arXiv subjects

Lester Mackey

Publications and source records attributed to Lester Mackey.

At least 73 records · Page 4Linked to original sources

arXiv2020

Stochastic Stein Discrepancies

Stein discrepancies (SDs) monitor convergence and non-convergence in approximate inference when exact integration and sampling are intractable. However, the computation of a Stein discrepancy can be prohibitive if the Stein operator - often a sum over likelihood terms or potentials - is expensive to evaluate. To address this deficiency, we show that stochastic Stein discrepancies (SSDs) based on subsampled approximations of the Stein operator inherit the convergence control properties of standard SDs with probability 1. Along the way, we establish the convergence of Stein variational gradient descent (SVGD) on unbounded domains, resolving an open question of Liu (2017). In our experiments with biased Markov chain Monte Carlo (MCMC) hyperparameter tuning, approximate MCMC sampler selection, and stochastic SVGD, SSDs deliver comparable inferences to standard SDs with orders of magnitude fewer likelihood evaluations.

stat.ML↗

arXiv2020

Model-specific Data Subsampling with Influence Functions

Model selection requires repeatedly evaluating models on a given dataset and measuring their relative performances. In modern applications of machine learning, the models being considered are increasingly more expensive to evaluate and the datasets of interest are increasing in size. As a result, the process of model selection is time-consuming and computationally inefficient. In this work, we develop a model-specific data subsampling strategy that improves over random sampling whenever training points have varying influence. Specifically, we leverage influence functions to guide our selection strategy, proving theoretically, and demonstrating empirically that our approach quickly selects high-quality models.

cs.LG↗

arXiv2020

Measuring Sample Quality with Kernels

Approximate Markov chain Monte Carlo (MCMC) offers the promise of more rapid sampling at the cost of more biased inference. Since standard MCMC diagnostics fail to detect these biases, researchers have developed computable Stein discrepancy measures that provably determine the convergence of a sample to its target distribution. This approach was recently combined with the theory of reproducing kernels to define a closed-form kernel Stein discrepancy (KSD) computable by summing kernel evaluations across pairs of sample points. We develop a theory of weak convergence for KSDs based on Stein's method, demonstrate that commonly used KSDs fail to detect non-convergence even for Gaussian targets, and show that kernels with slowly decaying tails provably determine convergence for a large class of target distributions. The resulting convergence-determining KSDs are suitable for comparing biased, exact, and deterministic sample sequences and simpler to compute and parallelize than alternative Stein discrepancies. We use our tools to compare biased samplers, select sampler hyperparameters, and improve upon existing KSD approaches to one-sample hypothesis testing and sample quality improvement.

stat.ML↗

arXiv2020

Stein Point Markov Chain Monte Carlo

An important task in machine learning and statistics is the approximation of a probability measure by an empirical measure supported on a discrete point set. Stein Points are a class of algorithms for this task, which proceed by sequentially minimising a Stein discrepancy between the empirical measure and the target and, hence, require the solution of a non-convex optimisation problem to obtain each new point. This paper removes the need to solve this optimisation problem by, instead, selecting each new point based on a Markov chain sample path. This significantly reduces the computational cost of Stein Points and leads to a suite of algorithms that are straightforward to implement. The new algorithms are illustrated on a set of challenging Bayesian inference problems, and rigorous theoretical guarantees of consistency are established.

stat.CO↗

arXiv2020

Single Point Transductive Prediction

Standard methods in supervised learning separate training and prediction: the model is fit independently of any test points it may encounter. However, can knowledge of the next test point $\mathbf{x}_{\star}$ be exploited to improve prediction accuracy? We address this question in the context of linear prediction, showing how techniques from semi-parametric inference can be used transductively to combat regularization bias. We first lower bound the $\mathbf{x}_{\star}$ prediction error of ridge regression and the Lasso, showing that they must incur significant bias in certain test directions. We then provide non-asymptotic upper bounds on the $\mathbf{x}_{\star}$ prediction error of two transductive prediction rules. We conclude by showing the efficacy of our methods on both synthetic and real data, highlighting the improvements single point transductive prediction can provide in settings with distribution shift.

