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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).

Data pruning, the combinatorial task of selecting a small and informative subset from a large dataset, is crucial for mitigating the enormous computational costs associated with training data-hungry modern deep learning models at scale. Since large-scale data collections are invariably noisy, developing data pruning strategies that remain robust even in the presence of corruption is critical in practice. Unfortunately, the existing heuristics for (robust) data pruning lack theoretical coherence and rely on heroic assumptions, that are, often unattainable, by the very nature of the problem setting. Moreover, these strategies often yield sub-optimal neural scaling laws even compared to random sampling, especially in scenarios involving strong corruption and aggressive pruning rates -- making provably robust data pruning an open challenge. In response, in this work, we propose Geometric Median ($\gm$) Matching -- a herding~\citep{welling2009herding} style greedy algorithm -- that yields a $k$-subset such that the mean of the subset approximates the geometric median of the (potentially) noisy dataset. Theoretically, we show that $\gm$ Matching enjoys an improved $\gO(1/k)$ scaling over $\gO(1/\sqrt{k})$ scaling of uniform sampling; while achieving the optimal breakdown point of 1/2 even under arbitrary corruption. Extensive experiments across popular deep learning benchmarks indicate that $\gm$ Matching consistently outperforms prior state-of-the-art; the gains become more profound at high rates of corruption and aggressive pruning rates; making $\gm$ Matching a strong baseline for future research in robust data pruning.

The performance of a model trained with \textit{noisy labels} is often improved by simply \textit{retraining} the model with its own predicted \textit{hard} labels (i.e., $1$/$0$ labels). Yet, a detailed theoretical characterization of this phenomenon is lacking. In this paper, we theoretically analyze retraining in a linearly separable setting with randomly corrupted labels given to us and prove that retraining can improve the population accuracy obtained by initially training with the given (noisy) labels. To the best of our knowledge, this is the first such theoretical result. Retraining finds application in improving training with local label differential privacy (DP) which involves training with noisy labels. We empirically show that retraining selectively on the samples for which the predicted label matches the given label significantly improves label DP training at \textit{no extra privacy cost}; we call this \textit{consensus-based retraining}. As an example, when training ResNet-18 on CIFAR-100 with $\epsilon=3$ label DP, we obtain $6.4\%$ improvement in accuracy with consensus-based retraining.

There is a notable dearth of results characterizing the preconditioning effect of Adam and showing how it may alleviate the curse of ill-conditioning -- an issue plaguing gradient descent (GD). In this work, we perform a detailed analysis of Adam's preconditioning effect for quadratic functions and quantify to what extent Adam can mitigate the dependence on the condition number of the Hessian. Our key finding is that Adam can suffer less from the condition number but at the expense of suffering a dimension-dependent quantity. Specifically, for a $d$-dimensional quadratic with a diagonal Hessian having condition number $\kappa$, we show that the effective condition number-like quantity controlling the iteration complexity of Adam without momentum is $\mathcal{O}(\min(d, \kappa))$. For a diagonally dominant Hessian, we obtain a bound of $\mathcal{O}(\min(d \sqrt{d \kappa}, \kappa))$ for the corresponding quantity. Thus, when $d < \mathcal{O}(\kappa^p)$ where $p = 1$ for a diagonal Hessian and $p = 1/3$ for a diagonally dominant Hessian, Adam can outperform GD (which has an $\mathcal{O}(\kappa)$ dependence). On the negative side, our results suggest that Adam can be worse than GD for a sufficiently non-diagonal Hessian even if $d \ll \mathcal{O}(\kappa^{1/3})$; we corroborate this with empirical evidence. Finally, we extend our analysis to functions satisfying per-coordinate Lipschitz smoothness and a modified version of the Polyak-\L ojasiewicz condition.

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.

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., Comment: ICLR 2024

Extreme multi-label classification (XMC) is a popular framework for solving many real-world problems that require accurate prediction from a very large number of potential output choices. A popular approach for dealing with the large label space is to arrange the labels into a shallow tree-based index and then learn an ML model to efficiently search this index via beam search. Existing methods initialize the tree index by clustering the label space into a few mutually exclusive clusters based on pre-defined features and keep it fixed throughout the training procedure. This approach results in a sub-optimal indexing structure over the label space and limits the search performance to the quality of choices made during the initialization of the index. In this paper, we propose a novel method ELIAS which relaxes the tree-based index to a specialized weighted graph-based index which is learned end-to-end with the final task objective. More specifically, ELIAS models the discrete cluster-to-label assignments in the existing tree-based index as soft learnable parameters that are learned jointly with the rest of the ML model. ELIAS achieves state-of-the-art performance on several large-scale extreme classification benchmarks with millions of labels. In particular, ELIAS can be up to 2.5% better at precision@1 and up to 4% better at recall@100 than existing XMC methods. A PyTorch implementation of ELIAS along with other resources is available at https://github.com/nilesh2797/ELIAS., Comment: 21 pages, 9 figures, NeurIPS 2022 camera-ready publication

Approximate K-Nearest Neighbor Search (AKNNS) has now become ubiquitous in modern applications, for example, as a fast search procedure with two tower deep learning models. Graph-based methods for AKNNS in particular have received great attention due to their superior performance. These methods rely on greedy graph search to traverse the data points as embedding vectors in a database. Under this greedy search scheme, we make a key observation: many distance computations do not influence search updates so these computations can be approximated without hurting performance. As a result, we propose FINGER, a fast inference method to achieve efficient graph search. FINGER approximates the distance function by estimating angles between neighboring residual vectors with low-rank bases and distribution matching. The approximated distance can be used to bypass unnecessary computations, which leads to faster searches. Empirically, accelerating a popular graph-based method named HNSW by FINGER is shown to outperform existing graph-based methods by 20%-60% across different benchmark datasets.

