptively sharing the super tickets across tasks benefits multi-task learning.

Siawpeng Er, Shihao Yang, Tuo Zhao

The global spread of COVID-19, the disease caused by the novel coronavirus SARS-CoV-2, has cast a significant threat to mankind. As the COVID-19 situation continues to evolve, predicting localized disease severity is crucial for advanced resource allocation. This paper proposes a method named COURAGE (COUnty aggRegation mixup AuGmEntation) to generate a short-term prediction of 2-week-ahead COVID-19 related deaths for each county in the United States, leveraging modern deep learning techniques. Specifically, our method adopts a self-attention model from Natural Language Processing, known as the transformer model, to capture both short-term and long-term dependencies within the time series while enjoying computational efficiency. Our model fully utilizes publicly available information of COVID-19 related confirmed cases, deaths, community mobility trends and demographic information, and can produce state-level prediction as an aggregation of the corresponding county-level predictions. Our numerical experiments demonstrate that our model achieves the state-of-the-art performance among the publicly available benchmark models.

Chen Liang, Haoming Jiang, Xiaodong Liu et al.

Existing curriculum learning approaches to Neural Machine Translation (NMT) require sampling sufficient amounts of "easy" samples from training data at the early training stage. This is not always achievable for low-resource languages where the amount of training data is limited. To address such limitation, we propose a novel token-wise curriculum learning approach that creates sufficient amounts of easy samples. Specifically, the model learns to predict a short sub-sequence from the beginning part of each target sentence at the early stage of training, and then the sub-sequence is gradually expanded as the training progresses. Such a new curriculum design is inspired by the cumulative effect of translation errors, which makes the latter tokens more difficult to predict than the beginning ones. Extensive experiments show that our approach can consistently outperform baselines on 5 language pairs, especially for low-resource languages. Combining our approach with sentence-level methods further improves the performance on high-resource languages.

Tianyi Liu, Yan Li, Song Wei et al.

Numerous empirical evidences have corroborated the importance of noise in nonconvex optimization problems. The theory behind such empirical observations, however, is still largely unknown. This paper studies this fundamental problem through investigating the nonconvex rectangular matrix factorization problem, which has infinitely many global minima due to rotation and scaling invariance. Hence, gradient descent (GD) can converge to any optimum, depending on the initialization. In contrast, we show that a perturbed form of GD with an arbitrary initialization converges to a global optimum that is uniquely determined by the injected noise. Our result implies that the noise imposes implicit bias towards certain optima. Numerical experiments are provided to support our theory.

Yujia Xie, Yixiu Mao, Simiao Zuo et al.

We consider a variant of regression problem, where the correspondence between input and output data is not available. Such shuffled data is commonly observed in many real world problems. Taking flow cytometry as an example, the measuring instruments may not be able to maintain the correspondence between the samples and the measurements. Due to the combinatorial nature of the problem, most existing methods are only applicable when the sample size is small, and limited to linear regression models. To overcome such bottlenecks, we propose a new computational framework -- ROBOT -- for the shuffled regression problem, which is applicable to large data and complex nonlinear models. Specifically, we reformulate the regression without correspondence as a continuous optimization problem. Then by exploiting the interaction between the regression model and the data correspondence, we develop a hypergradient approach based on differentiable programming techniques. Such a hypergradient approach essentially views the data correspondence as an operator of the regression, and therefore allows us to find a better descent direction for the model parameter by differentiating through the data correspondence. ROBOT can be further extended to the inexact correspondence setting, where there may not be an exact alignment between the input and output data. Thorough numerical experiments show that ROBOT achieves better performance than existing methods in both linear and nonlinear regression tasks, including real-world applications such as flow cytometry and multi-object tracking.

David Munzer, Siawpeng Er, Minshuo Chen et al.

We propose using machine learning models for the direct synthesis of on-chip electromagnetic (EM) passive structures to enable rapid or even automated designs and optimizations of RF/mm-Wave circuits. As a proof of concept, we demonstrate the direct synthesis of a 1:1 transformer on a 45nm SOI process using our proposed neural network model. Using pre-existing transformer s-parameter files and their geometric design training samples, the model predicts target geometric designs.

