Si Si

Justin Cui, Ruochen Wang, Si Si et al.

Dataset Distillation is a newly emerging area that aims to distill large datasets into much smaller and highly informative synthetic ones to accelerate training and reduce storage. Among various dataset distillation methods, trajectory-matching-based methods (MTT) have achieved SOTA performance in many tasks, e.g., on CIFAR-10/100. However, due to exorbitant memory consumption when unrolling optimization through SGD steps, MTT fails to scale to large-scale datasets such as ImageNet-1K. Can we scale this SOTA method to ImageNet-1K and does its effectiveness on CIFAR transfer to ImageNet-1K? To answer these questions, we first propose a procedure to exactly compute the unrolled gradient with constant memory complexity, which allows us to scale MTT to ImageNet-1K seamlessly with ~6x reduction in memory footprint. We further discover that it is challenging for MTT to handle datasets with a large number of classes, and propose a novel soft label assignment that drastically improves its convergence. The resulting algorithm sets new SOTA on ImageNet-1K: we can scale up to 50 IPCs (Image Per Class) on ImageNet-1K on a single GPU (all previous methods can only scale to 2 IPCs on ImageNet-1K), leading to the best accuracy (only 5.9% accuracy drop against full dataset training) while utilizing only 4.2% of the number of data points - an 18.2% absolute gain over prior SOTA. Our code is available at https://github.com/justincui03/tesla

Justin Cui, Ruochen Wang, Si Si et al.

Dataset Condensation is a newly emerging technique aiming at learning a tiny dataset that captures the rich information encoded in the original dataset. As the size of datasets contemporary machine learning models rely on becomes increasingly large, condensation methods become a prominent direction for accelerating network training and reducing data storage. Despite numerous methods have been proposed in this rapidly growing field, evaluating and comparing different condensation methods is non-trivial and still remains an open issue. The quality of condensed dataset are often shadowed by many critical contributing factors to the end performance, such as data augmentation and model architectures. The lack of a systematic way to evaluate and compare condensation methods not only hinders our understanding of existing techniques, but also discourages practical usage of the synthesized datasets. This work provides the first large-scale standardized benchmark on Dataset Condensation. It consists of a suite of evaluations to comprehensively reflect the generability and effectiveness of condensation methods through the lens of their generated dataset. Leveraging this benchmark, we conduct a large-scale study of current condensation methods, and report many insightful findings that open up new possibilities for future development. The benchmark library, including evaluators, baseline methods, and generated datasets, is open-sourced to facilitate future research and application.

Yihan Wang, Si Si, Daliang Li et al.

Pretrained large language models (LLMs) are general purpose problem solvers applicable to a diverse set of tasks with prompts. They can be further improved towards a specific task by fine-tuning on a specialized dataset. However, fine-tuning usually makes the model narrowly specialized on this dataset with reduced general in-context learning performances, which is undesirable whenever the fine-tuned model needs to handle additional tasks where no fine-tuning data is available. In this work, we first demonstrate that fine-tuning on a single task indeed decreases LLMs' general in-context learning performance. We discover one important cause of such forgetting, format specialization, where the model overfits to the format of the fine-tuned task.We further show that format specialization happens at the very beginning of fine-tuning. To solve this problem, we propose Prompt Tuning with MOdel Tuning (ProMoT), a simple yet effective two-stage fine-tuning framework that reduces format specialization and improves generalization.ProMoT offloads task-specific format learning into additional and removable parameters by first doing prompt tuning and then fine-tuning the model itself with this soft prompt attached. With experiments on several fine-tuning tasks and 8 in-context evaluation tasks, we show that ProMoT achieves comparable performance on fine-tuned tasks to standard fine-tuning, but with much less loss of in-context learning performances across a board range of out-of-domain evaluation tasks. More importantly, ProMoT can even enhance generalization on in-context learning tasks that are semantically related to the fine-tuned task, e.g. ProMoT on En-Fr translation significantly improves performance on other language pairs, and ProMoT on NLI improves performance on summarization. Experiments also show that ProMoT can improve the generalization performance of multi-task training.

Cho-Jui Hsieh, Si Si, Felix X. Yu et al.