Conventional methods for query autocompletion aim to predict which completed query a user will select from a list. A shortcoming of this approach is that users often do not know which query will provide the best retrieval performance on the current information retrieval system, meaning that any query autocompletion methods trained to mimic user behavior can lead to suboptimal query suggestions. To overcome this limitation, we propose a new approach that explicitly optimizes the query suggestions for downstream retrieval performance. We formulate this as a problem of ranking a set of rankings, where each query suggestion is represented by the downstream item ranking it produces. We then present a learning method that ranks query suggestions by the quality of their item rankings. The algorithm is based on a counterfactual learning approach that is able to leverage feedback on the items (e.g., clicks, purchases) to evaluate query suggestions through an unbiased estimator, thus avoiding the assumption that users write or select optimal queries. We establish theoretical support for the proposed approach and provide learning-theoretic guarantees. We also present empirical results on publicly available datasets, and demonstrate real-world applicability using data from an online shopping store.

Data augmentation is popular in the training of large neural networks; currently, however, there is no clear theoretical comparison between different algorithmic choices on how to use augmented data. In this paper, we take a step in this direction - we first present a simple and novel analysis for linear regression with label invariant augmentations, demonstrating that data augmentation consistency (DAC) is intrinsically more efficient than empirical risk minimization on augmented data (DA-ERM). The analysis is then extended to misspecified augmentations (i.e., augmentations that change the labels), which again demonstrates the merit of DAC over DA-ERM. Further, we extend our analysis to non-linear models (e.g., neural networks) and present generalization bounds. Finally, we perform experiments that make a clean and apples-to-apples comparison (i.e., with no extra modeling or data tweaks) between DAC and DA-ERM using CIFAR-100 and WideResNet; these together demonstrate the superior efficacy of DAC.

Learning on graphs has attracted significant attention in the learning community due to numerous real-world applications. In particular, graph neural networks (GNNs), which take numerical node features and graph structure as inputs, have been shown to achieve state-of-the-art performance on various graph-related learning tasks. Recent works exploring the correlation between numerical node features and graph structure via self-supervised learning have paved the way for further performance improvements of GNNs. However, methods used for extracting numerical node features from raw data are still graph-agnostic within standard GNN pipelines. This practice is sub-optimal as it prevents one from fully utilizing potential correlations between graph topology and node attributes. To mitigate this issue, we propose a new self-supervised learning framework, Graph Information Aided Node feature exTraction (GIANT). GIANT makes use of the eXtreme Multi-label Classification (XMC) formalism, which is crucial for fine-tuning the language model based on graph information, and scales to large datasets. We also provide a theoretical analysis that justifies the use of XMC over link prediction and motivates integrating XR-Transformers, a powerful method for solving XMC problems, into the GIANT framework. We demonstrate the superior performance of GIANT over the standard GNN pipeline on Open Graph Benchmark datasets: For example, we improve the accuracy of the top-ranked method GAMLP from $68.25\%$ to $69.67\%$, SGC from $63.29\%$ to $66.10\%$ and MLP from $47.24\%$ to $61.10\%$ on the ogbn-papers100M dataset by leveraging GIANT., Comment: Published in ICLR 2022

We propose a three-stage framework for forecasting high-dimensional time-series data. Our method first estimates parameters for each univariate time series. Next, we use these parameters to cluster the time series. These clusters can be viewed as multivariate time series, for which we then compute parameters. The forecasted values of a single time series can depend on the history of other time series in the same cluster, accounting for intra-cluster similarity while minimizing potential noise in predictions by ignoring inter-cluster effects. Our framework -- which we refer to as "cluster-and-conquer" -- is highly general, allowing for any time-series forecasting and clustering method to be used in each step. It is computationally efficient and embarrassingly parallel. We motivate our framework with a theoretical analysis in an idealized mixed linear regression setting, where we provide guarantees on the quality of the estimates. We accompany these guarantees with experimental results that demonstrate the advantages of our framework: when instantiated with simple linear autoregressive models, we are able to achieve state-of-the-art results on several benchmark datasets, sometimes outperforming deep-learning-based approaches., Comment: 25 pages, 3 figures

Extreme multi-label text classification (XMC) seeks to find relevant labels from an extreme large label collection for a given text input. Many real-world applications can be formulated as XMC problems, such as recommendation systems, document tagging and semantic search. Recently, transformer based XMC methods, such as X-Transformer and LightXML, have shown significant improvement over other XMC methods. Despite leveraging pre-trained transformer models for text representation, the fine-tuning procedure of transformer models on large label space still has lengthy computational time even with powerful GPUs. In this paper, we propose a novel recursive approach, XR-Transformer to accelerate the procedure through recursively fine-tuning transformer models on a series of multi-resolution objectives related to the original XMC objective function. Empirical results show that XR-Transformer takes significantly less training time compared to other transformer-based XMC models while yielding better state-of-the-art results. In particular, on the public Amazon-3M dataset with 3 million labels, XR-Transformer is not only 20x faster than X-Transformer but also improves the Precision@1 from 51% to 54%.

Partition-based methods are increasingly-used in extreme multi-label classification (XMC) problems due to their scalability to large output spaces (e.g., millions or more). However, existing methods partition the large label space into mutually exclusive clusters, which is sub-optimal when labels have multi-modality and rich semantics. For instance, the label "Apple" can be the fruit or the brand name, which leads to the following research question: can we disentangle these multi-modal labels with non-exclusive clustering tailored for downstream XMC tasks? In this paper, we show that the label assignment problem in partition-based XMC can be formulated as an optimization problem, with the objective of maximizing precision rates. This leads to an efficient algorithm to form flexible and overlapped label clusters, and a method that can alternatively optimizes the cluster assignments and the model parameters for partition-based XMC. Experimental results on synthetic and real datasets show that our method can successfully disentangle multi-modal labels, leading to state-of-the-art (SOTA) results on four XMC benchmarks.

