Rong Jin

Tian Zhou, PeiSong Niu, Xue Wang et al.

Although we have witnessed great success of pre-trained models in natural language processing (NLP) and computer vision (CV), limited progress has been made for general time series analysis. Unlike NLP and CV where a unified model can be used to perform different tasks, specially designed approach still dominates in each time series analysis task such as classification, anomaly detection, forecasting, and few-shot learning. The main challenge that blocks the development of pre-trained model for time series analysis is the lack of a large amount of data for training. In this work, we address this challenge by leveraging language or CV models, pre-trained from billions of tokens, for time series analysis. Specifically, we refrain from altering the self-attention and feedforward layers of the residual blocks in the pre-trained language or image model. This model, known as the Frozen Pretrained Transformer (FPT), is evaluated through fine-tuning on all major types of tasks involving time series. Our results demonstrate that pre-trained models on natural language or images can lead to a comparable or state-of-the-art performance in all main time series analysis tasks, as illustrated in Figure 1. We also found both theoretically and empirically that the self-attention module behaviors similarly to principle component analysis (PCA), an observation that helps explains how transformer bridges the domain gap and a crucial step towards understanding the universality of a pre-trained transformer.The code is publicly available at https://github.com/DAMO-DI-ML/One_Fits_All.

Zhishuai Guo, Rong Jin, Jiebo Luo et al.

In this paper, we tackle a novel federated learning (FL) problem for optimizing a family of X-risks, to which no existing FL algorithms are applicable. In particular, the objective has the form of $\mathbb E_{z\sim S_1} f(\mathbb E_{z'\sim S_2} \ell(w; z, z'))$, where two sets of data $S_1, S_2$ are distributed over multiple machines, $\ell(\cdot)$ is a pairwise loss that only depends on the prediction outputs of the input data pairs $(z, z')$, and $f(\cdot)$ is possibly a non-linear non-convex function. This problem has important applications in machine learning, e.g., AUROC maximization with a pairwise loss, and partial AUROC maximization with a compositional loss. The challenges for designing an FL algorithm for X-risks lie in the non-decomposability of the objective over multiple machines and the interdependency between different machines. To this end, we propose an active-passive decomposition framework that decouples the gradient's components with two types, namely active parts and passive parts, where the active parts depend on local data that are computed with the local model and the passive parts depend on other machines that are communicated/computed based on historical models and samples. Under this framework, we develop two provable FL algorithms (FeDXL) for handling linear and nonlinear $f$, respectively, based on federated averaging and merging. We develop a novel theoretical analysis to combat the latency of the passive parts and the interdependency between the local model parameters and the involved data for computing local gradient estimators. We establish both iteration and communication complexities and show that using the historical samples and models for computing the passive parts do not degrade the complexities. We conduct empirical studies of FeDXL for deep AUROC and partial AUROC maximization, and demonstrate their performance compared with several baselines.

YiFan Zhang, Xue Wang, Jian Liang et al.

A fundamental challenge for machine learning models is how to generalize learned models for out-of-distribution (OOD) data. Among various approaches, exploiting invariant features by Domain Adversarial Training (DAT) received widespread attention. Despite its success, we observe training instability from DAT, mostly due to over-confident domain discriminator and environment label noise. To address this issue, we proposed Environment Label Smoothing (ELS), which encourages the discriminator to output soft probability, which thus reduces the confidence of the discriminator and alleviates the impact of noisy environment labels. We demonstrate, both experimentally and theoretically, that ELS can improve training stability, local convergence, and robustness to noisy environment labels. By incorporating ELS with DAT methods, we are able to yield state-of-art results on a wide range of domain generalization/adaptation tasks, particularly when the environment labels are highly noisy.

Tian Zhou, Jianqing Zhu, Xue Wang et al.

