John Duchi

John Duchi, Alekh Agarwal, Martin Wainwright

The goal of decentralized optimization over a network is to optimize a global objective formed by a sum of local (possibly nonsmooth) convex functions using only local computation and communication. It arises in various application domains, including distributed tracking and localization, multi-agent co-ordination, estimation in sensor networks, and large-scale optimization in machine learning. We develop and analyze distributed algorithms based on dual averaging of subgradients, and we provide sharp bounds on their convergence rates as a function of the network size and topology. Our method of analysis allows for a clear separation between the convergence of the optimization algorithm itself and the effects of communication constraints arising from the network structure. In particular, we show that the number of iterations required by our algorithm scales inversely in the spectral gap of the network. The sharpness of this prediction is confirmed both by theoretical lower bounds and simulations for various networks. Our approach includes both the cases of deterministic optimization and communication, as well as problems with stochastic optimization and/or communication.

John Duchi, Saminul Haque, Rohith Kuditipudi

We design an $(\varepsilon, δ)$-differentially private algorithm to estimate the mean of a $d$-variate distribution, with unknown covariance $Σ$, that is adaptive to $Σ$. To within polylogarithmic factors, the estimator achieves optimal rates of convergence with respect to the induced Mahalanobis norm $||\cdot||_Σ$, takes time $\tilde{O}(n d^2)$ to compute, has near linear sample complexity for sub-Gaussian distributions, allows $Σ$ to be degenerate or low rank, and adaptively extends beyond sub-Gaussianity. Prior to this work, other methods required exponential computation time or the superlinear scaling $n = Ω(d^{3/2})$ to achieve non-trivial error with respect to the norm $||\cdot||_Σ$.

John Duchi, Vitaly Feldman, Lunjia Hu et al.

Recovering linear subspaces from data is a fundamental and important task in statistics and machine learning. Motivated by heterogeneity in Federated Learning settings, we study a basic formulation of this problem: the principal component analysis (PCA), with a focus on dealing with irregular noise. Our data come from $n$ users with user $i$ contributing data samples from a $d$-dimensional distribution with mean $μ_i$. Our goal is to recover the linear subspace shared by $μ_1,\ldots,μ_n$ using the data points from all users, where every data point from user $i$ is formed by adding an independent mean-zero noise vector to $μ_i$. If we only have one data point from every user, subspace recovery is information-theoretically impossible when the covariance matrices of the noise vectors can be non-spherical, necessitating additional restrictive assumptions in previous work. We avoid these assumptions by leveraging at least two data points from each user, which allows us to design an efficiently-computable estimator under non-spherical and user-dependent noise. We prove an upper bound for the estimation error of our estimator in general scenarios where the number of data points and amount of noise can vary across users, and prove an information-theoretic error lower bound that not only matches the upper bound up to a constant factor, but also holds even for spherical Gaussian noise. This implies that our estimator does not introduce additional estimation error (up to a constant factor) due to irregularity in the noise. We show additional results for a linear regression problem in a similar setup.

Karan Chadha, Junye Chen, John Duchi et al.

In this work, we study practical heuristics to improve the performance of prefix-tree based algorithms for differentially private heavy hitter detection. Our model assumes each user has multiple data points and the goal is to learn as many of the most frequent data points as possible across all users' data with aggregate and local differential privacy. We propose an adaptive hyperparameter tuning algorithm that improves the performance of the algorithm while satisfying computational, communication and privacy constraints. We explore the impact of different data-selection schemes as well as the impact of introducing deny lists during multiple runs of the algorithm. We test these improvements using extensive experimentation on the Reddit dataset~\cite{caldas2018leaf} on the task of learning the most frequent words.

Chen Cheng, Hilal Asi, John Duchi

The construction of most supervised learning datasets revolves around collecting multiple labels for each instance, then aggregating the labels to form a type of "gold-standard". We question the wisdom of this pipeline by developing a (stylized) theoretical model of this process and analyzing its statistical consequences, showing how access to non-aggregated label information can make training well-calibrated models more feasible than it is with gold-standard labels. The entire story, however, is subtle, and the contrasts between aggregated and fuller label information depend on the particulars of the problem, where estimators that use aggregated information exhibit robust but slower rates of convergence, while estimators that can effectively leverage all labels converge more quickly if they have fidelity to (or can learn) the true labeling process. The theory makes several predictions for real-world datasets, including when non-aggregate labels should improve learning performance, which we test to corroborate the validity of our predictions.