We consider the problem of semantic matching in product search: given a customer query, retrieve all semantically related products from a huge catalog of size 100 million, or more. Because of large catalog spaces and real-time latency constraints, semantic matching algorithms not only desire high recall but also need to have low latency. Conventional lexical matching approaches (e.g., Okapi-BM25) exploit inverted indices to achieve fast inference time, but fail to capture behavioral signals between queries and products. In contrast, embedding-based models learn semantic representations from customer behavior data, but the performance is often limited by shallow neural encoders due to latency constraints. Semantic product search can be viewed as an eXtreme Multi-label Classification (XMC) problem, where customer queries are input instances and products are output labels. In this paper, we aim to improve semantic product search by using tree-based XMC models where inference time complexity is logarithmic in the number of products. We consider hierarchical linear models with n-gram features for fast real-time inference. Quantitatively, our method maintains a low latency of 1.25 milliseconds per query and achieves a 65% improvement of Recall@100 (60.9% v.s. 36.8%) over a competing embedding-based DSSM model. Our model is robust to weight pruning with varying thresholds, which can flexibly meet different system requirements for online deployments. Qualitatively, our method can retrieve products that are complementary to existing product search system and add diversity to the match set., Comment: Accepted in KDD 2021 Applied Data Science Track

Geometric median (\textsc{Gm}) is a classical method in statistics for achieving a robust estimation of the uncorrupted data; under gross corruption, it achieves the optimal breakdown point of 0.5. However, its computational complexity makes it infeasible for robustifying stochastic gradient descent (SGD) for high-dimensional optimization problems. In this paper, we show that by applying \textsc{Gm} to only a judiciously chosen block of coordinates at a time and using a memory mechanism, one can retain the breakdown point of 0.5 for smooth non-convex problems, with non-asymptotic convergence rates comparable to the SGD with \textsc{Gm}.

There is a dearth of convergence results for differentially private federated learning (FL) with non-Lipschitz objective functions (i.e., when gradient norms are not bounded). The primary reason for this is that the clipping operation (i.e., projection onto an $\ell_2$ ball of a fixed radius called the clipping threshold) for bounding the sensitivity of the average update to each client's update introduces bias depending on the clipping threshold and the number of local steps in FL, and analyzing this is not easy. For Lipschitz functions, the Lipschitz constant serves as a trivial clipping threshold with zero bias. However, Lipschitzness does not hold in many practical settings; moreover, verifying it and computing the Lipschitz constant is hard. Thus, the choice of the clipping threshold is non-trivial and requires a lot of tuning in practice. In this paper, we provide the first convergence result for private FL on smooth \textit{convex} objectives \textit{for a general clipping threshold} -- \textit{without assuming Lipschitzness}. We also look at a simpler alternative to clipping (for bounding sensitivity) which is \textit{normalization} -- where we use only a scaled version of the unit vector along the client updates, completely discarding the magnitude information. {The resulting normalization-based private FL algorithm is theoretically shown to have better convergence than its clipping-based counterpart on smooth convex functions. We corroborate our theory with synthetic experiments as well as experiments on benchmarking datasets.

Extreme multi-label classification (XMC) aims to learn a model that can tag data points with a subset of relevant labels from an extremely large label set. Real world e-commerce applications like personalized recommendations and product advertising can be formulated as XMC problems, where the objective is to predict for a user a small subset of items from a catalog of several million products. For such applications, a common approach is to organize these labels into a tree, enabling training and inference times that are logarithmic in the number of labels. While training a model once a label tree is available is well studied, designing the structure of the tree is a difficult task that is not yet well understood, and can dramatically impact both model latency and statistical performance. Existing approaches to tree construction fall at an extreme point, either optimizing exclusively for statistical performance, or for latency. We propose an efficient information theory inspired algorithm to construct intermediary operating points that trade off between the benefits of both. Our algorithm enables interpolation between these objectives, which was not previously possible. We corroborate our theoretical analysis with numerical results, showing that on the Wiki-500K benchmark dataset our method can reduce a proxy for expected latency by up to 28% while maintaining the same accuracy as Parabel. On several datasets derived from e-commerce customer logs, our modified label tree is able to improve this expected latency metric by up to 20% while maintaining the same accuracy. Finally, we discuss challenges in realizing these latency improvements in deployed models., Comment: To appear in CIKM 2021

We consider the combinatorial bandits problem, where at each time step, the online learner selects a size-$k$ subset $s$ from the arms set $\mathcal{A}$, where $\left|\mathcal{A}\right| = n$, and observes a stochastic reward of each arm in the selected set $s$. The goal of the online learner is to minimize the regret, induced by not selecting $s^*$ which maximizes the expected total reward. Specifically, we focus on a challenging setting where 1) the reward distribution of an arm depends on the set $s$ it is part of, and crucially 2) there is \textit{no total order} for the arms in $\mathcal{A}$. In this paper, we formally present a reward model that captures set-dependent reward distribution and assumes no total order for arms. Correspondingly, we propose an Upper Confidence Bound (UCB) algorithm that maintains UCB for each individual arm and selects the arms with top-$k$ UCB. We develop a novel regret analysis and show an $O\left(\frac{k^2 n \log T}{\epsilon}\right)$ gap-dependent regret bound as well as an $O\left(k^2\sqrt{n T \log T}\right)$ gap-independent regret bound. We also provide a lower bound for the proposed reward model, which shows our proposed algorithm is near-optimal for any constant $k$. Empirical results on various reward models demonstrate the broad applicability of our algorithm.

We propose two linear bandits algorithms with per-step complexity sublinear in the number of arms $K$. The algorithms are designed for applications where the arm set is extremely large and slowly changing. Our key realization is that choosing an arm reduces to a maximum inner product search (MIPS) problem, which can be solved approximately without breaking regret guarantees. Existing approximate MIPS solvers run in sublinear time. We extend those solvers and present theoretical guarantees for online learning problems, where adaptivity (i.e., a later step depends on the feedback in previous steps) becomes a unique challenge. We then explicitly characterize the tradeoff between the per-step complexity and regret. For sufficiently large $K$, our algorithms have sublinear per-step complexity and $\tilde O(\sqrt{T})$ regret. Empirically, we evaluate our proposed algorithms in a synthetic environment and a real-world online movie recommendation problem. Our proposed algorithms can deliver a more than 72 times speedup compared to the linear time baselines while retaining similar regret., Comment: Accepted at ICML 2022