Xingguo Li, Tuo Zhao, Xiaoming Yuan et al.

This paper describes an R package named flare, which implements a family of new high dimensional regression methods (LAD Lasso, SQRT Lasso, $\ell_q$ Lasso, and Dantzig selector) and their extensions to sparse precision matrix estimation (TIGER and CLIME). These methods exploit different nonsmooth loss functions to gain modeling flexibility, estimation robustness, and tuning insensitiveness. The developed solver is based on the alternating direction method of multipliers (ADMM). The package flare is coded in double precision C, and called from R by a user-friendly interface. The memory usage is optimized by using the sparse matrix output. The experiments show that flare is efficient and can scale up to large problems.

Jason Ge, Xingguo Li, Haoming Jiang et al.

We describe a new library named picasso, which implements a unified framework of pathwise coordinate optimization for a variety of sparse learning problems (e.g., sparse linear regression, sparse logistic regression, sparse Poisson regression and scaled sparse linear regression) combined with efficient active set selection strategies. Besides, the library allows users to choose different sparsity-inducing regularizers, including the convex $\ell_1$, nonconvex MCP and SCAD regularizers. The library is coded in C++ and has user-friendly R and Python wrappers. Numerical experiments demonstrate that picasso can scale up to large problems efficiently.

Tuo Zhao, Han Liu, Kathryn Roeder et al.

We describe an R package named huge which provides easy-to-use functions for estimating high dimensional undirected graphs from data. This package implements recent results in the literature, including Friedman et al. (2007), Liu et al. (2009, 2012) and Liu et al. (2010). Compared with the existing graph estimation package glasso, the huge package provides extra features: (1) instead of using Fortan, it is written in C, which makes the code more portable and easier to modify; (2) besides fitting Gaussian graphical models, it also provides functions for fitting high dimensional semiparametric Gaussian copula models; (3) more functions like data-dependent model selection, data generation and graph visualization; (4) a minor convergence problem of the graphical lasso algorithm is corrected; (5) the package allows the user to apply both lossless and lossy screening rules to scale up large-scale problems, making a tradeoff between computational and statistical efficiency.

Minshuo Chen, Yu Bai, Jason D. Lee et al.

Deep neural networks can empirically perform efficient hierarchical learning, in which the layers learn useful representations of the data. However, how they make use of the intermediate representations are not explained by recent theories that relate them to "shallow learners" such as kernels. In this work, we demonstrate that intermediate neural representations add more flexibility to neural networks and can be advantageous over raw inputs. We consider a fixed, randomly initialized neural network as a representation function fed into another trainable network. When the trainable network is the quadratic Taylor model of a wide two-layer network, we show that neural representation can achieve improved sample complexities compared with the raw input: For learning a low-rank degree-$p$ polynomial ($p \geq 4$) in $d$ dimension, neural representation requires only $\tilde{O}(d^{\lceil p/2 \rceil})$ samples, while the best-known sample complexity upper bound for the raw input is $\tilde{O}(d^{p-1})$. We contrast our result with a lower bound showing that neural representations do not improve over the raw input (in the infinite width limit), when the trainable network is instead a neural tangent kernel. Our results characterize when neural representations are beneficial, and may provide a new perspective on why depth is important in deep learning.

Qianli Shen, Yan Li, Haoming Jiang et al.

Deep reinforcement learning (RL) has achieved great empirical successes in various domains. However, the large search space of neural networks requires a large amount of data, which makes the current RL algorithms not sample efficient. Motivated by the fact that many environments with continuous state space have smooth transitions, we propose to learn a smooth policy that behaves smoothly with respect to states. We develop a new framework -- \textbf{S}mooth \textbf{R}egularized \textbf{R}einforcement \textbf{L}earning ($\textbf{SR}^2\textbf{L}$), where the policy is trained with smoothness-inducing regularization. Such regularization effectively constrains the search space, and enforces smoothness in the learned policy. Moreover, our proposed framework can also improve the robustness of policy against measurement error in the state space, and can be naturally extended to distribubutionally robust setting. We apply the proposed framework to both on-policy (TRPO) and off-policy algorithm (DDPG). Through extensive experiments, we demonstrate that our method achieves improved sample efficiency and robustness.