Large language models (LLMs) have demonstrated remarkable capabilities in solving complex open-domain tasks, guided by comprehensive instructions and demonstrations provided in the form of prompts. However, these prompts can be lengthy, often comprising hundreds of lines and thousands of tokens, and their design often requires considerable human effort. Recent research has explored automatic prompt engineering for short prompts, typically consisting of one or a few sentences. However, the automatic design of long prompts remains a challenging problem due to its immense search space. In this paper, we investigate the performance of greedy algorithms and genetic algorithms for automatic long prompt engineering. We demonstrate that a simple greedy approach with beam search outperforms other methods in terms of search efficiency. Moreover, we introduce two novel techniques that utilize search history to enhance the effectiveness of LLM-based mutation in our search algorithm. Our results show that the proposed automatic long prompt engineering algorithm achieves an average of 9.2% accuracy gain on eight tasks in Big Bench Hard, highlighting the significance of automating prompt designs to fully harness the capabilities of LLMs.

Jixuan Leng, Si Si, Hsiang-Fu Yu et al.

Preference optimization is widely used to align Large Language Models (LLMs) with preference feedback. However, most existing methods train on a single positive-negative pair per prompt, discarding additional supervision available in preference datasets that typically contain multiple candidate responses. Motivated by this limitation, recent work explores group-wise preference optimization, which jointly contrasts multiple responses for the same prompt, but its empirical behavior and scalability remain underexplored due to the memory overhead of group-coupled objectives. In this work, we introduce a memory-efficient group-wise preference optimization algorithm that preserves gradients while decoupling samples during backpropagation, substantially reducing peak memory usage, which enables scalable training with larger group sizes. Across both offline and online alignment settings, we show that leveraging multiple responses consistently outperforms single-pair training. Furthermore, incorporating a negative log-likelihood (NLL) term on positive responses is critical for both performance gains and training stability.

Wei-Lin Chiang, Xuanqing Liu, Si Si et al.

Graph convolutional network (GCN) has been successfully applied to many graph-based applications; however, training a large-scale GCN remains challenging. Current SGD-based algorithms suffer from either a high computational cost that exponentially grows with number of GCN layers, or a large space requirement for keeping the entire graph and the embedding of each node in memory. In this paper, we propose Cluster-GCN, a novel GCN algorithm that is suitable for SGD-based training by exploiting the graph clustering structure. Cluster-GCN works as the following: at each step, it samples a block of nodes that associate with a dense subgraph identified by a graph clustering algorithm, and restricts the neighborhood search within this subgraph. This simple but effective strategy leads to significantly improved memory and computational efficiency while being able to achieve comparable test accuracy with previous algorithms. To test the scalability of our algorithm, we create a new Amazon2M data with 2 million nodes and 61 million edges which is more than 5 times larger than the previous largest publicly available dataset (Reddit). For training a 3-layer GCN on this data, Cluster-GCN is faster than the previous state-of-the-art VR-GCN (1523 seconds vs 1961 seconds) and using much less memory (2.2GB vs 11.2GB). Furthermore, for training 4 layer GCN on this data, our algorithm can finish in around 36 minutes while all the existing GCN training algorithms fail to train due to the out-of-memory issue. Furthermore, Cluster-GCN allows us to train much deeper GCN without much time and memory overhead, which leads to improved prediction accuracy---using a 5-layer Cluster-GCN, we achieve state-of-the-art test F1 score 99.36 on the PPI dataset, while the previous best result was 98.71 by [16]. Our codes are publicly available at https://github.com/google-research/google-research/tree/master/cluster_gcn.

Jui-Nan Yen, Si Si, Zhao Meng et al.

Low-rank adaption (LoRA) is a widely used parameter-efficient finetuning method for LLM that reduces memory requirements. However, current LoRA optimizers lack transformation invariance, meaning the actual updates to the weights depends on how the two LoRA factors are scaled or rotated. This deficiency leads to inefficient learning and sub-optimal solutions in practice. This paper introduces LoRA-RITE, a novel adaptive matrix preconditioning method for LoRA optimization, which can achieve transformation invariance and remain computationally efficient. We provide theoretical analysis to demonstrate the benefit of our method and conduct experiments on various LLM tasks with different models including Gemma 2B, 7B, and mT5-XXL. The results demonstrate consistent improvements against existing optimizers. For example, replacing Adam with LoRA-RITE during LoRA fine-tuning of Gemma-2B yielded 4.6\% accuracy gain on Super-Natural Instructions and 3.5\% accuracy gain across other four LLM benchmarks (HellaSwag, ArcChallenge, GSM8K, OpenBookQA).

Guangze Zheng, Shijie Lin, Haobo Zuo et al.