Query auto-completion (QAC) is a fundamental feature in search engines where the task is to suggest plausible completions of a prefix typed in the search bar. Previous queries in the user session can provide useful context for the user's intent and can be leveraged to suggest auto-completions that are more relevant while adhering to the user's prefix. Such session-aware QACs can be generated by recent sequence-to-sequence deep learning models; however, these generative approaches often do not meet the stringent latency requirements of responding to each user keystroke. Moreover, these generative approaches pose the risk of showing nonsensical queries. In this paper, we provide a solution to this problem: we take the novel approach of modeling session-aware QAC as an eXtreme Multi-Label Ranking (XMR) problem where the input is the previous query in the session and the user's current prefix, while the output space is the set of tens of millions of queries entered by users in the recent past. We adapt a popular XMR algorithm for this purpose by proposing several modifications to the key steps in the algorithm. The proposed modifications yield a 3.9x improvement in terms of Mean Reciprocal Rank (MRR) over the baseline XMR approach on a public search logs dataset. We are able to maintain an inference latency of less than 10 ms while still using session context. When compared against baseline models of acceptable latency, we observed a 33% improvement in MRR for short prefixes of up to 3 characters. Moreover, our model yielded a statistically significant improvement of 2.81% over a production QAC system in terms of suggestion acceptance rate, when deployed on the search bar of an online shopping store as part of an A/B test., Comment: Accepted in KDD 2021. Updated results for baseline XMR

We propose \texttt{FedGLOMO}, a novel federated learning (FL) algorithm with an iteration complexity of $\mathcal{O}(\epsilon^{-1.5})$ to converge to an $\epsilon$-stationary point (i.e., $\mathbb{E}[\|\nabla f(\bm{x})\|^2] \leq \epsilon$) for smooth non-convex functions -- under arbitrary client heterogeneity and compressed communication -- compared to the $\mathcal{O}(\epsilon^{-2})$ complexity of most prior works. Our key algorithmic idea that enables achieving this improved complexity is based on the observation that the convergence in FL is hampered by two sources of high variance: (i) the global server aggregation step with multiple local updates, exacerbated by client heterogeneity, and (ii) the noise of the local client-level stochastic gradients. By modeling the server aggregation step as a generalized gradient-type update, we propose a variance-reducing momentum-based global update at the server, which when applied in conjunction with variance-reduced local updates at the clients, enables \texttt{FedGLOMO} to enjoy an improved convergence rate. Moreover, we derive our results under a novel and more realistic client-heterogeneity assumption which we verify empirically -- unlike prior assumptions that are hard to verify. Our experiments illustrate the intrinsic variance reduction effect of \texttt{FedGLOMO}, which implicitly suppresses client-drift in heterogeneous data distribution settings and promotes communication efficiency.

Many large-scale applications amount to finding relevant results from an enormous output space of potential candidates. For example, finding the best matching product from a large catalog or suggesting related search phrases on a search engine. The size of the output space for these problems can range from millions to billions, and can even be infinite in some applications. Moreover, training data is often limited for the long-tail items in the output space. Fortunately, items in the output space are often correlated thereby presenting an opportunity to alleviate the data sparsity issue. In this paper, we propose the Prediction for Enormous and Correlated Output Spaces (PECOS) framework, a versatile and modular machine learning framework for solving prediction problems for very large output spaces, and apply it to the eXtreme Multilabel Ranking (XMR) problem: given an input instance, find and rank the most relevant items from an enormous but fixed and finite output space. We propose a three phase framework for PECOS: (i) in the first phase, PECOS organizes the output space using a semantic indexing scheme, (ii) in the second phase, PECOS uses the indexing to narrow down the output space by orders of magnitude using a machine learned matching scheme, and (iii) in the third phase, PECOS ranks the matched items using a final ranking scheme. The versatility and modularity of PECOS allows for easy plug-and-play of various choices for the indexing, matching, and ranking phases. We also develop very fast inference procedures which allow us to perform XMR predictions in real time; for example, inference takes less than 1 millisecond per input on the dataset with 2.8 million labels. The PECOS software is available at https://libpecos.org.

We study the problem of batch learning from bandit feedback in the setting of extremely large action spaces. Learning from extreme bandit feedback is ubiquitous in recommendation systems, in which billions of decisions are made over sets consisting of millions of choices in a single day, yielding massive observational data. In these large-scale real-world applications, supervised learning frameworks such as eXtreme Multi-label Classification (XMC) are widely used despite the fact that they incur significant biases due to the mismatch between bandit feedback and supervised labels. Such biases can be mitigated by importance sampling techniques, but these techniques suffer from impractical variance when dealing with a large number of actions. In this paper, we introduce a selective importance sampling estimator (sIS) that operates in a significantly more favorable bias-variance regime. The sIS estimator is obtained by performing importance sampling on the conditional expectation of the reward with respect to a small subset of actions for each instance (a form of Rao-Blackwellization). We employ this estimator in a novel algorithmic procedure -- named Policy Optimization for eXtreme Models (POXM) -- for learning from bandit feedback on XMC tasks. In POXM, the selected actions for the sIS estimator are the top-p actions of the logging policy, where p is adjusted from the data and is significantly smaller than the size of the action space. We use a supervised-to-bandit conversion on three XMC datasets to benchmark our POXM method against three competing methods: BanditNet, a previously applied partial matching pruning strategy, and a supervised learning baseline. Whereas BanditNet sometimes improves marginally over the logging policy, our experiments show that POXM systematically and significantly improves over all baselines.

The goal of co-clustering is to simultaneously identify a clustering of rows as well as columns of a two dimensional data matrix. A number of co-clustering techniques have been proposed including information-theoretic co-clustering and the minimum sum-squared residue co-clustering method. However, most existing co-clustering algorithms are designed to find pairwise disjoint and exhaustive co-clusters while many real-world datasets contain not only a large overlap between co-clusters but also outliers which should not belong to any co-cluster. In this paper, we formulate the problem of Non-Exhaustive, Overlapping Co-Clustering where both of the row and column clusters are allowed to overlap with each other and outliers for each dimension of the data matrix are not assigned to any cluster. To solve this problem, we propose intuitive objective functions, and develop an an efficient iterative algorithm which we call the NEO-CC algorithm. We theoretically show that the NEO-CC algorithm monotonically decreases the proposed objective functions. Experimental results show that the NEO-CC algorithm is able to effectively capture the underlying co-clustering structure of real-world data, and thus outperforms state-of-the-art clustering and co-clustering methods. This manuscript includes an extended analysis of [21].

