arXiv subjects

Daniel Hsu

Publications and source records attributed to Daniel Hsu.

At least 91 records · Page 5Linked to original sources

arXiv2018

Coding sets with asymmetric information

We study the following one-way asymmetric transmission problem, also a variant of model-based compressed sensing: a resource-limited encoder has to report a small set $S$ from a universe of $N$ items to a more powerful decoder (server). The distinguishing feature is asymmetric information: the subset $S$ is comprised of i.i.d. samples from a prior distribution $μ$, and $μ$ is only known to the decoder. The goal for the encoder is to encode $S$ obliviously, while achieving the information-theoretic bound of $|S| \cdot H(μ)$, i.e., the Shannon entropy bound. We first show that any such compression scheme must be {\em randomized}, if it gains non-trivially from the prior $μ$. This stands in contrast to the symmetric case (when both the encoder and decoder know $μ$), where the Huffman code provides a near-optimal deterministic solution. On the other hand, a rather simple argument shows that, when $|S|=k$, a random linear code achieves near-optimal communication rate of about $k\cdot H(μ)$ bits. Alas, the resulting scheme has prohibitive decoding time: about ${N\choose k} \approx (N/k)^k$. Our main result is a computationally efficient and linear coding scheme, which achieves an $O(\lg\lg N)$-competitive communication ratio compared to the optimal benchmark, and runs in $\text{poly}(N,k)$ time. Our "multi-level" coding scheme uses a combination of hashing and syndrome-decoding of Reed-Solomon codes, and relies on viewing the (unknown) prior $μ$ as a rather small convex combination of uniform ("flat") distributions.

cs.DS↗

arXiv2018

Non-Gaussian information from weak lensing data via deep learning

Weak lensing maps contain information beyond two-point statistics on small scales. Much recent work has tried to extract this information through a range of different observables or via nonlinear transformations of the lensing field. Here we train and apply a 2D convolutional neural network to simulated noiseless lensing maps covering 96 different cosmological models over a range of {$Ω_m,σ_8$}. Using the area of the confidence contour in the {$Ω_m,σ_8$} plane as a figure-of-merit, derived from simulated convergence maps smoothed on a scale of 1.0 arcmin, we show that the neural network yields $\approx 5 \times$ tighter constraints than the power spectrum, and $\approx 4 \times$ tighter than the lensing peaks. Such gains illustrate the extent to which weak lensing data encode cosmological information not accessible to the power spectrum or even other, non-Gaussian statistics such as lensing peaks.

astro-ph.CO↗

arXiv2017

Linear regression without correspondence

This article considers algorithmic and statistical aspects of linear regression when the correspondence between the covariates and the responses is unknown. First, a fully polynomial-time approximation scheme is given for the natural least squares optimization problem in any constant dimension. Next, in an average-case and noise-free setting where the responses exactly correspond to a linear function of i.i.d. draws from a standard multivariate normal distribution, an efficient algorithm based on lattice basis reduction is shown to exactly recover the unknown linear function in arbitrary dimension. Finally, lower bounds on the signal-to-noise ratio are established for approximate recovery of the unknown linear function by any estimator.

cs.LG↗

arXiv2017

Mixing time estimation in reversible Markov chains from a single sample path

The spectral gap $γ$ of a finite, ergodic, and reversible Markov chain is an important parameter measuring the asymptotic rate of convergence. In applications, the transition matrix $P$ may be unknown, yet one sample of the chain up to a fixed time $n$ may be observed. We consider here the problem of estimating $γ$ from this data. Let $π$ be the stationary distribution of $P$, and $π_\star = \min_x π(x)$. We show that if $n = \tilde{O}\bigl(\frac{1}{γπ_\star}\bigr)$, then $γ$ can be estimated to within multiplicative constants with high probability. When $π$ is uniform on $d$ states, this matches (up to logarithmic correction) a lower bound of $\tildeΩ\bigl(\frac{d}γ\bigr)$ steps required for precise estimation of $γ$. Moreover, we provide the first procedure for computing a fully data-dependent interval, from a single finite-length trajectory of the chain, that traps the mixing time $t_{\text{mix}}$ of the chain at a prescribed confidence level. The interval does not require the knowledge of any parameters of the chain. This stands in contrast to previous approaches, which either only provide point estimates, or require a reset mechanism, or additional prior knowledge. The interval is constructed around the relaxation time $t_{\text{relax}} = 1/γ$, which is strongly related to the mixing time, and the width of the interval converges to zero roughly at a $1/\sqrt{n}$ rate, where $n$ is the length of the sample path.