This work proposes the Lattice Boltzmann Model (LBM) to learn real-world pixel dynamicity for visual tracking. LBM decomposes visual representations into dynamic pixel lattices and solves pixel motion states through collision-streaming processes. Specifically, the high-dimensional distribution of the target pixels is acquired through a multilayer predict-update network to estimate the pixel positions and visibility. The predict stage formulates lattice collisions among the spatial neighborhood of target pixels and develops lattice streaming within the temporal visual context. The update stage rectifies the pixel distributions with online visual representations. Compared with existing methods, LBM demonstrates practical applicability in an online and real-time manner, which can efficiently adapt to real-world visual tracking tasks. Comprehensive evaluations of real-world point tracking benchmarks such as TAP-Vid and RoboTAP validate LBM's efficiency. A general evaluation of large-scale open-world object tracking benchmarks such as TAO, BFT, and OVT-B further demonstrates LBM's real-world practicality.

Ruochen Wang, Si Si, Felix Yu et al.

The trade-off between expressiveness and interpretability remains a core challenge when building human-centric predictive models for classification and decision-making. While symbolic rules offer interpretability, they often lack expressiveness, whereas neural networks excel in performance but are known for being black boxes. In this paper, we show a combination of Large Language Models (LLMs) and symbolic programs can bridge this gap. In the proposed LLM-based Symbolic Programs (LSPs), the pretrained LLM with natural language prompts provides a massive set of interpretable modules that can transform raw input into natural language concepts. Symbolic programs then integrate these modules into an interpretable decision rule. To train LSPs, we develop a divide-and-conquer approach to incrementally build the program from scratch, where the learning process of each step is guided by LLMs. To evaluate the effectiveness of LSPs in extracting interpretable and accurate knowledge from data, we introduce IL-Bench, a collection of diverse tasks, including both synthetic and real-world scenarios across different modalities. Empirical results demonstrate LSP's superior performance compared to traditional neurosymbolic programs and vanilla automatic prompt tuning methods. Moreover, as the knowledge learned by LSP is a combination of natural language descriptions and symbolic rules, it is easily transferable to humans (interpretable), and other LLMs, and generalizes well to out-of-distribution samples.

Yang Li, Si Si, Gang Li et al.

Attentional mechanisms are order-invariant. Positional encoding is a crucial component to allow attention-based deep model architectures such as Transformer to address sequences or images where the position of information matters. In this paper, we propose a novel positional encoding method based on learnable Fourier features. Instead of hard-coding each position as a token or a vector, we represent each position, which can be multi-dimensional, as a trainable encoding based on learnable Fourier feature mapping, modulated with a multi-layer perceptron. The representation is particularly advantageous for a spatial multi-dimensional position, e.g., pixel positions on an image, where $L_2$ distances or more complex positional relationships need to be captured. Our experiments based on several public benchmark tasks show that our learnable Fourier feature representation for multi-dimensional positional encoding outperforms existing methods by both improving the accuracy and allowing faster convergence.

Yuanhao Xiong, Xuanqing Liu, Li-Cheng Lan et al.

Many optimizers have been proposed for training deep neural networks, and they often have multiple hyperparameters, which make it tricky to benchmark their performance. In this work, we propose a new benchmarking protocol to evaluate both end-to-end efficiency (training a model from scratch without knowing the best hyperparameter) and data-addition training efficiency (the previously selected hyperparameters are used for periodically re-training the model with newly collected data). For end-to-end efficiency, unlike previous work that assumes random hyperparameter tuning, which over-emphasizes the tuning time, we propose to evaluate with a bandit hyperparameter tuning strategy. A human study is conducted to show that our evaluation protocol matches human tuning behavior better than the random search. For data-addition training, we propose a new protocol for assessing the hyperparameter sensitivity to data shift. We then apply the proposed benchmarking framework to 7 optimizers and various tasks, including computer vision, natural language processing, reinforcement learning, and graph mining. Our results show that there is no clear winner across all the tasks.

Hongge Chen, Si Si, Yang Li et al.

Multi-stage training and knowledge transfer, from a large-scale pretraining task to various finetuning tasks, have revolutionized natural language processing and computer vision resulting in state-of-the-art performance improvements. In this paper, we develop a multi-stage influence function score to track predictions from a finetuned model all the way back to the pretraining data. With this score, we can identify the pretraining examples in the pretraining task that contribute most to a prediction in the finetuning task. The proposed multi-stage influence function generalizes the original influence function for a single model in (Koh & Liang, 2017), thereby enabling influence computation through both pretrained and finetuned models. We study two different scenarios with the pretrained embeddings fixed or updated in the finetuning tasks. We test our proposed method in various experiments to show its effectiveness and potential applications.