The architecture of Transformer is based entirely on self-attention, and has been shown to outperform models that employ recurrence on sequence transduction tasks such as machine translation. The superior performance of Transformer has been attributed to propagating signals over shorter distances, between positions in the input and the output, compared to the recurrent architectures. We establish connections between the dynamics in Transformer and recurrent networks to argue that several factors including gradient flow along an ensemble of multiple weakly dependent paths play a paramount role in the success of Transformer. We then leverage the dynamics to introduce {\em Multiresolution Transformer Networks} as the first architecture that exploits hierarchical structure in data via self-attention. Our models significantly outperform state-of-the-art recurrent and hierarchical recurrent models on two real-world datasets for query suggestion, namely, \aol and \amazon. In particular, on AOL data, our model registers at least 20\% improvement on each precision score, and over 25\% improvement on the BLEU score with respect to the best performing recurrent model. We thus provide strong evidence that recurrence is not essential for modeling hierarchical structure., Comment: Initial version

We study the problem of inverting a deep generative model with ReLU activations. Inversion corresponds to finding a latent code vector that explains observed measurements as much as possible. In most prior works this is performed by attempting to solve a non-convex optimization problem involving the generator. In this paper we obtain several novel theoretical results for the inversion problem. We show that for the realizable case, single layer inversion can be performed exactly in polynomial time, by solving a linear program. Further, we show that for multiple layers, inversion is NP-hard and the pre-image set can be non-convex. For generative models of arbitrary depth, we show that exact recovery is possible in polynomial time with high probability, if the layers are expanding and the weights are randomly selected. Very recent work analyzed the same problem for gradient descent inversion. Their analysis requires significantly higher expansion (logarithmic in the latent dimension) while our proposed algorithm can provably reconstruct even with constant factor expansion. We also provide provable error bounds for different norms for reconstructing noisy observations. Our empirical validation demonstrates that we obtain better reconstructions when the latent dimension is large.

We propose a variant of the Frank-Wolfe algorithm for solving a class of sparse/low-rank optimization problems. Our formulation includes Elastic Net, regularized SVMs and phase retrieval as special cases. The proposed Primal-Dual Block Frank-Wolfe algorithm reduces the per-iteration cost while maintaining linear convergence rate. The per iteration cost of our method depends on the structural complexity of the solution (i.e. sparsity/low-rank) instead of the ambient dimension. We empirically show that our algorithm outperforms the state-of-the-art methods on (multi-class) classification tasks.

Deep neural networks have yielded superior performance in many applications; however, the gradient computation in a deep model with millions of instances lead to a lengthy training process even with modern GPU/TPU hardware acceleration. In this paper, we propose AutoAssist, a simple framework to accelerate training of a deep neural network. Typically, as the training procedure evolves, the amount of improvement in the current model by a stochastic gradient update on each instance varies dynamically. In AutoAssist, we utilize this fact and design a simple instance shrinking operation, which is used to filter out instances with relatively low marginal improvement to the current model; thus the computationally intensive gradient computations are performed on informative instances as much as possible. We prove that the proposed technique outperforms vanilla SGD with existing importance sampling approaches for linear SVM problems, and establish an O(1/k) convergence for strongly convex problems. In order to apply the proposed techniques to accelerate training of deep models, we propose to jointly train a very lightweight Assistant network in addition to the original deep network referred to as Boss. The Assistant network is designed to gauge the importance of a given instance with respect to the current Boss such that a shrinking operation can be applied in the batch generator. With careful design, we train the Boss and Assistant in a nonblocking and asynchronous fashion such that overhead is minimal. We demonstrate that AutoAssist reduces the number of epochs by 40% for training a ResNet to reach the same test accuracy on an image classification data set and saves 30% training time needed for a transformer model to yield the same BLEU scores on a translation dataset.

Machine learning (ML) techniques are enjoying rapidly increasing adoption. However, designing and implementing the systems that support ML models in real-world deployments remains a significant obstacle, in large part due to the radically different development and deployment profile of modern ML methods, and the range of practical concerns that come with broader adoption. We propose to foster a new systems machine learning research community at the intersection of the traditional systems and ML communities, focused on topics such as hardware systems for ML, software systems for ML, and ML optimized for metrics beyond predictive accuracy. To do this, we describe a new conference, MLSys, that explicitly targets research at the intersection of systems and machine learning with a program committee split evenly between experts in systems and ML, and an explicit focus on topics at the intersection of the two.

The adversarial training procedure proposed by Madry et al. (2018) is one of the most effective methods to defend against adversarial examples in deep neural networks (DNNs). In our paper, we shed some lights on the practicality and the hardness of adversarial training by showing that the effectiveness (robustness on test set) of adversarial training has a strong correlation with the distance between a test point and the manifold of training data embedded by the network. Test examples that are relatively far away from this manifold are more likely to be vulnerable to adversarial attacks. Consequentially, an adversarial training based defense is susceptible to a new class of attacks, the "blind-spot attack", where the input images reside in "blind-spots" (low density regions) of the empirical distribution of training data but is still on the ground-truth data manifold. For MNIST, we found that these blind-spots can be easily found by simply scaling and shifting image pixel values. Most importantly, for large datasets with high dimensional and complex data manifold (CIFAR, ImageNet, etc), the existence of blind-spots in adversarial training makes defending on any valid test examples difficult due to the curse of dimensionality and the scarcity of training data. Additionally, we find that blind-spots also exist on provable defenses including (Wong & Kolter, 2018) and (Sinha et al., 2018) because these trainable robustness certificates can only be practically optimized on a limited set of training data., Comment: Accepted by International Conference on Learning Representations (ICLR) 2019. Huan Zhang and Hongge Chen contributed equally