Kaixuan Huang, Yuqing Wang, Molei Tao et al.

Deep residual networks (ResNets) have demonstrated better generalization performance than deep feedforward networks (FFNets). However, the theory behind such a phenomenon is still largely unknown. This paper studies this fundamental problem in deep learning from a so-called "neural tangent kernel" perspective. Specifically, we first show that under proper conditions, as the width goes to infinity, training deep ResNets can be viewed as learning reproducing kernel functions with some kernel function. We then compare the kernel of deep ResNets with that of deep FFNets and discover that the class of functions induced by the kernel of FFNets is asymptotically not learnable, as the depth goes to infinity. In contrast, the class of functions induced by the kernel of ResNets does not exhibit such degeneracy. Our discovery partially justifies the advantages of deep ResNets over deep FFNets in generalization abilities. Numerical results are provided to support our claim.

Minshuo Chen, Yizhou Wang, Tianyi Liu et al.

Generative Adversarial Imitation Learning (GAIL) is a powerful and practical approach for learning sequential decision-making policies. Different from Reinforcement Learning (RL), GAIL takes advantage of demonstration data by experts (e.g., human), and learns both the policy and reward function of the unknown environment. Despite the significant empirical progresses, the theory behind GAIL is still largely unknown. The major difficulty comes from the underlying temporal dependency of the demonstration data and the minimax computational formulation of GAIL without convex-concave structure. To bridge such a gap between theory and practice, this paper investigates the theoretical properties of GAIL. Specifically, we show: (1) For GAIL with general reward parameterization, the generalization can be guaranteed as long as the class of the reward functions is properly controlled; (2) For GAIL, where the reward is parameterized as a reproducing kernel function, GAIL can be efficiently solved by stochastic first order optimization algorithms, which attain sublinear convergence to a stationary solution. To the best of our knowledge, these are the first results on statistical and computational guarantees of imitation learning with reward/policy function approximation. Numerical experiments are provided to support our analysis.

Haoming Jiang, Chen Liang, Chong Wang et al.

Many multi-domain neural machine translation (NMT) models achieve knowledge transfer by enforcing one encoder to learn shared embedding across domains. However, this design lacks adaptation to individual domains. To overcome this limitation, we propose a novel multi-domain NMT model using individual modules for each domain, on which we apply word-level, adaptive and layer-wise domain mixing. We first observe that words in a sentence are often related to multiple domains. Hence, we assume each word has a domain proportion, which indicates its domain preference. Then word representations are obtained by mixing their embedding in individual domains based on their domain proportions. We show this can be achieved by carefully designing multi-head dot-product attention modules for different domains, and eventually taking weighted averages of their parameters by word-level layer-wise domain proportions. Through this, we can achieve effective domain knowledge sharing, and capture fine-grained domain-specific knowledge as well. Our experiments show that our proposed model outperforms existing ones in several NMT tasks.

Minshuo Chen, Xingguo Li, Tuo Zhao

Recurrent Neural Networks (RNNs) have been widely applied to sequential data analysis. Due to their complicated modeling structures, however, the theory behind is still largely missing. To connect theory and practice, we study the generalization properties of vanilla RNNs as well as their variants, including Minimal Gated Unit (MGU), Long Short Term Memory (LSTM), and Convolutional (Conv) RNNs. Specifically, our theory is established under the PAC-Learning framework. The generalization bound is presented in terms of the spectral norms of the weight matrices and the total number of parameters. We also establish refined generalization bounds with additional norm assumptions, and draw a comparison among these bounds. We remark: (1) Our generalization bound for vanilla RNNs is significantly tighter than the best of existing results; (2) We are not aware of any other generalization bounds for MGU, LSTM, and Conv RNNs in the exiting literature; (3) We demonstrate the advantages of these variants in generalization.

Tianyi Liu, Minshuo Chen, Mo Zhou et al.