Chen Cheng, Gary Cheng, John Duchi

We study the problem of collaboratively learning least squares estimates for $m$ agents. Each agent observes a different subset of the features$\unicode{x2013}$e.g., containing data collected from sensors of varying resolution. Our goal is to determine how to coordinate the agents in order to produce the best estimator for each agent. We propose a distributed, semi-supervised algorithm Collab, consisting of three steps: local training, aggregation, and distribution. Our procedure does not require communicating the labeled data, making it communication efficient and useful in settings where the labeled data is inaccessible. Despite this handicap, our procedure is nearly asymptotically local minimax optimal$\unicode{x2013}$even among estimators allowed to communicate the labeled data such as imputation methods. We test our method on real and synthetic data.

Hilal Asi, Karan Chadha, Gary Cheng et al.

In non-private stochastic convex optimization, stochastic gradient methods converge much faster on interpolation problems -- problems where there exists a solution that simultaneously minimizes all of the sample losses -- than on non-interpolating ones; we show that generally similar improvements are impossible in the private setting. However, when the functions exhibit quadratic growth around the optimum, we show (near) exponential improvements in the private sample complexity. In particular, we propose an adaptive algorithm that improves the sample complexity to achieve expected error $α$ from $\frac{d}{\varepsilon \sqrtα}$ to $\frac{1}{α^ρ} + \frac{d}{\varepsilon} \log\left(\frac{1}α\right)$ for any fixed $ρ>0$, while retaining the standard minimax-optimal sample complexity for non-interpolation problems. We prove a lower bound that shows the dimension-dependent term is tight. Furthermore, we provide a superefficiency result which demonstrates the necessity of the polynomial term for adaptive algorithms: any algorithm that has a polylogarithmic sample complexity for interpolation problems cannot achieve the minimax-optimal rates for the family of non-interpolation problems.

Maxime Cauchois, John Duchi

The cost and scarcity of fully supervised labels in statistical machine learning encourage using partially labeled data for model validation as a cheaper and more accessible alternative. Effectively collecting and leveraging weakly supervised data for large-space structured prediction tasks thus becomes an important part of an end-to-end learning system. We propose a new computationally-friendly methodology to construct predictive sets using only partially labeled data on top of black-box predictive models. To do so, we introduce "probe" functions as a way to describe weakly supervised instances and define a false discovery proportion-type loss, both of which seamlessly adapt to partial supervision and structured prediction -- ranking, matching, segmentation, multilabel or multiclass classification. Our experiments highlight the validity of our predictive set construction as well as the attractiveness of a more flexible user-dependent loss framework.

Christopher Mohri, John Duchi, Tatsunori Hashimoto

We investigate data filtering for large model pretraining via new scaling studies that target the high compute, data-scarce regime. In spite of an apparently common belief that filtering data to include only high-quality information is essential, our experiments suggest that with enough compute, the best data filter is no data filter. We find that sufficiently trained large parameter models not only tolerate low-quality and distractor data, but in fact benefit from nominally ``poor'' data.

Karan Chadha, John Duchi, Rohith Kuditipudi

We consider the task of constructing confidence intervals with differential privacy. We propose two private variants of the non-parametric bootstrap, which privately compute the median of the results of multiple "little" bootstraps run on partitions of the data and give asymptotic bounds on the coverage error of the resulting confidence intervals. For a fixed differential privacy parameter $ε$, our methods enjoy the same error rates as that of the non-private bootstrap to within logarithmic factors in the sample size $n$. We empirically validate the performance of our methods for mean estimation, median estimation, and logistic regression with both real and synthetic data. Our methods achieve similar coverage accuracy to existing methods (and non-private baselines) while providing notably shorter ($\gtrsim 10$ times) confidence intervals than previous approaches.