stat.ML↗

arXiv2020

Minimax Estimation of Conditional Moment Models

We develop an approach for estimating models described via conditional moment restrictions, with a prototypical application being non-parametric instrumental variable regression. We introduce a min-max criterion function, under which the estimation problem can be thought of as solving a zero-sum game between a modeler who is optimizing over the hypothesis space of the target model and an adversary who identifies violating moments over a test function space. We analyze the statistical estimation rate of the resulting estimator for arbitrary hypothesis spaces, with respect to an appropriate analogue of the mean squared error metric, for ill-posed inverse problems. We show that when the minimax criterion is regularized with a second moment penalty on the test function and the test function space is sufficiently rich, then the estimation rate scales with the critical radius of the hypothesis and test function spaces, a quantity which typically gives tight fast rates. Our main result follows from a novel localized Rademacher analysis of statistical learning problems defined via minimax objectives. We provide applications of our main results for several hypothesis spaces used in practice such as: reproducing kernel Hilbert spaces, high dimensional sparse linear functions, spaces defined via shape constraints, ensemble estimators such as random forests, and neural networks. For each of these applications we provide computationally efficient optimization methods for solving the corresponding minimax problem (e.g. stochastic first-order heuristics for neural networks). In several applications, we show how our modified mean squared error rate, combined with conditions that bound the ill-posedness of the inverse problem, lead to mean squared error rates. We conclude with an extensive experimental analysis of the proposed methods.

econ.EM↗

arXiv2020

Approximate Cross-validation: Guarantees for Model Assessment and Selection

Cross-validation (CV) is a popular approach for assessing and selecting predictive models. However, when the number of folds is large, CV suffers from a need to repeatedly refit a learning procedure on a large number of training datasets. Recent work in empirical risk minimization (ERM) approximates the expensive refitting with a single Newton step warm-started from the full training set optimizer. While this can greatly reduce runtime, several open questions remain including whether these approximations lead to faithful model selection and whether they are suitable for non-smooth objectives. We address these questions with three main contributions: (i) we provide uniform non-asymptotic, deterministic model assessment guarantees for approximate CV; (ii) we show that (roughly) the same conditions also guarantee model selection performance comparable to CV; (iii) we provide a proximal Newton extension of the approximate CV framework for non-smooth prediction problems and develop improved assessment guarantees for problems such as l1-regularized ERM.

stat.ML↗

arXiv2020

Teacher-Student Compression with Generative Adversarial Networks

More accurate machine learning models often demand more computation and memory at test time, making them difficult to deploy on CPU- or memory-constrained devices. Teacher-student compression (TSC), also known as distillation, alleviates this burden by training a less expensive student model to mimic the expensive teacher model while maintaining most of the original accuracy. However, when fresh data is unavailable for the compression task, the teacher's training data is typically reused, leading to suboptimal compression. In this work, we propose to augment the compression dataset with synthetic data from a generative adversarial network (GAN) designed to approximate the training data distribution. Our GAN-assisted TSC (GAN-TSC) significantly improves student accuracy for expensive models such as large random forests and deep neural networks on both tabular and image datasets. Building on these results, we propose a comprehensive metric---the TSC Score---to evaluate the quality of synthetic datasets based on their induced TSC performance. The TSC Score captures both data diversity and class affinity, and we illustrate its benefits over the popular Inception Score in the context of image classification.

cs.LG↗

arXiv2020

Weighted Meta-Learning

Meta-learning leverages related source tasks to learn an initialization that can be quickly fine-tuned to a target task with limited labeled examples. However, many popular meta-learning algorithms, such as model-agnostic meta-learning (MAML), only assume access to the target samples for fine-tuning. In this work, we provide a general framework for meta-learning based on weighting the loss of different source tasks, where the weights are allowed to depend on the target samples. In this general setting, we provide upper bounds on the distance of the weighted empirical risk of the source tasks and expected target risk in terms of an integral probability metric (IPM) and Rademacher complexity, which apply to a number of meta-learning settings including MAML and a weighted MAML variant. We then develop a learning algorithm based on minimizing the error bound with respect to an empirical IPM, including a weighted MAML algorithm, $α$-MAML. Finally, we demonstrate empirically on several regression problems that our weighted meta-learning algorithm is able to find better initializations than uniformly-weighted meta-learning algorithms, such as MAML.