Residual Network (ResNet) is undoubtedly a milestone in deep learning. ResNet is equipped with shortcut connections between layers, and exhibits efficient training using simple first order algorithms. Despite of the great empirical success, the reason behind is far from being well understood. In this paper, we study a two-layer non-overlapping convolutional ResNet. Training such a network requires solving a non-convex optimization problem with a spurious local optimum. We show, however, that gradient descent combined with proper normalization, avoids being trapped by the spurious local optimum, and converges to a global optimum in polynomial time, when the weight of the first layer is initialized at 0, and that of the second layer is initialized arbitrarily in a ball. Numerical experiments are provided to support our theory.

Mo Zhou, Tianyi Liu, Yan Li et al.

Numerous empirical evidence has corroborated that the noise plays a crucial rule in effective and efficient training of neural networks. The theory behind, however, is still largely unknown. This paper studies this fundamental problem through training a simple two-layer convolutional neural network model. Although training such a network requires solving a nonconvex optimization problem with a spurious local optimum and a global optimum, we prove that perturbed gradient descent and perturbed mini-batch stochastic gradient algorithms in conjunction with noise annealing is guaranteed to converge to a global optimum in polynomial time with arbitrary initialization. This implies that the noise enables the algorithm to efficiently escape from the spurious local optimum. Numerical experiments are provided to support our theory.

Yujia Xie, Haoming Jiang, Feng Liu et al.

This paper proposes a new meta-learning method -- named HARMLESS (HAwkes Relational Meta LEarning method for Short Sequences) for learning heterogeneous point process models from short event sequence data along with a relational network. Specifically, we propose a hierarchical Bayesian mixture Hawkes process model, which naturally incorporates the relational information among sequences into point process modeling. Compared with existing methods, our model can capture the underlying mixed-community patterns of the relational network, which simultaneously encourages knowledge sharing among sequences and facilitates adaptive learning for each individual sequence. We further propose an efficient stochastic variational meta expectation maximization algorithm that can scale to large problems. Numerical experiments on both synthetic and real data show that HARMLESS outperforms existing methods in terms of predicting the future events.

Yan Li, Ethan X. Fang, Huan Xu et al.

Adversarial training is a principled approach for training robust neural networks. Despite of tremendous successes in practice, its theoretical properties still remain largely unexplored. In this paper, we provide new theoretical insights of gradient descent based adversarial training by studying its computational properties, specifically on its inductive bias. We take the binary classification task on linearly separable data as an illustrative example, where the loss asymptotically attains its infimum as the parameter diverges to infinity along certain directions. Specifically, we show that when the adversarial perturbation during training has bounded $\ell_2$-norm, the classifier learned by gradient descent based adversarial training converges in direction to the maximum $\ell_2$-norm margin classifier at the rate of $\tilde{\mathcal{O}}(1/\sqrt{T})$, significantly faster than the rate $\mathcal{O}(1/\log T)$ of training with clean data. In addition, when the adversarial perturbation during training has bounded $\ell_q$-norm for some $q\ge 1$, the resulting classifier converges in direction to a maximum mixed-norm margin classifier, which has a natural interpretation of robustness, as being the maximum $\ell_2$-norm margin classifier under worst-case $\ell_q$-norm perturbation to the data. Our findings provide theoretical backups for adversarial training that it indeed promotes robustness against adversarial perturbation.

Haoming Jiang, Zhehui Chen, Minshuo Chen et al.

Generative Adversarial Networks (GANs), though powerful, is hard to train. Several recent works (brock2016neural,miyato2018spectral) suggest that controlling the spectra of weight matrices in the discriminator can significantly improve the training of GANs. Motivated by their discovery, we propose a new framework for training GANs, which allows more flexible spectrum control (e.g., making the weight matrices of the discriminator have slow singular value decays). Specifically, we propose a new reparameterization approach for the weight matrices of the discriminator in GANs, which allows us to directly manipulate the spectra of the weight matrices through various regularizers and constraints, without intensively computing singular value decompositions. Theoretically, we further show that the spectrum control improves the generalization ability of GANs. Our experiments on CIFAR-10, STL-10, and ImageNet datasets confirm that compared to other methods, our proposed method is capable of generating images with competitive quality by utilizing spectral normalization and encouraging the slow singular value decay.

