John C. Duchi

Anastasios N. Angelopoulos, John C. Duchi, Tijana Zrnic · berkeley

We present PPI++: a computationally lightweight methodology for estimation and inference based on a small labeled dataset and a typically much larger dataset of machine-learning predictions. The methods automatically adapt to the quality of available predictions, yielding easy-to-compute confidence sets -- for parameters of any dimensionality -- that always improve on classical intervals using only the labeled data. PPI++ builds on prediction-powered inference (PPI), which targets the same problem setting, improving its computational and statistical efficiency. Real and synthetic experiments demonstrate the benefits of the proposed adaptations.

Yifan Zhu, John C. Duchi, Benjamin Van Roy

We prove that Thompson sampling exhibits $\tilde{O}(σd \sqrt{T} + d r \sqrt{\mathrm{Tr}(Σ_0)})$ Bayesian regret in the linear-Gaussian bandit with a $\mathcal{N}(μ_0, Σ_0)$ prior distribution on the coefficients, where $d$ is the dimension, $T$ is the time horizon, $r$ is the maximum $\ell_2$ norm of the actions, and $σ^2$ is the noise variance. In contrast to existing regret bounds, this shows that to within logarithmic factors, the prior-dependent ``burn-in'' term $d r \sqrt{\mathrm{Tr}(Σ_0)}$ decouples additively from the minimax (long run) regret $σd \sqrt{T}$. Previous regret bounds exhibit a multiplicative dependence on these terms. We establish these results via a new ``elliptical potential'' lemma, and also provide a lower bound indicating that the burn-in term is unavoidable.

John C. Duchi, Suyash Gupta, Kuanhao Jiang et al.

We address the challenge of constructing valid confidence intervals and sets in problems of prediction across multiple environments. We investigate two types of coverage suitable for these problems, extending the jackknife and split-conformal methods to show how to obtain distribution-free coverage in such non-traditional, potentially hierarchical data-generating scenarios. We demonstrate a novel resizing method to adapt to problem difficulty, which applies both to existing approaches for predictive inference and the methods we develop; this reduces prediction set sizes using limited information from the test environment, a key to the methods' practical performance, which we evaluate through neurochemical sensing and species classification datasets. Our contributions also include extensions for settings with non-real-valued responses, a theory of consistency for predictive inference in these general problems, and insights on the limits of conditional coverage.

Felipe Areces, John C. Duchi

The basic question of delineating those statistical problems that are solvable without making any assumptions on the underlying data distribution has long animated statistics and learning theory. This paper characterizes when a convex M-estimation or stochastic optimization problem is solvable in such an assumption-free setting, providing a precise dividing line between solvable and unsolvable problems. The conditions we identify show, perhaps surprisingly, that Lipschitz continuity of the loss being minimized is not necessary for distribution free minimization, and they are also distinct from classical characterizations of learnability in machine learning.

Hilal Asi, John C. Duchi, Kunal Talwar · apple-ml

We investigate differentially private estimators for individual parameters within larger parametric models. While generic private estimators exist, the estimators we provide repose on new local notions of estimand stability, and these notions allow procedures that provide private certificates of their own stability. By leveraging these private certificates, we provide computationally and statistical efficient mechanisms that release private statistics that are, at least asymptotically in the sample size, essentially unimprovable: they achieve instance optimal bounds. Additionally, we investigate the practicality of the algorithms both in simulated data and in real-world data from the American Community Survey and US Census, highlighting scenarios in which the new procedures are successful and identifying areas for future work.

Alnur Ali, Maxime Cauchois, John C. Duchi

The statistical machine learning community has demonstrated considerable resourcefulness over the years in developing highly expressive tools for estimation, prediction, and inference. The bedrock assumptions underlying these developments are that the data comes from a fixed population and displays little heterogeneity. But reality is significantly more complex: statistical models now routinely fail when released into real-world systems and scientific applications, where such assumptions rarely hold. Consequently, we pursue a different path in this paper vis-a-vis the well-worn trail of developing new methodology for estimation and prediction. In this paper, we develop tools and theory for detecting and identifying regions of the covariate space (subpopulations) where model performance has begun to degrade, and study intervening to fix these failures through refitting. We present empirical results with three real-world data sets -- including a time series involving forecasting the incidence of COVID-19 -- showing that our methodology generates interpretable results, is useful for tracking model performance, and can boost model performance through refitting. We complement these empirical results with theory proving that our methodology is minimax optimal for recovering anomalous subpopulations as well as refitting to improve accuracy in a structured normal means setting.

