arXiv subjects

Sewoong Oh

Publications and source records attributed to Sewoong Oh.

At least 127 records · Page 7Linked to original sources

arXiv2018

Estimating Mutual Information for Discrete-Continuous Mixtures

Estimating mutual information from observed samples is a basic primitive, useful in several machine learning tasks including correlation mining, information bottleneck clustering, learning a Chow-Liu tree, and conditional independence testing in (causal) graphical models. While mutual information is a well-defined quantity in general probability spaces, existing estimators can only handle two special cases of purely discrete or purely continuous pairs of random variables. The main challenge is that these methods first estimate the (differential) entropies of X, Y and the pair (X;Y) and add them up with appropriate signs to get an estimate of the mutual information. These 3H-estimators cannot be applied in general mixture spaces, where entropy is not well-defined. In this paper, we design a novel estimator for mutual information of discrete-continuous mixtures. We prove that the proposed estimator is consistent. We provide numerical experiments suggesting superiority of the proposed estimator compared to other heuristics of adding small continuous noise to all the samples and applying standard estimators tailored for purely continuous variables, and quantizing the samples and applying standard estimators tailored for purely discrete variables. This significantly widens the applicability of mutual information estimation in real-world applications, where some variables are discrete, some continuous, and others are a mixture between continuous and discrete components.

cs.IT↗

arXiv2018

Iterative Bayesian Learning for Crowdsourced Regression

Crowdsourcing platforms emerged as popular venues for purchasing human intelligence at low cost for large volume of tasks. As many low-paid workers are prone to give noisy answers, a common practice is to add redundancy by assigning multiple workers to each task and then simply average out these answers. However, to fully harness the wisdom of the crowd, one needs to learn the heterogeneous quality of each worker. We resolve this fundamental challenge in crowdsourced regression tasks, i.e., the answer takes continuous labels, where identifying good or bad workers becomes much more non-trivial compared to a classification setting of discrete labels. In particular, we introduce a Bayesian iterative scheme and show that it provably achieves the optimal mean squared error. Our evaluations on synthetic and real-world datasets support our theoretical results and show the superiority of the proposed scheme.

cs.LG↗

arXiv2018

Deepcode: Feedback Codes via Deep Learning

The design of codes for communicating reliably over a statistically well defined channel is an important endeavor involving deep mathematical research and wide-ranging practical applications. In this work, we present the first family of codes obtained via deep learning, which significantly beats state-of-the-art codes designed over several decades of research. The communication channel under consideration is the Gaussian noise channel with feedback, whose study was initiated by Shannon; feedback is known theoretically to improve reliability of communication, but no practical codes that do so have ever been successfully constructed. We break this logjam by integrating information theoretic insights harmoniously with recurrent-neural-network based encoders and decoders to create novel codes that outperform known codes by 3 orders of magnitude in reliability. We also demonstrate several desirable properties of the codes: (a) generalization to larger block lengths, (b) composability with known codes, (c) adaptation to practical constraints. This result also has broader ramifications for coding theory: even when the channel has a clear mathematical model, deep learning methodologies, when combined with channel-specific information-theoretic insights, can potentially beat state-of-the-art codes constructed over decades of mathematical research.

cs.LG↗

arXiv2018

Communication Algorithms via Deep Learning

Coding theory is a central discipline underpinning wireline and wireless modems that are the workhorses of the information age. Progress in coding theory is largely driven by individual human ingenuity with sporadic breakthroughs over the past century. In this paper we study whether it is possible to automate the discovery of decoding algorithms via deep learning. We study a family of sequential codes parameterized by recurrent neural network (RNN) architectures. We show that creatively designed and trained RNN architectures can decode well known sequential codes such as the convolutional and turbo codes with close to optimal performance on the additive white Gaussian noise (AWGN) channel, which itself is achieved by breakthrough algorithms of our times (Viterbi and BCJR decoders, representing dynamic programing and forward-backward algorithms). We show strong generalizations, i.e., we train at a specific signal to noise ratio and block length but test at a wide range of these quantities, as well as robustness and adaptivity to deviations from the AWGN setting.