Yang Li, Julien Amelot, Xin Zhou et al.

It has been of increasing interest in the field to develop automatic machineries to facilitate the design process. In this paper, we focus on assisting graphical user interface (UI) layout design, a crucial task in app development. Given a partial layout, which a designer has entered, our model learns to complete the layout by predicting the remaining UI elements with a correct position and dimension as well as the hierarchical structures. Such automation will significantly ease the effort of UI designers and developers. While we focus on interface layout prediction, our model can be generally applicable for other layout prediction problems that involve tree structures and 2-dimensional placements. Particularly, we design two versions of Transformer-based tree decoders: Pointer and Recursive Transformer, and experiment with these models on a public dataset. We also propose several metrics for measuring the accuracy of tree prediction and ground these metrics in the domain of user experience. These contribute a new task and methods to deep learning research.

Xuanqing Liu, Si Si, Xiaojin Zhu et al.

In this paper, we proposed a general framework for data poisoning attacks to graph-based semi-supervised learning (G-SSL). In this framework, we first unify different tasks, goals, and constraints into a single formula for data poisoning attack in G-SSL, then we propose two specialized algorithms to efficiently solve two important cases --- poisoning regression tasks under $\ell_2$-norm constraint and classification tasks under $\ell_0$-norm constraint. In the former case, we transform it into a non-convex trust region problem and show that our gradient-based algorithm with delicate initialization and update scheme finds the (globally) optimal perturbation. For the latter case, although it is an NP-hard integer programming problem, we propose a probabilistic solver that works much better than the classical greedy method. Lastly, we test our framework on real datasets and evaluate the robustness of G-SSL algorithms. For instance, on the MNIST binary classification problem (50000 training data with 50 labeled), flipping two labeled data is enough to make the model perform like random guess (around 50\% error).

Hongge Chen, Huan Zhang, Si Si et al.

We study the robustness verification problem for tree-based models, including decision trees, random forests (RFs) and gradient boosted decision trees (GBDTs). Formal robustness verification of decision tree ensembles involves finding the exact minimal adversarial perturbation or a guaranteed lower bound of it. Existing approaches find the minimal adversarial perturbation by a mixed integer linear programming (MILP) problem, which takes exponential time so is impractical for large ensembles. Although this verification problem is NP-complete in general, we give a more precise complexity characterization. We show that there is a simple linear time algorithm for verifying a single tree, and for tree ensembles, the verification problem can be cast as a max-clique problem on a multi-partite graph with bounded boxicity. For low dimensional problems when boxicity can be viewed as constant, this reformulation leads to a polynomial time algorithm. For general problems, by exploiting the boxicity of the graph, we develop an efficient multi-level verification algorithm that can give tight lower bounds on the robustness of decision tree ensembles, while allowing iterative improvement and any-time termination. OnRF/GBDT models trained on 10 datasets, our algorithm is hundreds of times faster than the previous approach that requires solving MILPs, and is able to give tight robustness verification bounds on large GBDTs with hundreds of deep trees.

Xuanqing Liu, Tesi Xiao, Si Si et al.

Neural Ordinary Differential Equation (Neural ODE) has been proposed as a continuous approximation to the ResNet architecture. Some commonly used regularization mechanisms in discrete neural networks (e.g. dropout, Gaussian noise) are missing in current Neural ODE networks. In this paper, we propose a new continuous neural network framework called Neural Stochastic Differential Equation (Neural SDE) network, which naturally incorporates various commonly used regularization mechanisms based on random noise injection. Our framework can model various types of noise injection frequently used in discrete networks for regularization purpose, such as dropout and additive/multiplicative noise in each block. We provide theoretical analysis explaining the improved robustness of Neural SDE models against input perturbations/adversarial attacks. Furthermore, we demonstrate that the Neural SDE network can achieve better generalization than the Neural ODE and is more resistant to adversarial and non-adversarial input perturbations.

Zhourong Chen, Yang Li, Samy Bengio et al.