Karan Chadha, Gary Cheng, John C. Duchi

We extend the Approximate-Proximal Point (aProx) family of model-based methods for solving stochastic convex optimization problems, including stochastic subgradient, proximal point, and bundle methods, to the minibatch and accelerated setting. To do so, we propose specific model-based algorithms and an acceleration scheme for which we provide non-asymptotic convergence guarantees, which are order-optimal in all problem-dependent constants and provide linear speedup in minibatch size, while maintaining the desirable robustness traits (e.g. to stepsize) of the aProx family. Additionally, we show improved convergence rates and matching lower bounds identifying new fundamental constants for "interpolation" problems, whose importance in statistical machine learning is growing; this, for example, gives a parallelization strategy for alternating projections. We corroborate our theoretical results with empirical testing to demonstrate the gains accurate modeling, acceleration, and minibatching provide.

Daniel Levy, Yair Carmon, John C. Duchi et al.

We propose and analyze algorithms for distributionally robust optimization of convex losses with conditional value at risk (CVaR) and $χ^2$ divergence uncertainty sets. We prove that our algorithms require a number of gradient evaluations independent of training set size and number of parameters, making them suitable for large-scale applications. For $χ^2$ uncertainty sets these are the first such guarantees in the literature, and for CVaR our guarantees scale linearly in the uncertainty level rather than quadratically as in previous work. We also provide lower bounds proving the worst-case optimality of our algorithms for CVaR and a penalized version of the $χ^2$ problem. Our primary technical contributions are novel bounds on the bias of batch robust risk estimation and the variance of a multilevel Monte Carlo gradient estimator due to [Blanchet & Glynn, 2015]. Experiments on MNIST and ImageNet confirm the theoretical scaling of our algorithms, which are 9--36 times more efficient than full-batch methods.

Maxime Cauchois, Suyash Gupta, Alnur Ali et al.

While the traditional viewpoint in machine learning and statistics assumes training and testing samples come from the same population, practice belies this fiction. One strategy -- coming from robust statistics and optimization -- is thus to build a model robust to distributional perturbations. In this paper, we take a different approach to describe procedures for robust predictive inference, where a model provides uncertainty estimates on its predictions rather than point predictions. We present a method that produces prediction sets (almost exactly) giving the right coverage level for any test distribution in an $f$-divergence ball around the training population. The method, based on conformal inference, achieves (nearly) valid coverage in finite samples, under only the condition that the training data be exchangeable. An essential component of our methodology is to estimate the amount of expected future data shift and build robustness to it; we develop estimators and prove their consistency for protection and validity of uncertainty estimates under shifts. By experimenting on several large-scale benchmark datasets, including Recht et al.'s CIFAR-v4 and ImageNet-V2 datasets, we provide complementary empirical results that highlight the importance of robust predictive validity.

Yossi Arjevani, Yair Carmon, John C. Duchi et al.

We design an algorithm which finds an $ε$-approximate stationary point (with $\|\nabla F(x)\|\le ε$) using $O(ε^{-3})$ stochastic gradient and Hessian-vector products, matching guarantees that were previously available only under a stronger assumption of access to multiple queries with the same random seed. We prove a lower bound which establishes that this rate is optimal and---surprisingly---that it cannot be improved using stochastic $p$th order methods for any $p\ge 2$, even when the first $p$ derivatives of the objective are Lipschitz. Together, these results characterize the complexity of non-convex stochastic optimization with second-order methods and beyond. Expanding our scope to the oracle complexity of finding $(ε,γ)$-approximate second-order stationary points, we establish nearly matching upper and lower bounds for stochastic second-order methods. Our lower bounds here are novel even in the noiseless case.

Hilal Asi, John C. Duchi

We develop two notions of instance optimality in differential privacy, inspired by classical statistical theory: one by defining a local minimax risk and the other by considering unbiased mechanisms and analogizing the Cramer-Rao bound, and we show that the local modulus of continuity of the estimand of interest completely determines these quantities. We also develop a complementary collection mechanisms, which we term the inverse sensitivity mechanisms, which are instance optimal (or nearly instance optimal) for a large class of estimands. Moreover, these mechanisms uniformly outperform the smooth sensitivity framework on each instance for several function classes of interest, including real-valued continuous functions. We carefully present two instantiations of the mechanisms for median and robust regression estimation with corresponding experiments.