Ming Yu, Zhuoran Yang, Tuo Zhao et al.

The success of machine learning methods heavily relies on having an appropriate representation for data at hand. Traditionally, machine learning approaches relied on user-defined heuristics to extract features encoding structural information about data. However, recently there has been a surge in approaches that learn how to encode the data automatically in a low dimensional space. Exponential family embedding provides a probabilistic framework for learning low-dimensional representation for various types of high-dimensional data. Though successful in practice, theoretical underpinnings for exponential family embeddings have not been established. In this paper, we study the Gaussian embedding model and develop the first theoretical results for exponential family embedding models. First, we show that, under mild condition, the embedding structure can be learned from one observation by leveraging the parameter sharing between different contexts even though the data are dependent with each other. Second, we study properties of two algorithms used for learning the embedding structure and establish convergence results for each of them. The first algorithm is based on a convex relaxation, while the other solved the non-convex formulation of the problem directly. Experiments demonstrate the effectiveness of our approach.

Zhehui Chen, Xingguo Li, Lin F. Yang et al.

We study constrained nonconvex optimization problems in machine learning, signal processing, and stochastic control. It is well-known that these problems can be rewritten to a minimax problem in a Lagrangian form. However, due to the lack of convexity, their landscape is not well understood and how to find the stable equilibria of the Lagrangian function is still unknown. To bridge the gap, we study the landscape of the Lagrangian function. Further, we define a special class of Lagrangian functions. They enjoy two properties: 1.Equilibria are either stable or unstable (Formal definition in Section 2); 2.Stable equilibria correspond to the global optima of the original problem. We show that a generalized eigenvalue (GEV) problem, including canonical correlation analysis and other problems, belongs to the class. Specifically, we characterize its stable and unstable equilibria by leveraging an invariant group and symmetric property (more details in Section 3). Motivated by these neat geometric structures, we propose a simple, efficient, and stochastic primal-dual algorithm solving the online GEV problem. Theoretically, we provide sufficient conditions, based on which we establish an asymptotic convergence rate and obtain the first sample complexity result for the online GEV problem by diffusion approximations, which are widely used in applied probability and stochastic control. Numerical results are provided to support our theory.

Yingxiang Yang, Adams Wei Yu, Zhaoran Wang et al.

We propose a nonparametric method for detecting nonlinear causal relationship within a set of multidimensional discrete time series, by using sparse additive models (SpAMs). We show that, when the input to the SpAM is a $β$-mixing time series, the model can be fitted by first approximating each unknown function with a linear combination of a set of B-spline bases, and then solving a group-lasso-type optimization problem with nonconvex regularization. Theoretically, we characterize the oracle statistical properties of the proposed sparse estimator in function estimation and model selection. Numerically, we propose an efficient pathwise iterative shrinkage thresholding algorithm (PISTA), which tames the nonconvexity and guarantees linear convergence towards the desired sparse estimator with high probability.

Tianyi Liu, Zhehui Chen, Enlu Zhou et al.

Momentum Stochastic Gradient Descent (MSGD) algorithm has been widely applied to many nonconvex optimization problems in machine learning, e.g., training deep neural networks, variational Bayesian inference, and etc. Despite its empirical success, there is still a lack of theoretical understanding of convergence properties of MSGD. To fill this gap, we propose to analyze the algorithmic behavior of MSGD by diffusion approximations for nonconvex optimization problems with strict saddle points and isolated local optima. Our study shows that the momentum helps escape from saddle points, but hurts the convergence within the neighborhood of optima (if without the step size annealing or momentum annealing). Our theoretical discovery partially corroborates the empirical success of MSGD in training deep neural networks.

Zhuoran Yang, Lin F. Yang, Ethan X. Fang et al.