Various deep learning models, especially some latest Transformer-based approaches, have greatly improved the state-of-art performance for long-term time series forecasting.However, those transformer-based models suffer a severe deterioration performance with prolonged input length, which prohibits them from using extended historical info.Moreover, these methods tend to handle complex examples in long-term forecasting with increased model complexity, which often leads to a significant increase in computation and less robustness in performance(e.g., overfitting). We propose a novel neural network architecture, called TreeDRNet, for more effective long-term forecasting. Inspired by robust regression, we introduce doubly residual link structure to make prediction more robust.Built upon Kolmogorov-Arnold representation theorem, we explicitly introduce feature selection, model ensemble, and a tree structure to further utilize the extended input sequence, which improves the robustness and representation power of TreeDRNet. Unlike previous deep models for sequential forecasting work, TreeDRNet is built entirely on multilayer perceptron and thus enjoys high computational efficiency. Our extensive empirical studies show that TreeDRNet is significantly more effective than state-of-the-art methods, reducing prediction errors by 20% to 40% for multivariate time series. In particular, TreeDRNet is over 10 times more efficient than transformer-based methods. The code will be released soon.

YiFan Zhang, Xue Wang, Tian Zhou et al.

Out-of-distribution (OOD) detection is essential for the reliability of ML models. Most existing methods for OOD detection learn a fixed decision criterion from a given in-distribution dataset and apply it universally to decide if a data point is OOD. Recent work~\cite{fang2022is} shows that given only in-distribution data, it is impossible to reliably detect OOD data without extra assumptions. Motivated by the theoretical result and recent exploration of test-time adaptation methods, we propose a Non-Parametric Test Time \textbf{Ada}ptation framework for \textbf{O}ut-Of-\textbf{D}istribution \textbf{D}etection (\abbr). Unlike conventional methods, \abbr utilizes online test samples for model adaptation during testing, enhancing adaptability to changing data distributions. The framework incorporates detected OOD instances into decision-making, reducing false positive rates, particularly when ID and OOD distributions overlap significantly. We demonstrate the effectiveness of \abbr through comprehensive experiments on multiple OOD detection benchmarks, extensive empirical studies show that \abbr significantly improves the performance of OOD detection over state-of-the-art methods. Specifically, \abbr reduces the false positive rate (FPR95) by $23.23\%$ on the CIFAR-10 benchmarks and $38\%$ on the ImageNet-1k benchmarks compared to the advanced methods. Lastly, we theoretically verify the effectiveness of \abbr.

Peisong Niu, Tian Zhou, Xue Wang et al.

In the burgeoning domain of Large Language Models (LLMs), there is a growing interest in applying LLM to time series forecasting, with multiple studies focused on leveraging textual prompts to further enhance the predictive prowess. This study aims to understand how and why the integration of textual prompts into LLM can effectively improve the prediction accuracy of time series, which is not obvious at the glance, given the significant domain gap between texts and time series. Our extensive examination leads us to believe that (a) adding text prompts is roughly equivalent to introducing additional adapters, and (b) It is the introduction of learnable parameters rather than textual information that aligns the LLM with the time series forecasting task, ultimately enhancing prediction accuracy. Inspired by this discovery, we developed four adapters that explicitly address the gap between LLM and time series, and further improve the prediction accuracy. Overall,our work highlights how textual prompts enhance LLM accuracy in time series forecasting and suggests new avenues for continually improving LLM-based time series analysis.

Yifan Hu, Jie Yang, Tian Zhou et al.

Although contrastive and other representation-learning methods have long been explored in vision and NLP, their adoption in modern time series forecasters remains limited. We believe they hold strong promise for this domain. To unlock this potential, we explicitly align past and future representations, thereby bridging the distributional gap between input histories and future targets. To this end, we introduce TimeAlign, a lightweight, plug-and-play framework that establishes a new representation paradigm, distinct from contrastive learning, by aligning auxiliary features via a simple reconstruction task and feeding them back into any base forecaster. Extensive experiments across eight benchmarks verify its superior performance. Further studies indicate that the gains arise primarily from correcting frequency mismatches between historical inputs and future outputs. Additionally, we provide two theoretical justifications for how reconstruction improves forecasting generalization and how alignment increases the mutual information between learned representations and predicted targets. The code is available at https://github.com/TROUBADOUR000/TimeAlign.

Chao Chen, Tian Zhou, Yanjun Zhao et al.