math.ST↗

arXiv2017

Time Series Compression Based on Adaptive Piecewise Recurrent Autoencoder

Time series account for a large proportion of the data stored in financial, medical and scientific databases. The efficient storage of time series is important in practical applications. In this paper, we propose a novel compression scheme for time series. The encoder and decoder are both composed by recurrent neural networks (RNN) such as long short-term memory (LSTM). There is an autoencoder between encoder and decoder, which encodes the hidden state and input together and decodes them at the decoder side. Moreover, we pre-process the original time series by partitioning it into segments with various lengths which have similar total variation. The experimental study shows that the proposed algorithm can achieve competitive compression ratio on real-world time series.

cs.NE↗

arXiv2017

Parameter identification in Markov chain choice models

This work studies the parameter identification problem for the Markov chain choice model of Blanchet, Gallego, and Goyal used in assortment planning. In this model, the product selected by a customer is determined by a Markov chain over the products, where the products in the offered assortment are absorbing states. The underlying parameters of the model were previously shown to be identifiable from the choice probabilities for the all-products assortment, together with choice probabilities for assortments of all-but-one products. Obtaining and estimating choice probabilities for such large assortments is not desirable in many settings. The main result of this work is that the parameters may be identified from assortments of sizes two and three, regardless of the total number of products. The result is obtained via a simple and efficient parameter recovery algorithm.

math.ST↗

arXiv2017

Greedy Approaches to Symmetric Orthogonal Tensor Decomposition

Finding the symmetric and orthogonal decomposition (SOD) of a tensor is a recurring problem in signal processing, machine learning and statistics. In this paper, we review, establish and compare the perturbation bounds for two natural types of incremental rank-one approximation approaches. Numerical experiments and open questions are also presented and discussed.

math.NA↗

arXiv2017

Successive Rank-One Approximations for Nearly Orthogonally Decomposable Symmetric Tensors

Many idealized problems in signal processing, machine learning and statistics can be reduced to the problem of finding the symmetric canonical decomposition of an underlying symmetric and orthogonally decomposable (SOD) tensor. Drawing inspiration from the matrix case, the successive rank-one approximations (SROA) scheme has been proposed and shown to yield this tensor decomposition exactly, and a plethora of numerical methods have thus been developed for the tensor rank-one approximation problem. In practice, however, the inevitable errors (say) from estimation, computation, and modeling necessitate that the input tensor can only be assumed to be a nearly SOD tensor---i.e., a symmetric tensor slightly perturbed from the underlying SOD tensor. This article shows that even in the presence of perturbation, SROA can still robustly recover the symmetric canonical decomposition of the underlying tensor. It is shown that when the perturbation error is small enough, the approximation errors do not accumulate with the iteration number. Numerical results are presented to support the theoretical findings.

math.NA↗

arXiv2017

Kernel Approximation Methods for Speech Recognition

We study large-scale kernel methods for acoustic modeling in speech recognition and compare their performance to deep neural networks (DNNs). We perform experiments on four speech recognition datasets, including the TIMIT and Broadcast News benchmark tasks, and compare these two types of models on frame-level performance metrics (accuracy, cross-entropy), as well as on recognition metrics (word/character error rate). In order to scale kernel methods to these large datasets, we use the random Fourier feature method of Rahimi and Recht (2007). We propose two novel techniques for improving the performance of kernel acoustic models. First, in order to reduce the number of random features required by kernel models, we propose a simple but effective method for feature selection. The method is able to explore a large number of non-linear features while maintaining a compact model more efficiently than existing approaches. Second, we present a number of frame-level metrics which correlate very strongly with recognition performance when computed on the heldout set; we take advantage of these correlations by monitoring these metrics during training in order to decide when to stop learning. This technique can noticeably improve the recognition performance of both DNN and kernel models, while narrowing the gap between them. Additionally, we show that the linear bottleneck method of Sainath et al. (2013) improves the performance of our kernel models significantly, in addition to speeding up training and making the models more compact. Together, these three methods dramatically improve the performance of kernel acoustic models, making their performance comparable to DNNs on the tasks we explored.

stat.ML↗

arXiv2016

Search Improves Label for Active Learning

We investigate active learning with access to two distinct oracles: Label (which is standard) and Search (which is not). The Search oracle models the situation where a human searches a database to seed or counterexample an existing solution. Search is stronger than Label while being natural to implement in many situations. We show that an algorithm using both oracles can provide exponentially large problem-dependent improvements over Label alone.

cs.LG↗

arXiv2016

Do dark matter halos explain lensing peaks?