stat.ML↗

arXiv2018

Attention-based Graph Neural Network for Semi-supervised Learning

Recently popularized graph neural networks achieve the state-of-the-art accuracy on a number of standard benchmark datasets for graph-based semi-supervised learning, improving significantly over existing approaches. These architectures alternate between a propagation layer that aggregates the hidden states of the local neighborhood and a fully-connected layer. Perhaps surprisingly, we show that a linear model, that removes all the intermediate fully-connected layers, is still able to achieve a performance comparable to the state-of-the-art models. This significantly reduces the number of parameters, which is critical for semi-supervised learning where number of labeled examples are small. This in turn allows a room for designing more innovative propagation layers. Based on this insight, we propose a novel graph neural network that removes all the intermediate fully-connected layers, and replaces the propagation layers with attention mechanisms that respect the structure of the graph. The attention mechanism allows us to learn a dynamic and adaptive local summary of the neighborhood to achieve more accurate predictions. In a number of experiments on benchmark citation networks datasets, we demonstrate that our approach outperforms competing methods. By examining the attention weights among neighbors, we show that our model provides some interesting insights on how neighbors influence each other.

stat.ML↗

arXiv2017

Discovering Potential Correlations via Hypercontractivity

Discovering a correlation from one variable to another variable is of fundamental scientific and practical interest. While existing correlation measures are suitable for discovering average correlation, they fail to discover hidden or potential correlations. To bridge this gap, (i) we postulate a set of natural axioms that we expect a measure of potential correlation to satisfy; (ii) we show that the rate of information bottleneck, i.e., the hypercontractivity coefficient, satisfies all the proposed axioms; (iii) we provide a novel estimator to estimate the hypercontractivity coefficient from samples; and (iv) we provide numerical experiments demonstrating that this proposed estimator discovers potential correlations among various indicators of WHO datasets, is robust in discovering gene interactions from gene expression time series data, and is statistically more powerful than the estimators for other correlation measures in binary hypothesis testing of canonical examples of potential correlations.

stat.ML↗

arXiv2017

Achieving Budget-optimality with Adaptive Schemes in Crowdsourcing

Crowdsourcing platforms provide marketplaces where task requesters can pay to get labels on their data. Such markets have emerged recently as popular venues for collecting annotations that are crucial in training machine learning models in various applications. However, as jobs are tedious and payments are low, errors are common in such crowdsourced labels. A common strategy to overcome such noise in the answers is to add redundancy by getting multiple answers for each task and aggregating them using some methods such as majority voting. For such a system, there is a fundamental question of interest: how can we maximize the accuracy given a fixed budget on how many responses we can collect on the crowdsourcing system. We characterize this fundamental trade-off between the budget (how many answers the requester can collect in total) and the accuracy in the estimated labels. In particular, we ask whether adaptive task assignment schemes lead to a more efficient trade-off between the accuracy and the budget. Adaptive schemes, where tasks are assigned adaptively based on the data collected thus far, are widely used in practical crowdsourcing systems to efficiently use a given fixed budget. However, existing theoretical analyses of crowdsourcing systems suggest that the gain of adaptive task assignments is minimal. To bridge this gap, we investigate this question under a strictly more general probabilistic model, which has been recently introduced to model practical crowdsourced annotations. Under this generalized Dawid-Skene model, we characterize the fundamental trade-off between budget and accuracy. We introduce a novel adaptive scheme that matches this fundamental limit. We further quantify the fundamental gap between adaptive and non-adaptive schemes, by comparing the trade-off with the one for non-adaptive schemes. Our analyses confirm that the gap is significant.

cs.LG↗

arXiv2017

Spectrum Estimation from a Few Entries

Singular values of a data in a matrix form provide insights on the structure of the data, the effective dimensionality, and the choice of hyper-parameters on higher-level data analysis tools. However, in many practical applications such as collaborative filtering and network analysis, we only get a partial observation. Under such scenarios, we consider the fundamental problem of recovering spectral properties of the underlying matrix from a sampling of its entries. We are particularly interested in directly recovering the spectrum, which is the set of singular values, and also in sample-efficient approaches for recovering a spectral sum function, which is an aggregate sum of the same function applied to each of the singular values. We propose first estimating the Schatten $k$-norms of a matrix, and then applying Chebyshev approximation to the spectral sum function or applying moment matching in Wasserstein distance to recover the singular values. The main technical challenge is in accurately estimating the Schatten norms from a sampling of a matrix. We introduce a novel unbiased estimator based on counting small structures in a graph and provide guarantees that match its empirical performance. Our theoretical analysis shows that Schatten norms can be recovered accurately from strictly smaller number of samples compared to what is needed to recover the underlying low-rank matrix. Numerical experiments suggest that we significantly improve upon a competing approach of using matrix completion methods.