Spatio-temporal forecasting is crucial in various fields and requires a careful balance between identifying subtle patterns and filtering out noise. Vector quantization (VQ) appears well-suited for this purpose, as it quantizes input vectors into a set of codebook vectors or patterns. Although VQ has shown promise in various computer vision tasks, it surprisingly falls short in enhancing the accuracy of spatio-temporal forecasting. We attribute this to two main issues: inaccurate optimization due to non-differentiability and limited representation power in hard-VQ. To tackle these challenges, we introduce Differentiable Sparse Soft-Vector Quantization (SVQ), the first VQ method to enhance spatio-temporal forecasting. SVQ balances detail preservation with noise reduction, offering full differentiability and a solid foundation in sparse regression. Our approach employs a two-layer MLP and an extensive codebook to streamline the sparse regression process, significantly cutting computational costs while simplifying training and improving performance. Empirical studies on five spatio-temporal benchmark datasets show SVQ achieves state-of-the-art results, including a 7.9% improvement on the WeatherBench-S temperature dataset and an average mean absolute error reduction of 9.4% in video prediction benchmarks (Human3.6M, KTH, and KittiCaltech), along with a 17.3% enhancement in image quality (LPIPS). Code is publicly available at https://github.com/Pachark/SVQ-Forecasting.

Tian Zhou, Ziqing Ma, Qingsong Wen et al.

Although Transformer-based methods have significantly improved state-of-the-art results for long-term series forecasting, they are not only computationally expensive but more importantly, are unable to capture the global view of time series (e.g. overall trend). To address these problems, we propose to combine Transformer with the seasonal-trend decomposition method, in which the decomposition method captures the global profile of time series while Transformers capture more detailed structures. To further enhance the performance of Transformer for long-term prediction, we exploit the fact that most time series tend to have a sparse representation in well-known basis such as Fourier transform, and develop a frequency enhanced Transformer. Besides being more effective, the proposed method, termed as Frequency Enhanced Decomposed Transformer ({\bf FEDformer}), is more efficient than standard Transformer with a linear complexity to the sequence length. Our empirical studies with six benchmark datasets show that compared with state-of-the-art methods, FEDformer can reduce prediction error by $14.8\%$ and $22.6\%$ for multivariate and univariate time series, respectively. Code is publicly available at https://github.com/MAZiqing/FEDformer.

Zhaoyuan Yin, Pichao Wang, Fan Wang et al.

Unsupervised semantic segmentation aims to obtain high-level semantic representation on low-level visual features without manual annotations. Most existing methods are bottom-up approaches that try to group pixels into regions based on their visual cues or certain predefined rules. As a result, it is difficult for these bottom-up approaches to generate fine-grained semantic segmentation when coming to complicated scenes with multiple objects and some objects sharing similar visual appearance. In contrast, we propose the first top-down unsupervised semantic segmentation framework for fine-grained segmentation in extremely complicated scenarios. Specifically, we first obtain rich high-level structured semantic concept information from large-scale vision data in a self-supervised learning manner, and use such information as a prior to discover potential semantic categories presented in target datasets. Secondly, the discovered high-level semantic categories are mapped to low-level pixel features by calculating the class activate map (CAM) with respect to certain discovered semantic representation. Lastly, the obtained CAMs serve as pseudo labels to train the segmentation module and produce the final semantic segmentation. Experimental results on multiple semantic segmentation benchmarks show that our top-down unsupervised segmentation is robust to both object-centric and scene-centric datasets under different semantic granularity levels, and outperforms all the current state-of-the-art bottom-up methods. Our code is available at \url{https://github.com/damo-cv/TransFGU}.

Qi Qi, Yi Xu, Rong Jin et al.

In this paper, we present a simple yet effective provable method (named ABSGD) for addressing the data imbalance or label noise problem in deep learning. Our method is a simple modification to momentum SGD where we assign an individual importance weight to each sample in the mini-batch. The individual-level weight of sampled data is systematically proportional to the exponential of a scaled loss value of the data, where the scaling factor is interpreted as the regularization parameter in the framework of distributionally robust optimization (DRO). Depending on whether the scaling factor is positive or negative, ABSGD is guaranteed to converge to a stationary point of an information-regularized min-max or min-min DRO problem, respectively. Compared with existing class-level weighting schemes, our method can capture the diversity between individual examples within each class. Compared with existing individual-level weighting methods using meta-learning that require three backward propagations for computing mini-batch stochastic gradients, our method is more efficient with only one backward propagation at each iteration as in standard deep learning methods. ABSGD is flexible enough to combine with other robust losses without any additional cost. Our empirical studies on several benchmark datasets demonstrate the effectiveness of the proposed method.\footnote{Code is available at:\url{https://github.com/qiqi-helloworld/ABSGD/}}

PeiSong Niu, Tian Zhou, Xue Wang et al.