Chen Cheng, John Duchi

We demonstrate that learning procedures that rely on aggregated labels, e.g., label information distilled from noisy responses, enjoy robustness properties impossible without data cleaning. This robustness appears in several ways. In the context of risk consistency -- when one takes the standard approach in machine learning of minimizing a surrogate (typically convex) loss in place of a desired task loss (such as the zero-one mis-classification error) -- procedures using label aggregation obtain stronger consistency guarantees than those even possible using raw labels. And while classical statistical scenarios of fitting perfectly-specified models suggest that incorporating all possible information -- modeling uncertainty in labels -- is statistically efficient, consistency fails for ``standard'' approaches as soon as a loss to be minimized is even slightly mis-specified. Yet procedures leveraging aggregated information still converge to optimal classifiers, highlighting how incorporating a fuller view of the data analysis pipeline, from collection to model-fitting to prediction time, can yield a more robust methodology by refining noisy signals.

Chen Cheng, John Duchi, Rohith Kuditipudi

We examine the necessity of interpolation in overparameterized models, that is, when achieving optimal predictive risk in machine learning problems requires (nearly) interpolating the training data. In particular, we consider simple overparameterized linear regression $y = X θ+ w$ with random design $X \in \mathbb{R}^{n \times d}$ under the proportional asymptotics $d/n \to γ\in (1, \infty)$. We precisely characterize how prediction (test) error necessarily scales with training error in this setting. An implication of this characterization is that as the label noise variance $σ^2 \to 0$, any estimator that incurs at least $\mathsf{c}σ^4$ training error for some constant $\mathsf{c}$ is necessarily suboptimal and will suffer growth in excess prediction error at least linear in the training error. Thus, optimal performance requires fitting training data to substantially higher accuracy than the inherent noise floor of the problem.

Maxime Cauchois, Suyash Gupta, Alnur Ali et al.

The expense of acquiring labels in large-scale statistical machine learning makes partially and weakly-labeled data attractive, though it is not always apparent how to leverage such data for model fitting or validation. We present a methodology to bridge the gap between partial supervision and validation, developing a conformal prediction framework to provide valid predictive confidence sets -- sets that cover a true label with a prescribed probability, independent of the underlying distribution -- using weakly labeled data. To do so, we introduce a (necessary) new notion of coverage and predictive validity, then develop several application scenarios, providing efficient algorithms for classification and several large-scale structured prediction problems. We corroborate the hypothesis that the new coverage definition allows for tighter and more informative (but valid) confidence sets through several experiments.

Gary Cheng, Karan Chadha, John Duchi

We propose an asymptotic framework to analyze the performance of (personalized) federated learning algorithms. In this new framework, we formulate federated learning as a multi-criterion objective, where the goal is to minimize each client's loss using information from all of the clients. We analyze a linear regression model where, for a given client, we may theoretically compare the performance of various algorithms in the high-dimensional asymptotic limit. This asymptotic multi-criterion approach naturally models the high-dimensional, many-device nature of federated learning. These tools make fairly precise predictions about the benefits of personalization and information sharing in federated scenarios -- at least in our (stylized) model -- including that Federated Averaging with simple client fine-tuning achieves the same asymptotic risk as the more intricate meta-learning and proximal-regularized approaches and outperforming Federated Averaging without personalization. We evaluate these predictions on federated versions of the EMNIST, CIFAR-100, Shakespeare, and Stack Overflow datasets, where the experiments corroborate the theoretical predictions, suggesting such frameworks may provide a useful guide to practical algorithmic development.

Hilal Asi, Daniel Levy, John Duchi

We develop algorithms for private stochastic convex optimization that adapt to the hardness of the specific function we wish to optimize. While previous work provide worst-case bounds for arbitrary convex functions, it is often the case that the function at hand belongs to a smaller class that enjoys faster rates. Concretely, we show that for functions exhibiting $κ$-growth around the optimum, i.e., $f(x) \ge f(x^*) + λκ^{-1} \|x-x^*\|_2^κ$ for $κ> 1$, our algorithms improve upon the standard ${\sqrt{d}}/{n\varepsilon}$ privacy rate to the faster $({\sqrt{d}}/{n\varepsilon})^{\tfracκ{κ- 1}}$. Crucially, they achieve these rates without knowledge of the growth constant $κ$ of the function. Our algorithms build upon the inverse sensitivity mechanism, which adapts to instance difficulty (Asi & Duchi, 2020), and recent localization techniques in private optimization (Feldman et al., 2020). We complement our algorithms with matching lower bounds for these function classes and demonstrate that our adaptive algorithm is \emph{simultaneously} (minimax) optimal over all $κ\ge 1+c$ whenever $c = Θ(1)$.