Existing nonconvex statistical optimization theory and methods crucially rely on the correct specification of the underlying "true" statistical models. To address this issue, we take a first step towards taming model misspecification by studying the high-dimensional sparse phase retrieval problem with misspecified link functions. In particular, we propose a simple variant of the thresholded Wirtinger flow algorithm that, given a proper initialization, linearly converges to an estimator with optimal statistical accuracy for a broad family of unknown link functions. We further provide extensive numerical experiments to support our theoretical findings.

Xingguo Li, Lin F. Yang, Jason Ge et al.

We propose a DC proximal Newton algorithm for solving nonconvex regularized sparse learning problems in high dimensions. Our proposed algorithm integrates the proximal Newton algorithm with multi-stage convex relaxation based on the difference of convex (DC) programming, and enjoys both strong computational and statistical guarantees. Specifically, by leveraging a sophisticated characterization of sparse modeling structures/assumptions (i.e., local restricted strong convexity and Hessian smoothness), we prove that within each stage of convex relaxation, our proposed algorithm achieves (local) quadratic convergence, and eventually obtains a sparse approximate local optimum with optimal statistical properties after only a few convex relaxations. Numerical experiments are provided to support our theory.

Lin F. Yang, Vladimir Braverman, Tuo Zhao et al.

Finding the reduced-dimensional structure is critical to understanding complex networks. Existing approaches such as spectral clustering are applicable only when the full network is explicitly observed. In this paper, we focus on the online factorization and partition of implicit large-scale networks based on observations from an associated random walk. We formulate this into a nonconvex stochastic factorization problem and propose an efficient and scalable stochastic generalized Hebbian algorithm. The algorithm is able to process dependent state-transition data dynamically generated by the underlying network and learn a low-dimensional representation for each vertex. By applying a diffusion approximation analysis, we show that the continuous-time limiting process of the stochastic algorithm converges globally to the "principal components" of the Markov chain and achieves a nearly optimal sample complexity. Once given the learned low-dimensional representations, we further apply clustering techniques to recover the network partition. We show that when the associated Markov process is lumpable, one can recover the partition exactly with high probability. We apply the proposed approach to model the traffic flow of Manhattan as city-wide random walks. By using our algorithm to analyze the taxi trip data, we discover a latent partition of the Manhattan city that closely matches the traffic dynamics.

Haotian Pang, Robert Vanderbei, Han Liu et al.

High dimensional sparse learning has imposed a great computational challenge to large scale data analysis. In this paper, we are interested in a broad class of sparse learning approaches formulated as linear programs parametrized by a {\em regularization factor}, and solve them by the parametric simplex method (PSM). Our parametric simplex method offers significant advantages over other competing methods: (1) PSM naturally obtains the complete solution path for all values of the regularization parameter; (2) PSM provides a high precision dual certificate stopping criterion; (3) PSM yields sparse solutions through very few iterations, and the solution sparsity significantly reduces the computational cost per iteration. Particularly, we demonstrate the superiority of PSM over various sparse learning approaches, including Dantzig selector for sparse linear regression, LAD-Lasso for sparse robust linear regression, CLIME for sparse precision matrix estimation, sparse differential network estimation, and sparse Linear Programming Discriminant (LPD) analysis. We then provide sufficient conditions under which PSM always outputs sparse solutions such that its computational performance can be significantly boosted. Thorough numerical experiments are provided to demonstrate the outstanding performance of the PSM method.

Zhehui Chen, Lin F. Yang, Chris J. Li et al.

Multiview representation learning is very popular for latent factor analysis. It naturally arises in many data analysis, machine learning, and information retrieval applications to model dependent structures among multiple data sources. For computational convenience, existing approaches usually formulate the multiview representation learning as convex optimization problems, where global optima can be obtained by certain algorithms in polynomial time. However, many pieces of evidence have corroborated that heuristic nonconvex approaches also have good empirical computational performance and convergence to the global optima, although there is a lack of theoretical justification. Such a gap between theory and practice motivates us to study a nonconvex formulation for multiview representation learning, which can be efficiently solved by a simple stochastic gradient descent (SGD) algorithm. We first illustrate the geometry of the nonconvex formulation; Then, we establish asymptotic global rates of convergence to the global optima by diffusion approximations. Numerical experiments are provided to support our theory.