Time series forecasting is essential for many practical applications, with the adoption of transformer-based models on the rise due to their impressive performance in NLP and CV. Transformers' key feature, the attention mechanism, dynamically fusing embeddings to enhance data representation, often relegating attention weights to a byproduct role. Yet, time series data, characterized by noise and non-stationarity, poses significant forecasting challenges. Our approach elevates attention weights as the primary representation for time series, capitalizing on the temporal relationships among data points to improve forecasting accuracy. Our study shows that an attention map, structured using global landmarks and local windows, acts as a robust kernel representation for data points, withstanding noise and shifts in distribution. Our method outperforms state-of-the-art models, reducing mean squared error (MSE) in multivariate time series forecasting by a notable 3.6% without altering the core neural network architecture. It serves as a versatile component that can readily replace recent patching based embedding schemes in transformer-based models, boosting their performance.

YiFan Zhang, Weiqi Chen, Zhaoyang Zhu et al.

Online updating of time series forecasting models aims to tackle the challenge of concept drifting by adjusting forecasting models based on streaming data. While numerous algorithms have been developed, most of them focus on model design and updating. In practice, many of these methods struggle with continuous performance regression in the face of accumulated concept drifts over time. To address this limitation, we present a novel approach, Concept \textbf{D}rift \textbf{D}etection an\textbf{D} \textbf{A}daptation (D3A), that first detects drifting conception and then aggressively adapts the current model to the drifted concepts after the detection for rapid adaption. To best harness the utility of historical data for model adaptation, we propose a data augmentation strategy introducing Gaussian noise into existing training instances. It helps mitigate the data distribution gap, a critical factor contributing to train-test performance inconsistency. The significance of our data augmentation process is verified by our theoretical analysis. Our empirical studies across six datasets demonstrate the effectiveness of D3A in improving model adaptation capability. Notably, compared to a simple Temporal Convolutional Network (TCN) baseline, D3A reduces the average Mean Squared Error (MSE) by $43.9\%$. For the state-of-the-art (SOTA) model, the MSE is reduced by $33.3\%$.

Yanjun Zhao, Tian Zhou, Chao Chen et al.

Time series analysis is vital for numerous applications, and transformers have become increasingly prominent in this domain. Leading methods customize the transformer architecture from NLP and CV, utilizing a patching technique to convert continuous signals into segments. Yet, time series data are uniquely challenging due to significant distribution shifts and intrinsic noise levels. To address these two challenges,we introduce the Sparse Vector Quantized FFN-Free Transformer (Sparse-VQ). Our methodology capitalizes on a sparse vector quantization technique coupled with Reverse Instance Normalization (RevIN) to reduce noise impact and capture sufficient statistics for forecasting, serving as an alternative to the Feed-Forward layer (FFN) in the transformer architecture. Our FFN-free approach trims the parameter count, enhancing computational efficiency and reducing overfitting. Through evaluations across ten benchmark datasets, including the newly introduced CAISO dataset, Sparse-VQ surpasses leading models with a 7.84% and 4.17% decrease in MAE for univariate and multivariate time series forecasting, respectively. Moreover, it can be seamlessly integrated with existing transformer-based models to elevate their performance.

Yun He, Xuxing Chen, Jiayi Xu et al.

In industrial recommendation systems, multi-task learning (learning multiple tasks simultaneously on a single model) is a predominant approach to save training/serving resources and improve recommendation performance via knowledge transfer between the joint learning tasks. However, multi-task learning often suffers from negative transfer: one or several tasks are less optimized than training them separately. To carefully balance the optimization, we propose a gradient balancing approach called MultiBalance, which is suitable for industrial-scale multi-task recommendation systems. It balances the per-task gradients to alleviate the negative transfer, while saving the huge cost for grid search or manual explorations for appropriate task weights. Moreover, compared with prior work that normally balance the per-task gradients of shared parameters, MultiBalance is more efficient since only requiring to access per-task gradients with respect to the shared feature representations. We conduct experiments on Meta's large-scale ads and feeds multi-task recommendation system, and observe that MultiBalance achieves significant gains (e.g., 0.738% improvement for normalized entropy (NE)) with neutral training cost in Queries Per Second (QPS), which is significantly more efficient than prior methods that balance per-task gradients of shared parameters with 70~80% QPS degradation.