Yossi Arjevani, Yair Carmon, John C. Duchi et al.

We lower bound the complexity of finding $ε$-stationary points (with gradient norm at most $ε$) using stochastic first-order methods. In a well-studied model where algorithms access smooth, potentially non-convex functions through queries to an unbiased stochastic gradient oracle with bounded variance, we prove that (in the worst case) any algorithm requires at least $ε^{-4}$ queries to find an $ε$ stationary point. The lower bound is tight, and establishes that stochastic gradient descent is minimax optimal in this model. In a more restrictive model where the noisy gradient estimates satisfy a mean-squared smoothness property, we prove a lower bound of $ε^{-3}$ queries, establishing the optimality of recently proposed variance reduction techniques.

Chen Cheng, Daniel Levy, John C. Duchi

We study computational and statistical consequences of problem geometry in stochastic and online optimization. By focusing on constraint set and gradient geometry, we characterize the problem families for which stochastic- and adaptive-gradient methods are (minimax) optimal and, conversely, when nonlinear updates -- such as those mirror descent employs -- are necessary for optimal convergence. When the constraint set is quadratically convex, diagonally pre-conditioned stochastic gradient methods are minimax optimal. We provide quantitative converses showing that the ``distance'' of the underlying constraints from quadratic convexity determines the sub-optimality of subgradient methods. These results apply, for example, to any $\ell_p$-ball for $p < 2$, and the computation/accuracy tradeoffs they demonstrate exhibit a striking analogy to those in Gaussian sequence models.

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

While adversarial training can improve robust accuracy (against an adversary), it sometimes hurts standard accuracy (when there is no adversary). Previous work has studied this tradeoff between standard and robust accuracy, but only in the setting where no predictor performs well on both objectives in the infinite data limit. In this paper, we show that even when the optimal predictor with infinite data performs well on both objectives, a tradeoff can still manifest itself with finite data. Furthermore, since our construction is based on a convex learning problem, we rule out optimization concerns, thus laying bare a fundamental tension between robustness and generalization. Finally, we show that robust self-training mostly eliminates this tradeoff by leveraging unlabeled data.

Yair Carmon, Aditi Raghunathan, Ludwig Schmidt et al.

We demonstrate, theoretically and empirically, that adversarial robustness can significantly benefit from semisupervised learning. Theoretically, we revisit the simple Gaussian model of Schmidt et al. that shows a sample complexity gap between standard and robust classification. We prove that unlabeled data bridges this gap: a simple semisupervised learning procedure (self-training) achieves high robust accuracy using the same number of labels required for achieving high standard accuracy. Empirically, we augment CIFAR-10 with 500K unlabeled images sourced from 80 Million Tiny Images and use robust self-training to outperform state-of-the-art robust accuracies by over 5 points in (i) $\ell_\infty$ robustness against several strong attacks via adversarial training and (ii) certified $\ell_2$ and $\ell_\infty$ robustness via randomized smoothing. On SVHN, adding the dataset's own extra training set with the labels removed provides gains of 4 to 10 points, within 1 point of the gain from using the extra labels.

Hilal Asi, John C. Duchi

Standard stochastic optimization methods are brittle, sensitive to stepsize choices and other algorithmic parameters, and they exhibit instability outside of well-behaved families of objectives. To address these challenges, we investigate models for stochastic minimization and learning problems that exhibit better robustness to problem families and algorithmic parameters. With appropriately accurate models---which we call the aProx family---stochastic methods can be made stable, provably convergent and asymptotically optimal; even modeling that the objective is nonnegative is sufficient for this stability. We extend these results beyond convexity to weakly convex objectives, which include compositions of convex losses with smooth functions common in modern machine learning applications. We highlight the importance of robustness and accurate modeling with a careful experimental evaluation of convergence time and algorithm sensitivity.

Yair Carmon, John C. Duchi, Aaron Sidford et al.