We have investigated a recently proposed halo-based model, Camelus, for predicting weak-lensing peak counts, and compared its results over a collection of 162 cosmologies with those from N-body simulations. While counts from both models agree for peaks with $\mathcal{S/N}>1$ (where $\mathcal{S/N}$ is the ratio of the peak height to the r.m.s. shape noise), we find $\approx 50\%$ fewer counts for peaks near $\mathcal{S/N}=0$ and significantly higher counts in the negative $\mathcal{S/N}$ tail. Adding shape noise reduces the differences to within $20\%$ for all cosmologies. We also found larger covariances that are more sensitive to cosmological parameters. As a result, credibility regions in the $\{Ω_m, σ_8\}$ are $\approx 30\%$ larger. Even though the credible contours are commensurate, each model draws its predictive power from different types of peaks. Low peaks, especially those with $2<\mathcal{S/N}<3$, convey important cosmological information in N-body data, as shown in \cite{DietrichHartlap, Kratochvil2010}, but \textsc{Camelus} constrains cosmology almost exclusively from high significance peaks $(\mathcal{S/N}>3)$. Our results confirm the importance of using a cosmology-dependent covariance with at least a 14\% improvement in parameter constraints. We identified the covariance estimation as the main driver behind differences in inference, and suggest possible ways to make Camelus even more useful as a highly accurate peak count emulator.

astro-ph.CO↗

arXiv2016

Global analysis of Expectation Maximization for mixtures of two Gaussians

Expectation Maximization (EM) is among the most popular algorithms for estimating parameters of statistical models. However, EM, which is an iterative algorithm based on the maximum likelihood principle, is generally only guaranteed to find stationary points of the likelihood objective, and these points may be far from any maximizer. This article addresses this disconnect between the statistical principles behind EM and its algorithmic properties. Specifically, it provides a global analysis of EM for specific models in which the observations comprise an i.i.d. sample from a mixture of two Gaussians. This is achieved by (i) studying the sequence of parameters from idealized execution of EM in the infinite sample limit, and fully characterizing the limit points of the sequence in terms of the initial parameters; and then (ii) based on this convergence analysis, establishing statistical consistency (or lack thereof) for the actual sequence of parameters produced by EM.

math.ST↗

arXiv2016

FairTest: Discovering Unwarranted Associations in Data-Driven Applications

In a world where traditional notions of privacy are increasingly challenged by the myriad companies that collect and analyze our data, it is important that decision-making entities are held accountable for unfair treatments arising from irresponsible data usage. Unfortunately, a lack of appropriate methodologies and tools means that even identifying unfair or discriminatory effects can be a challenge in practice. We introduce the unwarranted associations (UA) framework, a principled methodology for the discovery of unfair, discriminatory, or offensive user treatment in data-driven applications. The UA framework unifies and rationalizes a number of prior attempts at formalizing algorithmic fairness. It uniquely combines multiple investigative primitives and fairness metrics with broad applicability, granular exploration of unfair treatment in user subgroups, and incorporation of natural notions of utility that may account for observed disparities. We instantiate the UA framework in FairTest, the first comprehensive tool that helps developers check data-driven applications for unfair user treatment. It enables scalable and statistically rigorous investigation of associations between application outcomes (such as prices or premiums) and sensitive user attributes (such as race or gender). Furthermore, FairTest provides debugging capabilities that let programmers rule out potential confounders for observed unfair effects. We report on use of FairTest to investigate and in some cases address disparate impact, offensive labeling, and uneven rates of algorithmic error in four data-driven applications. As examples, our results reveal subtle biases against older populations in the distribution of error in a predictive health application and offensive racial labeling in an image tagger.