Zhishuai Guo, Yi Xu, Wotao Yin et al.

Since its invention in 2014, the Adam optimizer has received tremendous attention. On one hand, it has been widely used in deep learning and many variants have been proposed, while on the other hand their theoretical convergence property remains to be a mystery. It is far from satisfactory in the sense that some studies require strong assumptions about the updates, which are not necessarily applicable in practice, while other studies still follow the original problematic convergence analysis of Adam, which was shown to be not sufficient to ensure convergence. Although rigorous convergence analysis exists for Adam, they impose specific requirements on the update of the adaptive step size, which are not generic enough to cover many other variants of Adam. To address theses issues, in this extended abstract, we present a simple and generic proof of convergence for a family of Adam-style methods (including Adam, AMSGrad, Adabound, etc.). Our analysis only requires an increasing or large "momentum" parameter for the first-order moment, which is indeed the case used in practice, and a boundness condition on the adaptive factor of the step size, which applies to all variants of Adam under mild conditions of stochastic gradients. We also establish a variance diminishing result for the used stochastic gradient estimators. Indeed, our analysis of Adam is so simple and generic that it can be leveraged to establish the convergence for solving a broader family of non-convex optimization problems, including min-max, compositional, and bilevel optimization problems. For the full (earlier) version of this extended abstract, please refer to arXiv:2104.14840.

Zhishuai Guo, Yi Xu, Wotao Yin et al.

Although adaptive optimization algorithms have been successful in many applications, there are still some mysteries in terms of convergence analysis that have not been unraveled. This paper provides a novel non-convex analysis of adaptive optimization to uncover some of these mysteries. Our contributions are three-fold. First, we show that an increasing or large enough momentum parameter for the first-order moment used in practice is sufficient to ensure the convergence of adaptive algorithms whose adaptive scaling factors of the step size are bounded. Second, our analysis gives insights for practical implementations, e.g., increasing the momentum parameter in a stage-wise manner in accordance with stagewise decreasing step size would help improve the convergence. Third, the modular nature of our analysis allows its extension to solving other optimization problems, e.g., compositional, min-max and bilevel problems. As an interesting yet non-trivial use case, we present algorithms for solving non-convex min-max optimization and bilevel optimization that do not require using large batches of data to estimate gradients or double loops as the literature do. Our empirical studies corroborate our theoretical results.

Ming Lin, Xiaomin Song, Qi Qian et al.

Satellite-based positioning system such as GPS often suffers from large amount of noise that degrades the positioning accuracy dramatically especially in real-time applications. In this work, we consider a data-mining approach to enhance the GPS signal. We build a large-scale high precision GPS receiver grid system to collect real-time GPS signals for training. The Gaussian Process (GP) regression is chosen to model the vertical Total Electron Content (vTEC) distribution of the ionosphere of the Earth. Our experiments show that the noise in the real-time GPS signals often exceeds the breakdown point of the conventional robust regression methods resulting in sub-optimal system performance. We propose a three-step approach to address this challenge. In the first step we perform a set of signal validity tests to separate the signals into clean and dirty groups. In the second step, we train an initial model on the clean signals and then reweigting the dirty signals based on the residual error. A final model is retrained on both the clean signals and the reweighted dirty signals. In the theoretical analysis, we prove that the proposed three-step approach is able to tolerate much higher noise level than the vanilla robust regression methods if two reweighting rules are followed. We validate the superiority of the proposed method in our real-time high precision positioning system against several popular state-of-the-art robust regression methods. Our method achieves centimeter positioning accuracy in the benchmark region with probability $78.4\%$ , outperforming the second best baseline method by a margin of $8.3\%$. The benchmark takes 6 hours on 20,000 CPU cores or 14 years on a single CPU.