We show that a simple randomized sketch of the matrix multiplicative weight (MMW) update enjoys (in expectation) the same regret bounds as MMW, up to a small constant factor. Unlike MMW, where every step requires full matrix exponentiation, our steps require only a single product of the form $e^A b$, which the Lanczos method approximates efficiently. Our key technique is to view the sketch as a $\textit{randomized mirror projection}$, and perform mirror descent analysis on the $\textit{expected projection}$. Our sketch solves the online eigenvector problem, improving the best known complexity bounds by $Ω(\log^5 n)$. We also apply this sketch to semidefinite programming in saddle-point form, yielding a simple primal-dual scheme with guarantees matching the best in the literature.

Alon Kipnis, John C. Duchi

We consider the problem of estimating the mean of a symmetric log-concave distribution under the constraint that only a single bit per sample from this distribution is available to the estimator. We study the mean squared error as a function of the sample size (and hence the number of bits). We consider three settings: first, a centralized setting, where an encoder may release $n$ bits given a sample of size $n$, and for which there is no asymptotic penalty for quantization; second, an adaptive setting in which each bit is a function of the current observation and previously recorded bits, where we show that the optimal relative efficiency compared to the sample mean is precisely the efficiency of the median; lastly, we show that in a distributed setting where each bit is only a function of a local sample, no estimator can achieve optimal efficiency uniformly over the parameter space. We additionally complement our results in the adaptive setting by showing that \emph{one} round of adaptivity is sufficient to achieve optimal mean-square error.

Hilal Asi, John C. Duchi

We develop model-based methods for solving stochastic convex optimization problems, introducing the approximate-proximal point, or aProx, family, which includes stochastic subgradient, proximal point, and bundle methods. When the modeling approaches we propose are appropriately accurate, the methods enjoy stronger convergence and robustness guarantees than classical approaches, even though the model-based methods typically add little to no computational overhead over stochastic subgradient methods. For example, we show that improved models converge with probability 1 and enjoy optimal asymptotic normality results under weak assumptions; these methods are also adaptive to a natural class of what we term easy optimization problems, achieving linear convergence under appropriate strong growth conditions on the objective. Our substantial experimental investigation shows the advantages of more accurate modeling over standard subgradient methods across many smooth and non-smooth optimization problems.

Tatsunori B. Hashimoto, Steve Yadlowsky, John C. Duchi

We develop an algorithm for minimizing a function using $n$ batched function value measurements at each of $T$ rounds by using classifiers to identify a function's sublevel set. We show that sufficiently accurate classifiers can achieve linear convergence rates, and show that the convergence rate is tied to the difficulty of active learning sublevel sets. Further, we show that the bootstrap is a computationally efficient approximation to the necessary classification scheme. The end result is a computationally efficient derivative-free algorithm requiring no tuning that consistently outperforms other approaches on simulations, standard benchmarks, real-world DNA binding optimization, and airfoil design problems whenever batched function queries are natural.

Tatsunori B. Hashimoto, John C. Duchi, Percy Liang

Our goal is to extract meaningful transformations from raw images, such as varying the thickness of lines in handwriting or the lighting in a portrait. We propose an unsupervised approach to learn such transformations by attempting to reconstruct an image from a linear combination of transformations of its nearest neighbors. On handwritten digits and celebrity portraits, we show that even with linear transformations, our method generates visually high-quality modified images. Moreover, since our method is semiparametric and does not model the data distribution, the learned transformations extrapolate off the training data and can be applied to new types of images.

John C. Duchi, Sorathan Chaturapruek, Christopher Ré

We show that asymptotically, completely asynchronous stochastic gradient procedures achieve optimal (even to constant factors) convergence rates for the solution of convex optimization problems under nearly the same conditions required for asymptotic optimality of standard stochastic gradient procedures. Roughly, the noise inherent to the stochastic approximation scheme dominates any noise from asynchrony. We also give empirical evidence demonstrating the strong performance of asynchronous, parallel stochastic optimization schemes, demonstrating that the robustness inherent to stochastic approximation problems allows substantially faster parallel and asynchronous solution methods.

John C. Duchi, Michael I. Jordan, Martin J. Wainwright et al.