Hilal Asi, John Duchi, Alireza Fallah et al.

We study adaptive methods for differentially private convex optimization, proposing and analyzing differentially private variants of a Stochastic Gradient Descent (SGD) algorithm with adaptive stepsizes, as well as the AdaGrad algorithm. We provide upper bounds on the regret of both algorithms and show that the bounds are (worst-case) optimal. As a consequence of our development, we show that our private versions of AdaGrad outperform adaptive SGD, which in turn outperforms traditional SGD in scenarios with non-isotropic gradients where (non-private) Adagrad provably outperforms SGD. The major challenge is that the isotropic noise typically added for privacy dominates the signal in gradient geometry for high-dimensional problems; approaches to this that effectively optimize over lower-dimensional subspaces simply ignore the actual problems that varying gradient geometries introduce. In contrast, we study non-isotropic clipping and noise addition, developing a principled theoretical approach; the consequent procedures also enjoy significantly stronger empirical performance than prior approaches.

Annie Marsden, John Duchi, Gregory Valiant

We study probabilistic prediction games when the underlying model is misspecified, investigating the consequences of predicting using an incorrect parametric model. We show that for a broad class of loss functions and parametric families of distributions, the regret of playing a "proper" predictor -- one from the putative model class -- relative to the best predictor in the same model class has lower bound scaling at least as $\sqrt{γn}$, where $γ$ is a measure of the model misspecification to the true distribution in terms of total variation distance. In contrast, using an aggregation-based (improper) learner, one can obtain regret $d \log n$ for any underlying generating distribution, where $d$ is the dimension of the parameter; we exhibit instances in which this is unimprovable even over the family of all learners that may play distributions in the convex hull of the parametric family. These results suggest that simple strategies for aggregating multiple learners together should be more robust, and several experiments conform to this hypothesis.

Aman Sinha, Matthew O'Kelly, Russ Tedrake et al.

Learning-based methodologies increasingly find applications in safety-critical domains like autonomous driving and medical robotics. Due to the rare nature of dangerous events, real-world testing is prohibitively expensive and unscalable. In this work, we employ a probabilistic approach to safety evaluation in simulation, where we are concerned with computing the probability of dangerous events. We develop a novel rare-event simulation method that combines exploration, exploitation, and optimization techniques to find failure modes and estimate their rate of occurrence. We provide rigorous guarantees for the performance of our method in terms of both statistical and computational efficiency. Finally, we demonstrate the efficacy of our approach on a variety of scenarios, illustrating its usefulness as a tool for rapid sensitivity analysis and model comparison that are essential to developing and testing safety-critical autonomous systems.

John Duchi, Tatsunori Hashimoto, Hongseok Namkoong

While modern large-scale datasets often consist of heterogeneous subpopulations -- for example, multiple demographic groups or multiple text corpora -- the standard practice of minimizing average loss fails to guarantee uniformly low losses across all subpopulations. We propose a convex procedure that controls the worst-case performance over all subpopulations of a given size. Our procedure comes with finite-sample (nonparametric) convergence guarantees on the worst-off subpopulation. Empirically, we observe on lexical similarity, wine quality, and recidivism prediction tasks that our worst-case procedure learns models that do well against unseen subpopulations.

Maxime Cauchois, Suyash Gupta, John Duchi

We develop conformal prediction methods for constructing valid predictive confidence sets in multiclass and multilabel problems without assumptions on the data generating distribution. A challenge here is that typical conformal prediction methods---which give marginal validity (coverage) guarantees---provide uneven coverage, in that they address easy examples at the expense of essentially ignoring difficult examples. By leveraging ideas from quantile regression, we build methods that always guarantee correct coverage but additionally provide (asymptotically optimal) conditional coverage for both multiclass and multilabel prediction problems. To address the potential challenge of exponentially large confidence sets in multilabel prediction, we build tree-structured classifiers that efficiently account for interactions between labels. Our methods can be bolted on top of any classification model---neural network, random forest, boosted tree---to guarantee its validity. We also provide an empirical evaluation, simultaneously providing new validation methods, that suggests the more robust coverage of our confidence sets.

Aman Sinha, Matthew O'Kelly, Hongrui Zheng et al.