Yi Xu, Rong Jin, Tianbao Yang

Accelerated gradient (AG) methods are breakthroughs in convex optimization, improving the convergence rate of the gradient descent method for optimization with smooth functions. However, the analysis of AG methods for non-convex optimization is still limited. It remains an open question whether AG methods from convex optimization can accelerate the convergence of the gradient descent method for finding local minimum of non-convex optimization problems. This paper provides an affirmative answer to this question. In particular, we analyze two renowned variants of AG methods (namely Polyak's Heavy Ball method and Nesterov's Accelerated Gradient method) for extracting the negative curvature from random noise, which is central to escaping from saddle points. By leveraging the proposed AG methods for extracting the negative curvature, we present a new AG algorithm with double loops for non-convex optimization~\footnote{this is in contrast to a single-loop AG algorithm proposed in a recent manuscript~\citep{AGNON}, which directly analyzed the Nesterov's AG method for non-convex optimization and appeared online on November 29, 2017. However, we emphasize that our work is an independent work, which is inspired by our earlier work~\citep{NEON17} and is based on a different novel analysis.}, which converges to second-order stationary point $\x$ such that $\|\nabla f(\x)\|\leq ε$ and $\nabla^2 f(\x)\geq -\sqrtε I$ with $\widetilde O(1/ε^{1.75})$ iteration complexity, improving that of gradient descent method by a factor of $ε^{-0.25}$ and matching the best iteration complexity of second-order Hessian-free methods for non-convex optimization.

Yi Xu, Rong Jin, Tianbao Yang

Two classes of methods have been proposed for escaping from saddle points with one using the second-order information carried by the Hessian and the other adding the noise into the first-order information. The existing analysis for algorithms using noise in the first-order information is quite involved and hides the essence of added noise, which hinder further improvements of these algorithms. In this paper, we present a novel perspective of noise-adding technique, i.e., adding the noise into the first-order information can help extract the negative curvature from the Hessian matrix, and provide a formal reasoning of this perspective by analyzing a simple first-order procedure. More importantly, the proposed procedure enables one to design purely first-order stochastic algorithms for escaping from non-degenerate saddle points with a much better time complexity (almost linear time in terms of the problem's dimensionality). In particular, we develop a {\bf first-order stochastic algorithm} based on our new technique and an existing algorithm that only converges to a first-order stationary point to enjoy a time complexity of {$\widetilde O(d/ε^{3.5})$ for finding a nearly second-order stationary point $\bf{x}$ such that $\|\nabla F(bf{x})\|\leq ε$ and $\nabla^2 F(bf{x})\geq -\sqrtεI$ (in high probability), where $F(\cdot)$ denotes the objective function and $d$ is the dimensionality of the problem. To the best of our knowledge, this is the best theoretical result of first-order algorithms for stochastic non-convex optimization, which is even competitive with if not better than existing stochastic algorithms hinging on the second-order information.

Cong Leng, Hao Li, Shenghuo Zhu et al.

Although deep learning models are highly effective for various learning tasks, their high computational costs prohibit the deployment to scenarios where either memory or computational resources are limited. In this paper, we focus on compressing and accelerating deep models with network weights represented by very small numbers of bits, referred to as extremely low bit neural network. We model this problem as a discretely constrained optimization problem. Borrowing the idea from Alternating Direction Method of Multipliers (ADMM), we decouple the continuous parameters from the discrete constraints of network, and cast the original hard problem into several subproblems. We propose to solve these subproblems using extragradient and iterative quantization algorithms that lead to considerably faster convergency compared to conventional optimization methods. Extensive experiments on image recognition and object detection verify that the proposed algorithm is more effective than state-of-the-art approaches when coming to extremely low bit neural network.

Lijun Zhang, Tianbao Yang, Rong Jin et al.

In this paper, we develop a randomized algorithm and theory for learning a sparse model from large-scale and high-dimensional data, which is usually formulated as an empirical risk minimization problem with a sparsity-inducing regularizer. Under the assumption that there exists a (approximately) sparse solution with high classification accuracy, we argue that the dual solution is also sparse or approximately sparse. The fact that both primal and dual solutions are sparse motivates us to develop a randomized approach for a general convex-concave optimization problem. Specifically, the proposed approach combines the strength of random projection with that of sparse learning: it utilizes random projection to reduce the dimensionality, and introduces $\ell_1$-norm regularization to alleviate the approximation error caused by random projection. Theoretical analysis shows that under favored conditions, the randomized algorithm can accurately recover the optimal solutions to the convex-concave optimization problem (i.e., recover both the primal and dual solutions).