Adversarial examples are carefully constructed modifications to an input that completely change the output of a classifier but are imperceptible to humans. Despite these successful attacks for continuous data (such as image and audio samples), generating adversarial examples for discrete structures such as text has proven significantly more challenging. In this paper we formulate the attacks with discrete input on a set function as an optimization task. We prove that this set function is submodular for some popular neural network text classifiers under simplifying assumption. This finding guarantees a $1-1/e$ approximation factor for attacks that use the greedy algorithm. Meanwhile, we show how to use the gradient of the attacked classifier to guide the greedy search. Empirical studies with our proposed optimization scheme show significantly improved attack ability and efficiency, on three different text classification tasks over various baselines. We also use a joint sentence and word paraphrasing technique to maintain the original semantics and syntax of the text. This is validated by a human subject evaluation in subjective metrics on the quality and semantic coherence of our generated adversarial text., Comment: In SysML 2019

The goal of a recommendation system is to predict the interest of a user in a given item by exploiting the existing set of ratings as well as certain user/item features. A standard approach to modeling this problem is Inductive Matrix Completion where the predicted rating is modeled as an inner product of the user and the item features projected onto a latent space. In order to learn the parameters effectively from a small number of observed ratings, the latent space is constrained to be low-dimensional which implies that the parameter matrix is constrained to be low-rank. However, such bilinear modeling of the ratings can be limiting in practice and non-linear prediction functions can lead to significant improvements. A natural approach to introducing non-linearity in the prediction function is to apply a non-linear activation function on top of the projected user/item features. Imposition of non-linearities further complicates an already challenging problem that has two sources of non-convexity: a) low-rank structure of the parameter matrix, and b) non-linear activation function. We show that one can still solve the non-linear Inductive Matrix Completion problem using gradient descent type methods as long as the solution is initialized well. That is, close to the optima, the optimization function is strongly convex and hence admits standard optimization techniques, at least for certain activation functions, such as Sigmoid and tanh. We also highlight the importance of the activation function and show how ReLU can behave significantly differently than say a sigmoid function. Finally, we apply our proposed technique to recommendation systems and semi-supervised clustering, and show that our method can lead to much better performance than standard linear Inductive Matrix Completion methods.

Verifying the robustness property of a general Rectified Linear Unit (ReLU) network is an NP-complete problem [Katz, Barrett, Dill, Julian and Kochenderfer CAV17]. Although finding the exact minimum adversarial distortion is hard, giving a certified lower bound of the minimum distortion is possible. Current available methods of computing such a bound are either time-consuming or delivering low quality bounds that are too loose to be useful. In this paper, we exploit the special structure of ReLU networks and provide two computationally efficient algorithms Fast-Lin and Fast-Lip that are able to certify non-trivial lower bounds of minimum distortions, by bounding the ReLU units with appropriate linear functions Fast-Lin, or by bounding the local Lipschitz constant Fast-Lip. Experiments show that (1) our proposed methods deliver bounds close to (the gap is 2-3X) exact minimum distortion found by Reluplex in small MNIST networks while our algorithms are more than 10,000 times faster; (2) our methods deliver similar quality of bounds (the gap is within 35% and usually around 10%; sometimes our bounds are even better) for larger networks compared to the methods based on solving linear programming problems but our algorithms are 33-14,000 times faster; (3) our method is capable of solving large MNIST and CIFAR networks up to 7 layers with more than 10,000 neurons within tens of seconds on a single CPU core. In addition, we show that, in fact, there is no polynomial time algorithm that can approximately find the minimum $\ell_1$ adversarial distortion of a ReLU network with a $0.99\ln n$ approximation ratio unless $\mathsf{NP}$=$\mathsf{P}$, where $n$ is the number of neurons in the network., Comment: Tsui-Wei Weng and Huan Zhang contributed equally

Vanishing and exploding gradients are two of the main obstacles in training deep neural networks, especially in capturing long range dependencies in recurrent neural networks~(RNNs). In this paper, we present an efficient parametrization of the transition matrix of an RNN that allows us to stabilize the gradients that arise in its training. Specifically, we parameterize the transition matrix by its singular value decomposition(SVD), which allows us to explicitly track and control its singular values. We attain efficiency by using tools that are common in numerical linear algebra, namely Householder reflectors for representing the orthogonal matrices that arise in the SVD. By explicitly controlling the singular values, our proposed Spectral-RNN method allows us to easily solve the exploding gradient problem and we observe that it empirically solves the vanishing gradient issue to a large extent. We note that the SVD parameterization can be used for any rectangular weight matrix, hence it can be easily extended to any deep neural network, such as a multi-layer perceptron. Theoretically, we demonstrate that our parameterization does not lose any expressive power, and show how it controls generalization of RNN for the classification task. %, and show how it potentially makes the optimization process easier. Our extensive experimental results also demonstrate that the proposed framework converges faster, and has good generalization, especially in capturing long range dependencies, as shown on the synthetic addition and copy tasks, as well as on MNIST and Penn Tree Bank data sets., Comment: main text 13 pages, 22 pages including reference and appendix

It is a known fact that training recurrent neural networks for tasks that have long term dependencies is challenging. One of the main reasons is the vanishing or exploding gradient problem, which prevents gradient information from propagating to early layers. In this paper we propose a simple recurrent architecture, the Fourier Recurrent Unit (FRU), that stabilizes the gradients that arise in its training while giving us stronger expressive power. Specifically, FRU summarizes the hidden states $h^{(t)}$ along the temporal dimension with Fourier basis functions. This allows gradients to easily reach any layer due to FRU's residual learning structure and the global support of trigonometric functions. We show that FRU has gradient lower and upper bounds independent of temporal dimension. We also show the strong expressivity of sparse Fourier basis, from which FRU obtains its strong expressive power. Our experimental study also demonstrates that with fewer parameters the proposed architecture outperforms other recurrent architectures on many tasks.