Xingguo Li, Junwei Lu, Raman Arora et al.

We propose a general theory for studying the \xl{landscape} of nonconvex \xl{optimization} with underlying symmetric structures \tz{for a class of machine learning problems (e.g., low-rank matrix factorization, phase retrieval, and deep linear neural networks)}. In specific, we characterize the locations of stationary points and the null space of Hessian matrices \xl{of the objective function} via the lens of invariant groups\removed{for associated optimization problems, including low-rank matrix factorization, phase retrieval, and deep linear neural networks}. As a major motivating example, we apply the proposed general theory to characterize the global \xl{landscape} of the \xl{nonconvex optimization in} low-rank matrix factorization problem. In particular, we illustrate how the rotational symmetry group gives rise to infinitely many nonisolated strict saddle points and equivalent global minima of the objective function. By explicitly identifying all stationary points, we divide the entire parameter space into three regions: ($\cR_1$) the region containing the neighborhoods of all strict saddle points, where the objective has negative curvatures; ($\cR_2$) the region containing neighborhoods of all global minima, where the objective enjoys strong convexity along certain directions; and ($\cR_3$) the complement of the above regions, where the gradient has sufficiently large magnitudes. We further extend our result to the matrix sensing problem. Such global landscape implies strong global convergence guarantees for popular iterative algorithms with arbitrary initial solutions.

Lin F. Yang, R. Arora, V. Braverman et al.

We use differential equations based approaches to provide some {\it \textbf{physics}} insights into analyzing the dynamics of popular optimization algorithms in machine learning. In particular, we study gradient descent, proximal gradient descent, coordinate gradient descent, proximal coordinate gradient, and Newton's methods as well as their Nesterov's accelerated variants in a unified framework motivated by a natural connection of optimization algorithms to physical systems. Our analysis is applicable to more general algorithms and optimization problems {\it \textbf{beyond}} convexity and strong convexity, e.g. Polyak-Łojasiewicz and error bound conditions (possibly nonconvex).

Xingguo Li, Tuo Zhao, Raman Arora et al.

The cyclic block coordinate descent-type (CBCD-type) methods, which performs iterative updates for a few coordinates (a block) simultaneously throughout the procedure, have shown remarkable computational performance for solving strongly convex minimization problems. Typical applications include many popular statistical machine learning methods such as elastic-net regression, ridge penalized logistic regression, and sparse additive regression. Existing optimization literature has shown that for strongly convex minimization, the CBCD-type methods attain iteration complexity of $\mathcal{O}(p\log(1/ε))$, where $ε$ is a pre-specified accuracy of the objective value, and $p$ is the number of blocks. However, such iteration complexity explicitly depends on $p$, and therefore is at least $p$ times worse than the complexity $\mathcal{O}(\log(1/ε))$ of gradient descent (GD) methods. To bridge this theoretical gap, we propose an improved convergence analysis for the CBCD-type methods. In particular, we first show that for a family of quadratic minimization problems, the iteration complexity $\mathcal{O}(\log^2(p)\cdot\log(1/ε))$ of the CBCD-type methods matches that of the GD methods in term of dependency on $p$, up to a $\log^2 p$ factor. Thus our complexity bounds are sharper than the existing bounds by at least a factor of $p/\log^2(p)$. We also provide a lower bound to confirm that our improved complexity bounds are tight (up to a $\log^2 (p)$ factor), under the assumption that the largest and smallest eigenvalues of the Hessian matrix do not scale with $p$. Finally, we generalize our analysis to other strongly convex minimization problems beyond quadratic ones.

Xingguo Li, Haoming Jiang, Jarvis Haupt et al.