Tianbao Yang, Lijun Zhang, Rong Jin et al.

In this paper, we study randomized reduction methods, which reduce high-dimensional features into low-dimensional space by randomized methods (e.g., random projection, random hashing), for large-scale high-dimensional classification. Previous theoretical results on randomized reduction methods hinge on strong assumptions about the data, e.g., low rank of the data matrix or a large separable margin of classification, which hinder their applications in broad domains. To address these limitations, we propose dual-sparse regularized randomized reduction methods that introduce a sparse regularizer into the reduced dual problem. Under a mild condition that the original dual solution is a (nearly) sparse vector, we show that the resulting dual solution is close to the original dual solution and concentrates on its support set. In numerical experiments, we present an empirical study to support the analysis and we also present a novel application of the dual-sparse regularized randomized reduction methods to reducing the communication cost of distributed learning from large-scale high-dimensional data.

Lijun Zhang, Tianbao Yang, Rong Jin et al.

Traditional algorithms for stochastic optimization require projecting the solution at each iteration into a given domain to ensure its feasibility. When facing complex domains, such as positive semi-definite cones, the projection operation can be expensive, leading to a high computational cost per iteration. In this paper, we present a novel algorithm that aims to reduce the number of projections for stochastic optimization. The proposed algorithm combines the strength of several recent developments in stochastic optimization, including mini-batch, extra-gradient, and epoch gradient descent, in order to effectively explore the smoothness and strong convexity. We show, both in expectation and with a high probability, that when the objective function is both smooth and strongly convex, the proposed algorithm achieves the optimal $O(1/T)$ rate of convergence with only $O(\log T)$ projections. Our empirical study verifies the theoretical result.

Mehrdad Mahdavi, Tianbao Yang, Rong Jin

We develop an improved bound for the approximation error of the Nyström method under the assumption that there is a large eigengap in the spectrum of kernel matrix. This is based on the empirical observation that the eigengap has a significant impact on the approximation error of the Nyström method. Our approach is based on the concentration inequality of integral operator and the theory of matrix perturbation. Our analysis shows that when there is a large eigengap, we can improve the approximation error of the Nyström method from $O(N/m^{1/4})$ to $O(N/m^{1/2})$ when measured in Frobenius norm, where $N$ is the size of the kernel matrix, and $m$ is the number of sampled columns.

Liu Yang, Rong Jin, Rahul Sukthankar

Distance metric learning is an important component for many tasks, such as statistical classification and content-based image retrieval. Existing approaches for learning distance metrics from pairwise constraints typically suffer from two major problems. First, most algorithms only offer point estimation of the distance metric and can therefore be unreliable when the number of training examples is small. Second, since these algorithms generally select their training examples at random, they can be inefficient if labeling effort is limited. This paper presents a Bayesian framework for distance metric learning that estimates a posterior distribution for the distance metric from labeled pairwise constraints. We describe an efficient algorithm based on the variational method for the proposed Bayesian approach. Furthermore, we apply the proposed Bayesian framework to active distance metric learning by selecting those unlabeled example pairs with the greatest uncertainty in relative distance. Experiments in classification demonstrate that the proposed framework achieves higher classification accuracy and identifies more informative training examples than the non-Bayesian approach and state-of-the-art distance metric learning algorithms.

Kaizhu Huang, Rong Jin, Zenglin Xu et al.

Most existing distance metric learning methods assume perfect side information that is usually given in pairwise or triplet constraints. Instead, in many real-world applications, the constraints are derived from side information, such as users' implicit feedbacks and citations among articles. As a result, these constraints are usually noisy and contain many mistakes. In this work, we aim to learn a distance metric from noisy constraints by robust optimization in a worst-case scenario, to which we refer as robust metric learning. We formulate the learning task initially as a combinatorial optimization problem, and show that it can be elegantly transformed to a convex programming problem. We present an efficient learning algorithm based on smooth optimization [7]. It has a worst-case convergence rate of O(1/{\surd}{\varepsilon}) for smooth optimization problems, where {\varepsilon} is the desired error of the approximate solution. Finally, our empirical study with UCI data sets demonstrate the effectiveness of the proposed method in comparison to state-of-the-art methods.