Balancing performance and safety is crucial to deploying autonomous vehicles in multi-agent environments. In particular, autonomous racing is a domain that penalizes safe but conservative policies, highlighting the need for robust, adaptive strategies. Current approaches either make simplifying assumptions about other agents or lack robust mechanisms for online adaptation. This work makes algorithmic contributions to both challenges. First, to generate a realistic, diverse set of opponents, we develop a novel method for self-play based on replica-exchange Markov chain Monte Carlo. Second, we propose a distributionally robust bandit optimization procedure that adaptively adjusts risk aversion relative to uncertainty in beliefs about opponents' behaviors. We rigorously quantify the tradeoffs in performance and robustness when approximating these computations in real-time motion-planning, and we demonstrate our methods experimentally on autonomous vehicles that achieve scaled speeds comparable to Formula One racecars.

Aditi Raghunathan, Sang Michael Xie, Fanny Yang et al.

Adversarial training augments the training set with perturbations to improve the robust error (over worst-case perturbations), but it often leads to an increase in the standard error (on unperturbed test inputs). Previous explanations for this tradeoff rely on the assumption that no predictor in the hypothesis class has low standard and robust error. In this work, we precisely characterize the effect of augmentation on the standard error in linear regression when the optimal linear predictor has zero standard and robust error. In particular, we show that the standard error could increase even when the augmented perturbations have noiseless observations from the optimal linear predictor. We then prove that the recently proposed robust self-training (RST) estimator improves robust error without sacrificing standard error for noiseless linear regression. Empirically, for neural networks, we find that RST with different adversarial training methods improves both standard and robust error for random and adversarial rotations and adversarial $\ell_\infty$ perturbations in CIFAR-10.

Hilal Asi, John Duchi, Omid Javidbakht

Differential Privacy (DP) provides strong guarantees on the risk of compromising a user's data in statistical learning applications, though these strong protections make learning challenging and may be too stringent for some use cases. To address this, we propose element level differential privacy, which extends differential privacy to provide protection against leaking information about any particular "element" a user has, allowing better utility and more robust results than classical DP. By carefully choosing these "elements," it is possible to provide privacy protections at a desired granularity. We provide definitions, associated privacy guarantees, and analysis to identify the tradeoffs with the new definition; we also develop several private estimation and learning methodologies, providing careful examples for item frequency and M-estimation (empirical risk minimization) with concomitant privacy and utility analysis. We complement our theoretical and methodological advances with several real-world applications, estimating histograms and fitting several large-scale prediction models, including deep networks.

Yu Bai, John Duchi, Song Mei

We study a family of (potentially non-convex) constrained optimization problems with convex composite structure. Through a novel analysis of non-smooth geometry, we show that proximal-type algorithms applied to exact penalty formulations of such problems exhibit local linear convergence under a quadratic growth condition, which the compositional structure we consider ensures. The main application of our results is to low-rank semidefinite optimization with Burer-Monteiro factorizations. We precisely identify the conditions for quadratic growth in the factorized problem via structures in the semidefinite problem, which could be of independent interest for understanding matrix factorization.

Abhishek Bhowmick, John Duchi, Julien Freudiger et al.

In large-scale statistical learning, data collection and model fitting are moving increasingly toward peripheral devices---phones, watches, fitness trackers---away from centralized data collection. Concomitant with this rise in decentralized data are increasing challenges of maintaining privacy while allowing enough information to fit accurate, useful statistical models. This motivates local notions of privacy---most significantly, local differential privacy, which provides strong protections against sensitive data disclosures---where data is obfuscated before a statistician or learner can even observe it, providing strong protections to individuals' data. Yet local privacy as traditionally employed may prove too stringent for practical use, especially in modern high-dimensional statistical and machine learning problems. Consequently, we revisit the types of disclosures and adversaries against which we provide protections, considering adversaries with limited prior information and ensuring that with high probability, ensuring they cannot reconstruct an individual's data within useful tolerances. By reconceptualizing these protections, we allow more useful data release---large privacy parameters in local differential privacy---and we design new (minimax) optimal locally differentially private mechanisms for statistical learning problems for \emph{all} privacy levels. We thus present practicable approaches to large-scale locally private model training that were previously impossible, showing theoretically and empirically that we can fit large-scale image classification and language models with little degradation in utility.

Matthew O'Kelly, Aman Sinha, Hongseok Namkoong et al.