The concept of conditional computation for deep nets has been proposed previously to improve model performance by selectively using only parts of the model conditioned on the sample it is processing. In this paper, we investigate input-dependent dynamic filter selection in deep convolutional neural networks (CNNs). The problem is interesting because the idea of forcing different parts of the model to learn from different types of samples may help us acquire better filters in CNNs, improve the model generalization performance and potentially increase the interpretability of model behavior. We propose a novel yet simple framework called GaterNet, which involves a backbone and a gater network. The backbone network is a regular CNN that performs the major computation needed for making a prediction, while a global gater network is introduced to generate binary gates for selectively activating filters in the backbone network based on each input. Extensive experiments on CIFAR and ImageNet datasets show that our models consistently outperform the original models with a large margin. On CIFAR-10, our model also improves upon state-of-the-art results.

Patrick H. Chen, Si Si, Sanjiv Kumar et al.

Neural language models have been widely used in various NLP tasks, including machine translation, next word prediction and conversational agents. However, it is challenging to deploy these models on mobile devices due to their slow prediction speed, where the bottleneck is to compute top candidates in the softmax layer. In this paper, we introduce a novel softmax layer approximation algorithm by exploiting the clustering structure of context vectors. Our algorithm uses a light-weight screening model to predict a much smaller set of candidate words based on the given context, and then conducts an exact softmax only within that subset. Training such a procedure end-to-end is challenging as traditional clustering methods are discrete and non-differentiable, and thus unable to be used with back-propagation in the training process. Using the Gumbel softmax, we are able to train the screening model end-to-end on the training set to exploit data distribution. The algorithm achieves an order of magnitude faster inference than the original softmax layer for predicting top-$k$ words in various tasks such as beam search in machine translation or next words prediction. For example, for machine translation task on German to English dataset with around 25K vocabulary, we can achieve 20.4 times speed up with 98.9\% precision@1 and 99.3\% precision@5 with the original softmax layer prediction, while state-of-the-art ~\citep{MSRprediction} only achieves 6.7x speedup with 98.7\% precision@1 and 98.1\% precision@5 for the same task.

Yang Li, Lukasz Kaiser, Samy Bengio et al.

Existing attention mechanisms are trained to attend to individual items in a collection (the memory) with a predefined, fixed granularity, e.g., a word token or an image grid. We propose area attention: a way to attend to areas in the memory, where each area contains a group of items that are structurally adjacent, e.g., spatially for a 2D memory such as images, or temporally for a 1D memory such as natural language sentences. Importantly, the shape and the size of an area are dynamically determined via learning, which enables a model to attend to information with varying granularity. Area attention can easily work with existing model architectures such as multi-head attention for simultaneously attending to multiple areas in the memory. We evaluate area attention on two tasks: neural machine translation (both character and token-level) and image captioning, and improve upon strong (state-of-the-art) baselines in all the cases. These improvements are obtainable with a basic form of area attention that is parameter free.

Patrick H. Chen, Si Si, Yang Li et al.

Model compression is essential for serving large deep neural nets on devices with limited resources or applications that require real-time responses. As a case study, a state-of-the-art neural language model usually consists of one or more recurrent layers sandwiched between an embedding layer used for representing input tokens and a softmax layer for generating output tokens. For problems with a very large vocabulary size, the embedding and the softmax matrices can account for more than half of the model size. For instance, the bigLSTM model achieves state-of- the-art performance on the One-Billion-Word (OBW) dataset with around 800k vocabulary, and its word embedding and softmax matrices use more than 6GBytes space, and are responsible for over 90% of the model parameters. In this paper, we propose GroupReduce, a novel compression method for neural language models, based on vocabulary-partition (block) based low-rank matrix approximation and the inherent frequency distribution of tokens (the power-law distribution of words). The experimental results show our method can significantly outperform traditional compression methods such as low-rank approximation and pruning. On the OBW dataset, our method achieved 6.6 times compression rate for the embedding and softmax matrices, and when combined with quantization, our method can achieve 26 times compression rate, which translates to a factor of 12.8 times compression for the entire model with very little degradation in perplexity.

Si Si, Sanjiv Kumar, Yang Li

Use of nonlinear feature maps via kernel approximation has led to success in many online learning tasks. As a popular kernel approximation method, Nyström approximation, has been well investigated, and various landmark points selection methods have been proposed to improve the approximation quality. However, these improved Nyström methods cannot be directly applied to the online learning setting as they need to access the entire dataset to learn the landmark points, while we need to update model on-the-fly in the online setting. To address this challenge, we propose Adaptive Nyström approximation for solving nonlinear online learning problems. The key idea is to adaptively modify the landmark points via online kmeans and adjust the model accordingly via solving least square problem followed by a gradient descent step. We show that the resulting algorithm outperforms state-of-the-art online learning methods under the same budget.