cs.CY↗

arXiv2016

Greedy bi-criteria approximations for $k$-medians and $k$-means

This paper investigates the following natural greedy procedure for clustering in the bi-criterion setting: iteratively grow a set of centers, in each round adding the center from a candidate set that maximally decreases clustering cost. In the case of $k$-medians and $k$-means, the key results are as follows. $\bullet$ When the method considers all data points as candidate centers, then selecting $\mathcal{O}(k\log(1/\varepsilon))$ centers achieves cost at most $2+\varepsilon$ times the optimal cost with $k$ centers. $\bullet$ Alternatively, the same guarantees hold if each round samples $\mathcal{O}(k/\varepsilon^5)$ candidate centers proportionally to their cluster cost (as with $\texttt{kmeans++}$, but holding centers fixed). $\bullet$ In the case of $k$-means, considering an augmented set of $n^{\lceil1/\varepsilon\rceil}$ candidate centers gives $1+\varepsilon$ approximation with $\mathcal{O}(k\log(1/\varepsilon))$ centers, the entire algorithm taking $\mathcal{O}(dk\log(1/\varepsilon)n^{1+\lceil1/\varepsilon\rceil})$ time, where $n$ is the number of data points in $\mathbb{R}^d$. $\bullet$ In the case of Euclidean $k$-medians, generating a candidate set via $n^{\mathcal{O}(1/\varepsilon^2)}$ executions of stochastic gradient descent with adaptively determined constraint sets will once again give approximation $1+\varepsilon$ with $\mathcal{O}(k\log(1/\varepsilon))$ centers in $dk\log(1/\varepsilon)n^{\mathcal{O}(1/\varepsilon^2)}$ time. Ancillary results include: guarantees for cluster costs based on powers of metrics; a brief, favorable empirical evaluation against $\texttt{kmeans++}$; data-dependent bounds allowing $1+\varepsilon$ in the first two bullets above, for example with $k$-medians over finite metric spaces.

cs.DS↗

arXiv2016

Kernel ridge vs. principal component regression: minimax bounds and adaptability of regularization operators

Regularization is an essential element of virtually all kernel methods for nonparametric regression problems. A critical factor in the effectiveness of a given kernel method is the type of regularization that is employed. This article compares and contrasts members from a general class of regularization techniques, which notably includes ridge regression and principal component regression. We derive an explicit finite-sample risk bound for regularization-based estimators that simultaneously accounts for (i) the structure of the ambient function space, (ii) the regularity of the true regression function, and (iii) the adaptability (or qualification) of the regularization. A simple consequence of this upper bound is that the risk of the regularization-based estimators matches the minimax rate in a variety of settings. The general bound also illustrates how some regularization techniques are more adaptable than others to favorable regularity properties that the true regression function may possess. This, in particular, demonstrates a striking difference between kernel ridge regression and kernel principal component regression. Our theoretical results are supported by numerical experiments.

math.ST↗

arXiv2016

Learning Sparse Low-Threshold Linear Classifiers

We consider the problem of learning a non-negative linear classifier with a $1$-norm of at most $k$, and a fixed threshold, under the hinge-loss. This problem generalizes the problem of learning a $k$-monotone disjunction. We prove that we can learn efficiently in this setting, at a rate which is linear in both $k$ and the size of the threshold, and that this is the best possible rate. We provide an efficient online learning algorithm that achieves the optimal rate, and show that in the batch case, empirical risk minimization achieves this rate as well. The rates we show are tighter than the uniform convergence rate, which grows with $k^2$.

stat.ML↗

arXiv2016

Loss minimization and parameter estimation with heavy tails

This work studies applications and generalizations of a simple estimation technique that provides exponential concentration under heavy-tailed distributions, assuming only bounded low-order moments. We show that the technique can be used for approximate minimization of smooth and strongly convex losses, and specifically for least squares linear regression. For instance, our $d$-dimensional estimator requires just $\tilde{O}(d\log(1/δ))$ random samples to obtain a constant factor approximation to the optimal least squares loss with probability $1-δ$, without requiring the covariates or noise to be bounded or subgaussian. We provide further applications to sparse linear regression and low-rank covariance matrix estimation with similar allowances on the noise and covariate distributions. The core technique is a generalization of the median-of-means estimator to arbitrary metric spaces.

cs.LG↗

arXiv2015

Mixing Time Estimation in Reversible Markov Chains from a Single Sample Path

This article provides the first procedure for computing a fully data-dependent interval that traps the mixing time $t_{\text{mix}}$ of a finite reversible ergodic Markov chain at a prescribed confidence level. The interval is computed from a single finite-length sample path from the Markov chain, and does not require the knowledge of any parameters of the chain. This stands in contrast to previous approaches, which either only provide point estimates, or require a reset mechanism, or additional prior knowledge. The interval is constructed around the relaxation time $t_{\text{relax}}$, which is strongly related to the mixing time, and the width of the interval converges to zero roughly at a $\sqrt{n}$ rate, where $n$ is the length of the sample path. Upper and lower bounds are given on the number of samples required to achieve constant-factor multiplicative accuracy. The lower bounds indicate that, unless further restrictions are placed on the chain, no procedure can achieve this accuracy level before seeing each state at least $Ω(t_{\text{relax}})$ times on the average. Finally, future directions of research are identified.

cs.LG↗