stat.ML↗

arXiv2017

Density Functional Estimators with k-Nearest Neighbor Bandwidths

Estimating expected polynomials of density functions from samples is a basic problem with numerous applications in statistics and information theory. Although kernel density estimators are widely used in practice for such functional estimation problems, practitioners are left on their own to choose an appropriate bandwidth for each application in hand. Further, kernel density estimators suffer from boundary biases, which are prevalent in real world data with lower dimensional structures. We propose using the fixed-k nearest neighbor distances for the bandwidth, which adaptively adjusts to local geometry. Further, we propose a novel estimator based on local likelihood density estimators, that mitigates the boundary biases. Although such a choice of fixed-k nearest neighbor distances to bandwidths results in inconsistent estimators, we provide a simple debiasing scheme that precomputes the asymptotic bias and divides off this term. With this novel correction, we show consistency of this debiased estimator. We provide numerical experiments suggesting that it improves upon competing state-of-the-art methods.

cs.IT↗

arXiv2017

Optimal Inference in Crowdsourced Classification via Belief Propagation

Crowdsourcing systems are popular for solving large-scale labelling tasks with low-paid workers. We study the problem of recovering the true labels from the possibly erroneous crowdsourced labels under the popular Dawid-Skene model. To address this inference problem, several algorithms have recently been proposed, but the best known guarantee is still significantly larger than the fundamental limit. We close this gap by introducing a tighter lower bound on the fundamental limit and proving that Belief Propagation (BP) exactly matches this lower bound. The guaranteed optimality of BP is the strongest in the sense that it is information-theoretically impossible for any other algorithm to correctly label a larger fraction of the tasks. Experimental results suggest that BP is close to optimal for all regimes considered and improves upon competing state-of-the-art algorithms.

cs.LG↗

arXiv2016

Data-driven Rank Breaking for Efficient Rank Aggregation

Rank aggregation systems collect ordinal preferences from individuals to produce a global ranking that represents the social preference. Rank-breaking is a common practice to reduce the computational complexity of learning the global ranking. The individual preferences are broken into pairwise comparisons and applied to efficient algorithms tailored for independent paired comparisons. However, due to the ignored dependencies in the data, naive rank-breaking approaches can result in inconsistent estimates. The key idea to produce accurate and consistent estimates is to treat the pairwise comparisons unequally, depending on the topology of the collected data. In this paper, we provide the optimal rank-breaking estimator, which not only achieves consistency but also achieves the best error bound. This allows us to characterize the fundamental tradeoff between accuracy and complexity. Further, the analysis identifies how the accuracy depends on the spectral gap of a corresponding comparison graph.

cs.LG↗

arXiv2016

Breaking the Bandwidth Barrier: Geometrical Adaptive Entropy Estimation

Estimators of information theoretic measures such as entropy and mutual information are a basic workhorse for many downstream applications in modern data science. State of the art approaches have been either geometric (nearest neighbor (NN) based) or kernel based (with a globally chosen bandwidth). In this paper, we combine both these approaches to design new estimators of entropy and mutual information that outperform state of the art methods. Our estimator uses local bandwidth choices of $k$-NN distances with a finite $k$, independent of the sample size. Such a local and data dependent choice improves performance in practice, but the bandwidth is vanishing at a fast rate, leading to a non-vanishing bias. We show that the asymptotic bias of the proposed estimator is universal; it is independent of the underlying distribution. Hence, it can be pre-computed and subtracted from the estimate. As a byproduct, we obtain a unified way of obtaining both kernel and NN estimators. The corresponding theoretical contribution relating the asymptotic geometry of nearest neighbors to order statistics is of independent mathematical interest.

cs.IT↗

arXiv2016

Hiding the Rumor Source

Anonymous social media platforms like Secret, Yik Yak, and Whisper have emerged as important tools for sharing ideas without the fear of judgment. Such anonymous platforms are also important in nations under authoritarian rule, where freedom of expression and the personal safety of message authors may depend on anonymity. Whether for fear of judgment or retribution, it is sometimes crucial to hide the identities of users who post sensitive messages. In this paper, we consider a global adversary who wishes to identify the author of a message; it observes either a snapshot of the spread of a message at a certain time, sampled timestamp metadata, or both. Recent advances in rumor source detection show that existing messaging protocols are vulnerable against such an adversary. We introduce a novel messaging protocol, which we call adaptive diffusion, and show that under the snapshot adversarial model, adaptive diffusion spreads content fast and achieves perfect obfuscation of the source when the underlying contact network is an infinite regular tree. That is, all users with the message are nearly equally likely to have been the origin of the message. When the contact network is an irregular tree, we characterize the probability of maximum likelihood detection by proving a concentration result over Galton-Watson trees. Experiments on a sampled Facebook network demonstrate that adaptive diffusion effectively hides the location of the source even when the graph is finite, irregular and has cycles.