In this paper, we consider parameter recovery for non-overlapping convolutional neural networks (CNNs) with multiple kernels. We show that when the inputs follow Gaussian distribution and the sample size is sufficiently large, the squared loss of such CNNs is $\mathit{~locally~strongly~convex}$ in a basin of attraction near the global optima for most popular activation functions, like ReLU, Leaky ReLU, Squared ReLU, Sigmoid and Tanh. The required sample complexity is proportional to the dimension of the input and polynomial in the number of kernels and a condition number of the parameters. We also show that tensor methods are able to initialize the parameters to the local strong convex region. Hence, for most smooth activations, gradient descent following tensor initialization is guaranteed to converge to the global optimal with time that is linear in input dimension, logarithmic in precision and polynomial in other factors. To the best of our knowledge, this is the first work that provides recovery guarantees for CNNs with multiple kernels under polynomial sample and computational complexities., Comment: arXiv admin note: text overlap with arXiv:1706.03175

In this paper, we consider regression problems with one-hidden-layer neural networks (1NNs). We distill some properties of activation functions that lead to $\mathit{local~strong~convexity}$ in the neighborhood of the ground-truth parameters for the 1NN squared-loss objective. Most popular nonlinear activation functions satisfy the distilled properties, including rectified linear units (ReLUs), leaky ReLUs, squared ReLUs and sigmoids. For activation functions that are also smooth, we show $\mathit{local~linear~convergence}$ guarantees of gradient descent under a resampling rule. For homogeneous activations, we show tensor methods are able to initialize the parameters to fall into the local strong convexity region. As a result, tensor initialization followed by gradient descent is guaranteed to recover the ground truth with sample complexity $ d \cdot \log(1/\epsilon) \cdot \mathrm{poly}(k,\lambda )$ and computational complexity $n\cdot d \cdot \mathrm{poly}(k,\lambda) $ for smooth homogeneous activations with high probability, where $d$ is the dimension of the input, $k$ ($k\leq d$) is the number of hidden nodes, $\lambda$ is a conditioning property of the ground-truth parameter matrix between the input layer and the hidden layer, $\epsilon$ is the targeted precision and $n$ is the number of samples. To the best of our knowledge, this is the first work that provides recovery guarantees for 1NNs with both sample complexity and computational complexity $\mathit{linear}$ in the input dimension and $\mathit{logarithmic}$ in the precision., Comment: ICML 2017

A considerable amount of clustering algorithms take instance-feature matrices as their inputs. As such, they cannot directly analyze time series data due to its temporal nature, usually unequal lengths, and complex properties. This is a great pity since many of these algorithms are effective, robust, efficient, and easy to use. In this paper, we bridge this gap by proposing an efficient representation learning framework that is able to convert a set of time series with various lengths to an instance-feature matrix. In particular, we guarantee that the pairwise similarities between time series are well preserved after the transformation, thus the learned feature representation is particularly suitable for the time series clustering task. Given a set of $n$ time series, we first construct an $n\times n$ partially-observed similarity matrix by randomly sampling $\mathcal{O}(n \log n)$ pairs of time series and computing their pairwise similarities. We then propose an efficient algorithm that solves a non-convex and NP-hard problem to learn new features based on the partially-observed similarity matrix. By conducting extensive empirical studies, we show that the proposed framework is more effective, efficient, and flexible, compared to other state-of-the-art time series clustering methods.

Maximum Inner Product Search (MIPS) is an important task in many machine learning applications such as the prediction phase of a low-rank matrix factorization model for a recommender system. There have been some works on how to perform MIPS in sub-linear time recently. However, most of them do not have the flexibility to control the trade-off between search efficient and search quality. In this paper, we study the MIPS problem with a computational budget. By carefully studying the problem structure of MIPS, we develop a novel Greedy-MIPS algorithm, which can handle budgeted MIPS by design. While simple and intuitive, Greedy-MIPS yields surprisingly superior performance compared to state-of-the-art approaches. As a specific example, on a candidate set containing half a million vectors of dimension 200, Greedy-MIPS runs 200x faster than the naive approach while yielding search results with the top-5 precision greater than 75\%.

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.

We present a novel k-way high-dimensional graphical model called the Generalized Root Model (GRM) that explicitly models dependencies between variable sets of size k > 2---where k = 2 is the standard pairwise graphical model. This model is based on taking the k-th root of the original sufficient statistics of any univariate exponential family with positive sufficient statistics, including the Poisson and exponential distributions. As in the recent work with square root graphical (SQR) models [Inouye et al. 2016]---which was restricted to pairwise dependencies---we give the conditions of the parameters that are needed for normalization using the radial conditionals similar to the pairwise case [Inouye et al. 2016]. In particular, we show that the Poisson GRM has no restrictions on the parameters and the exponential GRM only has a restriction akin to negative definiteness. We develop a simple but general learning algorithm based on L1-regularized node-wise regressions. We also present a general way of numerically approximating the log partition function and associated derivatives of the GRM univariate node conditionals---in contrast to [Inouye et al. 2016], which only provided algorithm for estimating the exponential SQR. To illustrate GRM, we model word counts with a Poisson GRM and show the associated k-sized variable sets. We finish by discussing methods for reducing the parameter space in various situations.

Stochastic variational inference (SVI), the state-of-the-art algorithm for scaling variational inference to large-datasets, is inherently serial. Moreover, it requires the parameters to fit in the memory of a single processor; this is problematic when the number of parameters is in billions. In this paper, we propose extreme stochastic variational inference (ESVI), an asynchronous and lock-free algorithm to perform variational inference for mixture models on massive real world datasets. ESVI overcomes the limitations of SVI by requiring that each processor only access a subset of the data and a subset of the parameters, thus providing data and model parallelism simultaneously. We demonstrate the effectiveness of ESVI by running Latent Dirichlet Allocation (LDA) on UMBC-3B, a dataset that has a vocabulary of 3 million and a token size of 3 billion. In our experiments, we found that ESVI not only outperforms VI and SVI in wallclock-time, but also achieves a better quality solution. In addition, we propose a strategy to speed up computation and save memory when fitting large number of topics.

We develop Square Root Graphical Models (SQR), a novel class of parametric graphical models that provides multivariate generalizations of univariate exponential family distributions. Previous multivariate graphical models [Yang et al. 2015] did not allow positive dependencies for the exponential and Poisson generalizations. However, in many real-world datasets, variables clearly have positive dependencies. For example, the airport delay time in New York---modeled as an exponential distribution---is positively related to the delay time in Boston. With this motivation, we give an example of our model class derived from the univariate exponential distribution that allows for almost arbitrary positive and negative dependencies with only a mild condition on the parameter matrix---a condition akin to the positive definiteness of the Gaussian covariance matrix. Our Poisson generalization allows for both positive and negative dependencies without any constraints on the parameter values. We also develop parameter estimation methods using node-wise regressions with $\ell_1$ regularization and likelihood approximation methods using sampling. Finally, we demonstrate our exponential generalization on a synthetic dataset and a real-world dataset of airport delay times.