Many machine learning techniques sacrifice convenient computational structures to gain estimation robustness and modeling flexibility. However, by exploring the modeling structures, we find these "sacrifices" do not always require more computational efforts. To shed light on such a "free-lunch" phenomenon, we study the square-root-Lasso (SQRT-Lasso) type regression problem. Specifically, we show that the nonsmooth loss functions of SQRT-Lasso type regression ease tuning effort and gain adaptivity to inhomogeneous noise, but is not necessarily more challenging than Lasso in computation. We can directly apply proximal algorithms (e.g. proximal gradient descent, proximal Newton, and proximal Quasi-Newton algorithms) without worrying the nonsmoothness of the loss function. Theoretically, we prove that the proximal algorithms combined with the pathwise optimization scheme enjoy fast convergence guarantees with high probability. Numerical results are provided to support our theory.

Davood Hajinezhad, Mingyi Hong, Tuo Zhao et al.

We study a stochastic and distributed algorithm for nonconvex problems whose objective consists of a sum of $N$ nonconvex $L_i/N$-smooth functions, plus a nonsmooth regularizer. The proposed NonconvEx primal-dual SpliTTing (NESTT) algorithm splits the problem into $N$ subproblems, and utilizes an augmented Lagrangian based primal-dual scheme to solve it in a distributed and stochastic manner. With a special non-uniform sampling, a version of NESTT achieves $ε$-stationary solution using $\mathcal{O}((\sum_{i=1}^N\sqrt{L_i/N})^2/ε)$ gradient evaluations, which can be up to $\mathcal{O}(N)$ times better than the (proximal) gradient descent methods. It also achieves Q-linear convergence rate for nonconvex $\ell_1$ penalized quadratic problems with polyhedral constraints. Further, we reveal a fundamental connection between primal-dual based methods and a few primal only methods such as IAG/SAG/SAGA.

Xingguo Li, Raman Arora, Han Liu et al.

We propose a stochastic variance reduced optimization algorithm for solving sparse learning problems with cardinality constraints. Sufficient conditions are provided, under which the proposed algorithm enjoys strong linear convergence guarantees and optimal estimation accuracy in high dimensions. We further extend the proposed algorithm to an asynchronous parallel variant with a near linear speedup. Numerical experiments demonstrate the efficiency of our algorithm in terms of both parameter estimation and computational performance.

Tuo Zhao, Han Liu, Tong Zhang

The pathwise coordinate optimization is one of the most important computational frameworks for high dimensional convex and nonconvex sparse learning problems. It differs from the classical coordinate optimization algorithms in three salient features: {\it warm start initialization}, {\it active set updating}, and {\it strong rule for coordinate preselection}. Such a complex algorithmic structure grants superior empirical performance, but also poses significant challenge to theoretical analysis. To tackle this long lasting problem, we develop a new theory showing that these three features play pivotal roles in guaranteeing the outstanding statistical and computational performance of the pathwise coordinate optimization framework. Particularly, we analyze the existing pathwise coordinate optimization algorithms and provide new theoretical insights into them. The obtained insights further motivate the development of several modifications to improve the pathwise coordinate optimization framework, which guarantees linear convergence to a unique sparse local optimum with optimal statistical properties in parameter estimation and support recovery. This is the first result on the computational and statistical guarantees of the pathwise coordinate optimization framework in high dimensions. Thorough numerical experiments are provided to support our theory.

Han Liu, Lie Wang, Tuo Zhao

We propose a calibrated multivariate regression method named CMR for fitting high dimensional multivariate regression models. Compared with existing methods, CMR calibrates regularization for each regression task with respect to its noise level so that it simultaneously attains improved finite-sample performance and tuning insensitiveness. Theoretically, we provide sufficient conditions under which CMR achieves the optimal rate of convergence in parameter estimation. Computationally, we propose an efficient smoothed proximal gradient algorithm with a worst-case numerical rate of convergence $\cO(1/ε)$, where $ε$ is a pre-specified accuracy of the objective function value. We conduct thorough numerical simulations to illustrate that CMR consistently outperforms other high dimensional multivariate regression methods. We also apply CMR to solve a brain activity prediction problem and find that it is as competitive as a handcrafted model created by human experts. The R package \texttt{camel} implementing the proposed method is available on the Comprehensive R Archive Network \url{http://cran.r-project.org/web/packages/camel/}.