stat.ML↗

arXiv2020

Importance Sampling via Local Sensitivity

Given a loss function $F:\mathcal{X} \rightarrow \R^+$ that can be written as the sum of losses over a large set of inputs $a_1,\ldots, a_n$, it is often desirable to approximate $F$ by subsampling the input points. Strong theoretical guarantees require taking into account the importance of each point, measured by how much its individual loss contributes to $F(x)$. Maximizing this importance over all $x \in \mathcal{X}$ yields the \emph{sensitivity score} of $a_i$. Sampling with probabilities proportional to these scores gives strong guarantees, allowing one to approximately minimize of $F$ using just the subsampled points. Unfortunately, sensitivity sampling is difficult to apply since (1) it is unclear how to efficiently compute the sensitivity scores and (2) the sample size required is often impractically large. To overcome both obstacles we introduce \emph{local sensitivity}, which measures data point importance in a ball around some center $x_0$. We show that the local sensitivity can be efficiently estimated using the \emph{leverage scores} of a quadratic approximation to $F$ and that the sample size required to approximate $F$ around $x_0$ can be bounded. We propose employing local sensitivity sampling in an iterative optimization method and analyze its convergence when $F$ is smooth and convex.

math.OC↗

arXiv2020

Stochastic Runge-Kutta Accelerates Langevin Monte Carlo and Beyond

Sampling with Markov chain Monte Carlo methods often amounts to discretizing some continuous-time dynamics with numerical integration. In this paper, we establish the convergence rate of sampling algorithms obtained by discretizing smooth Itô diffusions exhibiting fast Wasserstein-$2$ contraction, based on local deviation properties of the integration scheme. In particular, we study a sampling algorithm constructed by discretizing the overdamped Langevin diffusion with the method of stochastic Runge-Kutta. For strongly convex potentials that are smooth up to a certain order, its iterates converge to the target distribution in $2$-Wasserstein distance in $\tilde{\mathcal{O}}(dε^{-2/3})$ iterations. This improves upon the best-known rate for strongly log-concave sampling based on the overdamped Langevin equation using only the gradient oracle without adjustment. In addition, we extend our analysis of stochastic Runge-Kutta methods to uniformly dissipative diffusions with possibly non-convex potentials and show they achieve better rates compared to the Euler-Maruyama scheme in terms of the dependence on tolerance $ε$. Numerical studies show that these algorithms lead to better stability and lower asymptotic errors.

stat.ML↗

arXiv2020

Accelerating Rescaled Gradient Descent: Fast Optimization of Smooth Functions

We present a family of algorithms, called descent algorithms, for optimizing convex and non-convex functions. We also introduce a new first-order algorithm, called rescaled gradient descent (RGD), and show that RGD achieves a faster convergence rate than gradient descent provided the function is strongly smooth -- a natural generalization of the standard smoothness assumption on the objective function. When the objective function is convex, we present two novel frameworks for "accelerating" descent methods, one in the style of Nesterov and the other in the style of Monteiro and Svaiter, using a single Lyapunov. Rescaled gradient descent can be accelerated under the same strong smoothness assumption using both frameworks. We provide several examples of strongly smooth loss functions in machine learning and numerical experiments that verify our theoretical findings. We also present several extensions of our novel Lyapunov framework, including deriving optimal universal tensor methods and extending our framework to the coordinate setting.

math.OC↗

arXiv2020

Accurate Inference for Adaptive Linear Models

Estimators computed from adaptively collected data do not behave like their non-adaptive brethren. Rather, the sequential dependence of the collection policy can lead to severe distributional biases that persist even in the infinite data limit. We develop a general method -- $\mathbf{W}$-decorrelation -- for transforming the bias of adaptive linear regression estimators into variance. The method uses only coarse-grained information about the data collection policy and does not need access to propensity scores or exact knowledge of the policy. We bound the finite-sample bias and variance of the $\mathbf{W}$-estimator and develop asymptotically correct confidence intervals based on a novel martingale central limit theorem. We then demonstrate the empirical benefits of the generic $\mathbf{W}$-decorrelation procedure in two different adaptive data settings: the multi-armed bandit and the autoregressive time series.

stat.ML↗

arXiv2019

Global Non-convex Optimization with Discretized Diffusions

An Euler discretization of the Langevin diffusion is known to converge to the global minimizers of certain convex and non-convex optimization problems. We show that this property holds for any suitably smooth diffusion and that different diffusions are suitable for optimizing different classes of convex and non-convex functions. This allows us to design diffusions suitable for globally optimizing convex and non-convex functions not covered by the existing Langevin theory. Our non-asymptotic analysis delivers computable optimization and integration error bounds based on easily accessed properties of the objective and chosen diffusion. Central to our approach are new explicit Stein factor bounds on the solutions of Poisson equations. We complement these results with improved optimization guarantees for targets other than the standard Gibbs measure.