Clustering is one of the most fundamental and important tasks in data mining. Traditional clustering algorithms, such as K-means, assign every data point to exactly one cluster. However, in real-world datasets, the clusters may overlap with each other. Furthermore, often, there are outliers that should not belong to any cluster. We recently proposed the NEO-K-Means (Non-Exhaustive, Overlapping K-Means) objective as a way to address both issues in an integrated fashion. Optimizing this discrete objective is NP-hard, and even though there is a convex relaxation of the objective, straightforward convex optimization approaches are too expensive for large datasets. A practical alternative is to use a low-rank factorization of the solution matrix in the convex formulation. The resulting optimization problem is non-convex, and we can locally optimize the objective function using an augmented Lagrangian method. In this paper, we consider two fast multiplier methods to accelerate the convergence of an augmented Lagrangian scheme: a proximal method of multipliers and an alternating direction method of multipliers (ADMM). For the proximal augmented Lagrangian or proximal method of multipliers, we show a convergence result for the non-convex case with bound-constrained subproblems. These methods are up to 13 times faster---with no change in quality---compared with a standard augmented Lagrangian method on problems with over 10,000 variables and bring runtimes down from over an hour to around 5 minutes., Comment: 9 pages. 2 figures

High-dimensional time series prediction is needed in applications as diverse as demand forecasting and climatology. Often, such applications require methods that are both highly scalable, and deal with noisy data in terms of corruptions or missing values. Classical time series methods usually fall short of handling both these issues. In this paper, we propose to adapt matrix matrix completion approaches that have previously been successfully applied to large scale noisy data, but which fail to adequately model high-dimensional time series due to temporal dependencies. We present a novel temporal regularized matrix factorization (TRMF) framework which supports data-driven temporal dependency learning and enables forecasting ability to our new matrix factorization approach. TRMF is highly general, and subsumes many existing matrix factorization approaches for time series data. We make interesting connections to graph regularized matrix factorization methods in the context of learning the dependencies. Experiments on both real and synthetic data show that TRMF outperforms several existing approaches for common time series tasks.

In this paper we consider the collaborative ranking setting: a pool of users each provides a small number of pairwise preferences between $d$ possible items; from these we need to predict preferences of the users for items they have not yet seen. We do so by fitting a rank $r$ score matrix to the pairwise data, and provide two main contributions: (a) we show that an algorithm based on convex optimization provides good generalization guarantees once each user provides as few as $O(r\log^2 d)$ pairwise comparisons -- essentially matching the sample complexity required in the related matrix completion setting (which uses actual numerical as opposed to pairwise information), and (b) we develop a large-scale non-convex implementation, which we call AltSVM, that trains a factored form of the matrix via alternating minimization (which we show reduces to alternating SVM problems), and scales and parallelizes very well to large problem settings. It also outperforms common baselines on many moderately large popular collaborative filtering datasets in both NDCG and in other measures of ranking performance.

We provide a general theoretical analysis of expected out-of-sample utility, also referred to as decision-theoretic classification, for non-decomposable binary classification metrics such as F-measure and Jaccard coefficient. Our key result is that the expected out-of-sample utility for many performance metrics is provably optimized by a classifier which is equivalent to a signed thresholding of the conditional probability of the positive class. Our analysis bridges a gap in the literature on binary classification, revealed in light of recent results for non-decomposable metrics in population utility maximization style classification. Our results identify checkable properties of a performance metric which are sufficient to guarantee a probability ranking principle. We propose consistent estimators for optimal expected out-of-sample classification. As a consequence of the probability ranking principle, computational requirements can be reduced from exponential to cubic complexity in the general case, and further reduced to quadratic complexity in special cases. We provide empirical results on simulated and benchmark datasets evaluating the performance of the proposed algorithms for decision-theoretic classification and comparing them to baseline and state-of-the-art methods in population utility maximization for non-decomposable metrics.

Stochastic Dual Coordinate Descent (SDCD) has become one of the most efficient ways to solve the family of $\ell_2$-regularized empirical risk minimization problems, including linear SVM, logistic regression, and many others. The vanilla implementation of DCD is quite slow; however, by maintaining primal variables while updating dual variables, the time complexity of SDCD can be significantly reduced. Such a strategy forms the core algorithm in the widely-used LIBLINEAR package. In this paper, we parallelize the SDCD algorithms in LIBLINEAR. In recent research, several synchronized parallel SDCD algorithms have been proposed, however, they fail to achieve good speedup in the shared memory multi-core setting. In this paper, we propose a family of asynchronous stochastic dual coordinate descent algorithms (ASDCD). Each thread repeatedly selects a random dual variable and conducts coordinate updates using the primal variables that are stored in the shared memory. We analyze the convergence properties when different locking/atomic mechanisms are applied. For implementation with atomic operations, we show linear convergence under mild conditions. For implementation without any atomic operations or locking, we present the first {\it backward error analysis} for ASDCD under the multi-core environment, showing that the converged solution is the exact solution for a primal problem with perturbed regularizer. Experimental results show that our methods are much faster than previous parallel coordinate descent solvers.

Community detection is an important task in network analysis. A community (also referred to as a cluster) is a set of cohesive vertices that have more connections inside the set than outside. In many social and information networks, these communities naturally overlap. For instance, in a social network, each vertex in a graph corresponds to an individual who usually participates in multiple communities. In this paper, we propose an efficient overlapping community detection algorithm using a seed expansion approach. The key idea of our algorithm is to find good seeds, and then greedily expand these seeds based on a community metric. Within this seed expansion method, we investigate the problem of how to determine good seed nodes in a graph. In particular, we develop new seeding strategies for a personalized PageRank clustering scheme that optimizes the conductance community score. Experimental results show that our seed expansion algorithm outperforms other state-of-the-art overlapping community detection methods in terms of producing cohesive clusters and identifying ground-truth communities. We also show that our new seeding strategies are better than existing strategies, and are thus effective in finding good overlapping communities in real-world networks.