Huan Zhang, Si Si, Cho-Jui Hsieh

In this paper, we present a novel massively parallel algorithm for accelerating the decision tree building procedure on GPUs (Graphics Processing Units), which is a crucial step in Gradient Boosted Decision Tree (GBDT) and random forests training. Previous GPU based tree building algorithms are based on parallel multi-scan or radix sort to find the exact tree split, and thus suffer from scalability and performance issues. We show that using a histogram based algorithm to approximately find the best split is more efficient and scalable on GPU. By identifying the difference between classical GPU-based image histogram construction and the feature histogram construction in decision tree training, we develop a fast feature histogram building kernel on GPU with carefully designed computational and memory access sequence to reduce atomic update conflict and maximize GPU utilization. Our algorithm can be used as a drop-in replacement for histogram construction in popular tree boosting systems to improve their scalability. As an example, to train GBDT on epsilon dataset, our method using a main-stream GPU is 7-8 times faster than histogram based algorithm on CPU in LightGBM and 25 times faster than the exact-split finding algorithm in XGBoost on a dual-socket 28-core Xeon server, while achieving similar prediction accuracy.

Cho-Jui Hsieh, Si Si, Inderjit S. Dhillon

Kernel machines often yield superior predictive performance on various tasks; however, they suffer from severe computational challenges. In this paper, we show how to overcome the important challenge of speeding up kernel machines. In particular, we develop a parallel block minimization framework for solving kernel machines, including kernel SVM and kernel logistic regression. Our framework proceeds by dividing the problem into smaller subproblems by forming a block-diagonal approximation of the Hessian matrix. The subproblems are then solved approximately in parallel. After that, a communication efficient line search procedure is developed to ensure sufficient reduction of the objective function value at each iteration. We prove global linear convergence rate of the proposed method with a wide class of subproblem solvers, and our analysis covers strongly convex and some non-strongly convex functions. We apply our algorithm to solve large-scale kernel SVM problems on distributed systems, and show a significant improvement over existing parallel solvers. As an example, on the covtype dataset with half-a-million samples, our algorithm can obtain an approximate solution with 96% accuracy in 20 seconds using 32 machines, while all the other parallel kernel SVM solvers require more than 2000 seconds to achieve a solution with 95% accuracy. Moreover, our algorithm can scale to very large data sets, such as the kdd algebra dataset with 8 million samples and 20 million features.

Rashish Tandon, Si Si, Pradeep Ravikumar et al.

In this paper, we investigate a divide and conquer approach to Kernel Ridge Regression (KRR). Given n samples, the division step involves separating the points based on some underlying disjoint partition of the input space (possibly via clustering), and then computing a KRR estimate for each partition. The conquering step is simple: for each partition, we only consider its own local estimate for prediction. We establish conditions under which we can give generalization bounds for this estimator, as well as achieve optimal minimax rates. We also show that the approximation error component of the generalization error is lesser than when a single KRR estimate is fit on the data: thus providing both statistical and computational advantages over a single KRR estimate over the entire data (or an averaging over random partitions as in other recent work, [30]). Lastly, we provide experimental validation for our proposed estimator and our assumptions.

Cho-Jui Hsieh, Si Si, Inderjit S. Dhillon

The kernel support vector machine (SVM) is one of the most widely used classification methods; however, the amount of computation required becomes the bottleneck when facing millions of samples. In this paper, we propose and analyze a novel divide-and-conquer solver for kernel SVMs (DC-SVM). In the division step, we partition the kernel SVM problem into smaller subproblems by clustering the data, so that each subproblem can be solved independently and efficiently. We show theoretically that the support vectors identified by the subproblem solution are likely to be support vectors of the entire kernel SVM problem, provided that the problem is partitioned appropriately by kernel clustering. In the conquer step, the local solutions from the subproblems are used to initialize a global coordinate descent solver, which converges quickly as suggested by our analysis. By extending this idea, we develop a multilevel Divide-and-Conquer SVM algorithm with adaptive clustering and early prediction strategy, which outperforms state-of-the-art methods in terms of training speed, testing accuracy, and memory usage. As an example, on the covtype dataset with half-a-million samples, DC-SVM is 7 times faster than LIBSVM in obtaining the exact SVM solution (to within $10^{-6}$ relative error) which achieves 96.15% prediction accuracy. Moreover, with our proposed early prediction strategy, DC-SVM achieves about 96% accuracy in only 12 minutes, which is more than 100 times faster than LIBSVM.