While recent developments in autonomous vehicle (AV) technology highlight substantial progress, we lack tools for rigorous and scalable testing. Real-world testing, the $\textit{de facto}$ evaluation environment, places the public in danger, and, due to the rare nature of accidents, will require billions of miles in order to statistically validate performance claims. We implement a simulation framework that can test an entire modern autonomous driving system, including, in particular, systems that employ deep-learning perception and control algorithms. Using adaptive importance-sampling methods to accelerate rare-event probability evaluation, we estimate the probability of an accident under a base distribution governing standard traffic behavior. We demonstrate our framework on a highway scenario, accelerating system evaluation by $2$-$20$ times over naive Monte Carlo sampling methods and $10$-$300 \mathsf{P}$ times (where $\mathsf{P}$ is the number of processors) over real-world testing.

John Duchi, Hongseok Namkoong

A common goal in statistics and machine learning is to learn models that can perform well against distributional shifts, such as latent heterogeneous subpopulations, unknown covariate shifts, or unmodeled temporal effects. We develop and analyze a distributionally robust stochastic optimization (DRO) framework that learns a model providing good performance against perturbations to the data-generating distribution. We give a convex formulation for the problem, providing several convergence guarantees. We prove finite-sample minimax upper and lower bounds, showing that distributional robustness sometimes comes at a cost in convergence rates. We give limit theorems for the learned parameters, where we fully specify the limiting distribution so that confidence intervals can be computed. On real tasks including generalizing to unknown subpopulations, fine-grained recognition, and providing good tail performance, the distributionally robust approach often exhibits improved performance.

Riccardo Volpi, Hongseok Namkoong, Ozan Sener et al.

We are concerned with learning models that generalize well to different \emph{unseen} domains. We consider a worst-case formulation over data distributions that are near the source domain in the feature space. Only using training data from a single source distribution, we propose an iterative procedure that augments the dataset with examples from a fictitious target domain that is "hard" under the current model. We show that our iterative scheme is an adaptive data augmentation method where we append adversarial examples at each iteration. For softmax losses, we show that our method is a data-dependent regularization scheme that behaves differently from classical regularizers that regularize towards zero (e.g., ridge or lasso). On digit recognition and semantic segmentation tasks, our method learns models improve performance across a range of a priori unknown target domains.

Aman Sinha, Hongseok Namkoong, Riccardo Volpi et al.

Neural networks are vulnerable to adversarial examples and researchers have proposed many heuristic attack and defense mechanisms. We address this problem through the principled lens of distributionally robust optimization, which guarantees performance under adversarial input perturbations. By considering a Lagrangian penalty formulation of perturbing the underlying data distribution in a Wasserstein ball, we provide a training procedure that augments model parameter updates with worst-case perturbations of training data. For smooth losses, our procedure provably achieves moderate levels of robustness with little computational or statistical cost relative to empirical risk minimization. Furthermore, our statistical guarantees allow us to efficiently certify robustness for the population loss. For imperceptible perturbations, our method matches or outperforms heuristic approaches.

John Duchi, Feng Ruan

We study local complexity measures for stochastic convex optimization problems, providing a local minimax theory analogous to that of Hájek and Le Cam for classical statistical problems. We give complementary optimality results, developing fully online methods that adaptively achieve optimal convergence guarantees. Our results provide function-specific lower bounds and convergence results that make precise a correspondence between statistical difficulty and the geometric notion of tilt-stability from optimization. As part of this development, we show how variants of Nesterov's dual averaging---a stochastic gradient-based procedure---guarantee finite time identification of constraints in optimization problems, while stochastic gradient procedures fail. Additionally, we highlight a gap between problems with linear and nonlinear constraints: standard stochastic-gradient-based procedures are suboptimal even for the simplest nonlinear constraints, necessitating the development of asymptotically optimal Riemannian stochastic gradient methods.