stat.ML↗

arXiv2019

Improving Subseasonal Forecasting in the Western U.S. with Machine Learning

Water managers in the western United States (U.S.) rely on longterm forecasts of temperature and precipitation to prepare for droughts and other wet weather extremes. To improve the accuracy of these longterm forecasts, the U.S. Bureau of Reclamation and the National Oceanic and Atmospheric Administration (NOAA) launched the Subseasonal Climate Forecast Rodeo, a year-long real-time forecasting challenge in which participants aimed to skillfully predict temperature and precipitation in the western U.S. two to four weeks and four to six weeks in advance. Here we present and evaluate our machine learning approach to the Rodeo and release our SubseasonalRodeo dataset, collected to train and evaluate our forecasting system. Our system is an ensemble of two regression models. The first integrates the diverse collection of meteorological measurements and dynamic model forecasts in the SubseasonalRodeo dataset and prunes irrelevant predictors using a customized multitask model selection procedure. The second uses only historical measurements of the target variable (temperature or precipitation) and introduces multitask nearest neighbor features into a weighted local linear regression. Each model alone is significantly more accurate than the debiased operational U.S. Climate Forecasting System (CFSv2), and our ensemble skill exceeds that of the top Rodeo competitor for each target variable and forecast horizon. Moreover, over 2011-2018, an ensemble of our regression models and debiased CFSv2 improves debiased CFSv2 skill by 40-50% for temperature and 129-169% for precipitation. We hope that both our dataset and our methods will help to advance the state of the art in subseasonal forecasting.

stat.AP↗

arXiv2019

Measuring Sample Quality with Stein's Method

To improve the efficiency of Monte Carlo estimation, practitioners are turning to biased Markov chain Monte Carlo procedures that trade off asymptotic exactness for computational speed. The reasoning is sound: a reduction in variance due to more rapid sampling can outweigh the bias introduced. However, the inexactness creates new challenges for sampler and parameter selection, since standard measures of sample quality like effective sample size do not account for asymptotic bias. To address these challenges, we introduce a new computable quality measure based on Stein's method that quantifies the maximum discrepancy between sample and target expectations over a large class of test functions. We use our tool to compare exact, biased, and deterministic sample sequences and illustrate applications to hyperparameter selection, convergence rate assessment, and quantifying bias-variance tradeoffs in posterior inference.

stat.ML↗

arXiv2018

Measuring Sample Quality with Diffusions

Stein's method for measuring convergence to a continuous target distribution relies on an operator characterizing the target and Stein factor bounds on the solutions of an associated differential equation. While such operators and bounds are readily available for a diversity of univariate targets, few multivariate targets have been analyzed. We introduce a new class of characterizing operators based on Ito diffusions and develop explicit multivariate Stein factor bounds for any target with a fast-coupling Ito diffusion. As example applications, we develop computable and convergence-determining diffusion Stein discrepancies for log-concave, heavy-tailed, and multimodal targets and use these quality measures to select the hyperparameters of biased Markov chain Monte Carlo (MCMC) samplers, compare random and deterministic quadrature rules, and quantify bias-variance tradeoffs in approximate MCMC. Our results establish a near-linear relationship between diffusion Stein discrepancies and Wasserstein distances, improving upon past work even for strongly log-concave targets. The exposed relationship between Stein factors and Markov process coupling may be of independent interest.

stat.ML↗

arXiv2018

Orthogonal Machine Learning: Power and Limitations

Double machine learning provides $\sqrt{n}$-consistent estimates of parameters of interest even when high-dimensional or nonparametric nuisance parameters are estimated at an $n^{-1/4}$ rate. The key is to employ Neyman-orthogonal moment equations which are first-order insensitive to perturbations in the nuisance parameters. We show that the $n^{-1/4}$ requirement can be improved to $n^{-1/(2k+2)}$ by employing a $k$-th order notion of orthogonality that grants robustness to more complex or higher-dimensional nuisance parameters. In the partially linear regression setting popular in causal inference, we show that we can construct second-order orthogonal moments if and only if the treatment residual is not normally distributed. Our proof relies on Stein's lemma and may be of independent interest. We conclude by demonstrating the robustness benefits of an explicit doubly-orthogonal estimation procedure for treatment effect.

cs.LG↗