cs.CR↗

arXiv2016

Computational and Statistical Tradeoffs in Learning to Rank

For massive and heterogeneous modern datasets, it is of fundamental interest to provide guarantees on the accuracy of estimation when computational resources are limited. In the application of learning to rank, we provide a hierarchy of rank-breaking mechanisms ordered by the complexity in thus generated sketch of the data. This allows the number of data points collected to be gracefully traded off against computational resources available, while guaranteeing the desired level of accuracy. Theoretical guarantees on the proposed generalized rank-breaking implicitly provide such trade-offs, which can be explicitly characterized under certain canonical scenarios on the structure of the data.

cs.LG↗

arXiv2016

Demystifying Fixed k-Nearest Neighbor Information Estimators

Estimating mutual information from i.i.d. samples drawn from an unknown joint density function is a basic statistical problem of broad interest with multitudinous applications. The most popular estimator is one proposed by Kraskov and Stögbauer and Grassberger (KSG) in 2004, and is nonparametric and based on the distances of each sample to its $k^{\rm th}$ nearest neighboring sample, where $k$ is a fixed small integer. Despite its widespread use (part of scientific software packages), theoretical properties of this estimator have been largely unexplored. In this paper we demonstrate that the estimator is consistent and also identify an upper bound on the rate of convergence of the bias as a function of number of samples. We argue that the superior performance benefits of the KSG estimator stems from a curious "correlation boosting" effect and build on this intuition to modify the KSG estimator in novel ways to construct a superior estimator. As a byproduct of our investigations, we obtain nearly tight rates of convergence of the $\ell_2$ error of the well known fixed $k$ nearest neighbor estimator of differential entropy by Kozachenko and Leonenko.

cs.LG↗

arXiv2016

Conditional Dependence via Shannon Capacity: Axioms, Estimators and Applications

We conduct an axiomatic study of the problem of estimating the strength of a known causal relationship between a pair of variables. We propose that an estimate of causal strength should be based on the conditional distribution of the effect given the cause (and not on the driving distribution of the cause), and study dependence measures on conditional distributions. Shannon capacity, appropriately regularized, emerges as a natural measure under these axioms. We examine the problem of calculating Shannon capacity from the observed samples and propose a novel fixed-$k$ nearest neighbor estimator, and demonstrate its consistency. Finally, we demonstrate an application to single-cell flow-cytometry, where the proposed estimators significantly reduce sample complexity.

cs.IT↗

arXiv2016

Top-$K$ Ranking from Pairwise Comparisons: When Spectral Ranking is Optimal

We explore the top-$K$ rank aggregation problem. Suppose a collection of items is compared in pairs repeatedly, and we aim to recover a consistent ordering that focuses on the top-$K$ ranked items based on partially revealed preference information. We investigate the Bradley-Terry-Luce model in which one ranks items according to their perceived utilities modeled as noisy observations of their underlying true utilities. Our main contributions are two-fold. First, in a general comparison model where item pairs to compare are given a priori, we attain an upper and lower bound on the sample size for reliable recovery of the top-$K$ ranked items. Second, more importantly, extending the result to a random comparison model where item pairs to compare are chosen independently with some probability, we show that in slightly restricted regimes, the gap between the derived bounds reduces to a constant factor, hence reveals that a spectral method can achieve the minimax optimality on the (order-wise) sample size required for top-$K$ ranking. That is to say, we demonstrate a spectral method alone to be sufficient to achieve the optimality and advantageous in terms of computational complexity, as it does not require an additional stage of maximum likelihood estimation that a state-of-the-art scheme employs to achieve the optimality. We corroborate our main results by numerical experiments.

cs.LG↗

arXiv2015

The Composition Theorem for Differential Privacy

Sequential querying of differentially private mechanisms degrades the overall privacy level. In this paper, we answer the fundamental question of characterizing the level of overall privacy degradation as a function of the number of queries and the privacy levels maintained by each privatization mechanism. Our solution is complete: we prove an upper bound on the overall privacy level and construct a sequence of privatization mechanisms that achieves this bound. The key innovation is the introduction of an operational interpretation of differential privacy (involving hypothesis testing) and the use of new data processing inequalities. Our result improves over the state-of-the-art, and has immediate applications in several problems studied in the literature including differentially private multi-party computation.

cs.DS↗