John Duchi, Peter Glynn, Hongseok Namkoong

We study statistical inference and distributionally robust solution methods for stochastic optimization problems, focusing on confidence intervals for optimal values and solutions that achieve exact coverage asymptotically. We develop a generalized empirical likelihood framework---based on distributional uncertainty sets constructed from nonparametric $f$-divergence balls---for Hadamard differentiable functionals, and in particular, stochastic optimization problems. As consequences of this theory, we provide a principled method for choosing the size of distributional uncertainty regions to provide one- and two-sided confidence intervals that achieve exact coverage. We also give an asymptotic expansion for our distributionally robust formulation, showing how robustification regularizes problems by their variance. Finally, we show that optimizers of the distributionally robust formulations we study enjoy (essentially) the same consistency properties as those in classical sample average approximations. Our general approach applies to quickly mixing stationary sequences, including geometrically ergodic Harris recurrent Markov chains.

John Duchi, Hongseok Namkoong

We develop an approach to risk minimization and stochastic optimization that provides a convex surrogate for variance, allowing near-optimal and computationally efficient trading between approximation and estimation error. Our approach builds off of techniques for distributionally robust optimization and Owen's empirical likelihood, and we provide a number of finite-sample and asymptotic results characterizing the theoretical performance of the estimator. In particular, we show that our procedure comes with certificates of optimality, achieving (in some scenarios) faster rates of convergence than empirical risk minimization by virtue of automatically balancing bias and variance. We give corroborating empirical evidence showing that in practice, the estimator indeed trades between variance and absolute performance on a training sample, improving out-of-sample (test) performance over standard empirical risk minimization for a number of classification problems.

Aditi Raghunathan, Roy Frostig, John Duchi et al.

In structured prediction problems where we have indirect supervision of the output, maximum marginal likelihood faces two computational obstacles: non-convexity of the objective and intractability of even a single gradient computation. In this paper, we bypass both obstacles for a class of what we call linear indirectly-supervised problems. Our approach is simple: we solve a linear system to estimate sufficient statistics of the model, which we then use to estimate parameters via convex optimization. We analyze the statistical properties of our approach and show empirically that it is effective in two settings: learning with local privacy constraints and learning from low-cost count-based annotations.

Yuancheng Zhu, Sabyasachi Chatterjee, John Duchi et al.

We extend the traditional worst-case, minimax analysis of stochastic convex optimization by introducing a localized form of minimax complexity for individual functions. Our main result gives function-specific lower and upper bounds on the number of stochastic subgradient evaluations needed to optimize either the function or its "hardest local alternative" to a given numerical precision. The bounds are expressed in terms of a localized and computational analogue of the modulus of continuity that is central to statistical minimax analysis. We show how the computational modulus of continuity can be explicitly calculated in concrete cases, and relates to the curvature of the function at the optimum. We also prove a superefficiency result that demonstrates it is a meaningful benchmark, acting as a computational analogue of the Fisher information in statistical estimation. The nature and practical implications of the results are demonstrated in simulations.

Varun Ganapathi, David Vickrey, John Duchi et al.

Parameter estimation in Markov random fields (MRFs) is a difficult task, in which inference over the network is run in the inner loop of a gradient descent procedure. Replacing exact inference with approximate methods such as loopy belief propagation (LBP) can suffer from poor convergence. In this paper, we provide a different approach for combining MRF learning and Bethe approximation. We consider the dual of maximum likelihood Markov network learning - maximizing entropy with moment matching constraints - and then approximate both the objective and the constraints in the resulting optimization problem. Unlike previous work along these lines (Teh & Welling, 2003), our formulation allows parameter sharing between features in a general log-linear model, parameter regularization and conditional training. We show that piecewise training (Sutton & McCallum, 2005) is a very restricted special case of this formulation. We study two optimization strategies: one based on a single convex approximation and one that uses repeated convex approximations. We show results on several real-world networks that demonstrate that these algorithms can significantly outperform learning with loopy and piecewise. Our results also provide a framework for analyzing the trade-offs of different relaxations of the entropy objective and of the constraints.

John Duchi, Stephen Gould, Daphne Koller

Gaussian Markov random fields (GMRFs) are useful in a broad range of applications. In this paper we tackle the problem of learning a sparse GMRF in a high-dimensional space. Our approach uses the l1-norm as a regularization on the inverse covariance matrix. We utilize a novel projected gradient method, which is faster than previous methods in practice and equal to the best performing of these in asymptotic complexity. We also extend the l1-regularized objective to the problem of sparsifying entire blocks within the inverse covariance matrix. Our methods generalize fairly easily to this case, while other methods do not. We demonstrate that our extensions give better generalization performance on two real domains--biological network analysis and a 2D-shape modeling image task.