Large data sets often require performing distributed statistical estimation, with a full data set split across multiple machines and limited communication between machines. To study such scenarios, we define and study some refinements of the classical minimax risk that apply to distributed settings, comparing to the performance of estimators with access to the entire data. Lower bounds on these quantities provide a precise characterization of the minimum amount of communication required to achieve the centralized minimax risk. We study two classes of distributed protocols: one in which machines send messages independently over channels without feedback, and a second allowing for interactive communication, in which a central server broadcasts the messages from a given machine to all other machines. We establish lower bounds for a variety of problems, including location estimation in several families and parameter estimation in different types of regression models. Our results include a novel class of quantitative data-processing inequalities used to characterize the effects of limited communication.

John C. Duchi, Michael I. Jordan, Martin J. Wainwright et al.

We consider derivative-free algorithms for stochastic and non-stochastic convex optimization problems that use only function values rather than gradients. Focusing on non-asymptotic bounds on convergence rates, we show that if pairs of function values are available, algorithms for $d$-dimensional optimization that use gradient estimates based on random perturbations suffer a factor of at most $\sqrt{d}$ in convergence rate over traditional stochastic gradient methods. We establish such results for both smooth and non-smooth cases, sharpening previous analyses that suggested a worse dimension dependence, and extend our results to the case of multiple ($m \ge 2$) evaluations. We complement our algorithmic development with information-theoretic lower bounds on the minimax convergence rate of such problems, establishing the sharpness of our achievable results up to constant (sometimes logarithmic) factors.

John C. Duchi, Michael I. Jordan, Martin J. Wainwright

We provide a detailed study of the estimation of probability distributions---discrete and continuous---in a stringent setting in which data is kept private even from the statistician. We give sharp minimax rates of convergence for estimation in these locally private settings, exhibiting fundamental tradeoffs between privacy and convergence rate, as well as providing tools to allow movement along the privacy-statistical efficiency continuum. One of the consequences of our results is that Warner's classical work on randomized response is an optimal way to perform survey sampling while maintaining privacy of the respondents.

Yuchen Zhang, John C. Duchi, Martin J. Wainwright

We establish optimal convergence rates for a decomposition-based scalable approach to kernel ridge regression. The method is simple to describe: it randomly partitions a dataset of size N into m subsets of equal size, computes an independent kernel ridge regression estimator for each subset, then averages the local solutions into a global predictor. This partitioning leads to a substantial reduction in computation time versus the standard approach of performing kernel ridge regression on all N samples. Our two main theorems establish that despite the computational speed-up, statistical optimality is retained: as long as m is not too large, the partition-based estimator achieves the statistical minimax rate over all estimators using the set of N samples. As concrete examples, our theory guarantees that the number of processors m may grow nearly linearly for finite-rank kernels and Gaussian kernels and polynomially in N for Sobolev spaces, which in turn allows for substantial reductions in computational cost. We conclude with experiments on both simulated data and a music-prediction task that complement our theoretical results, exhibiting the computational and statistical benefits of our approach.

John C. Duchi, Michael I. Jordan, Martin J. Wainwright

Working under a model of privacy in which data remains private even from the statistician, we study the tradeoff between privacy guarantees and the utility of the resulting statistical estimators. We prove bounds on information-theoretic quantities, including mutual information and Kullback-Leibler divergence, that depend on the privacy guarantees. When combined with standard minimax techniques, including the Le Cam, Fano, and Assouad methods, these inequalities allow for a precise characterization of statistical rates under local privacy constraints. We provide a treatment of several canonical families of problems: mean estimation, parameter estimation in fixed-design regression, multinomial probability estimation, and nonparametric density estimation. For all of these families, we provide lower and upper bounds that match up to constant factors, and exhibit new (optimal) privacy-preserving mechanisms and computationally efficient estimators that achieve the bounds.

John C. Duchi, Michael I. Jordan, Martin J. Wainwright

We study statistical risk minimization problems under a privacy model in which the data is kept confidential even from the learner. In this local privacy framework, we establish sharp upper and lower bounds on the convergence rates of statistical estimation procedures. As a consequence, we exhibit a precise tradeoff between the amount of privacy the data preserves and the utility, as measured by convergence rate, of any statistical estimator or learning procedure.

Yuchen Zhang, John C. Duchi, Martin Wainwright

We analyze two communication-efficient algorithms for distributed statistical optimization on large-scale data sets. The first algorithm is a standard averaging method that distributes the $N$ data samples evenly to $\nummac$ machines, performs separate minimization on each subset, and then averages the estimates. We provide a sharp analysis of this average mixture algorithm, showing that under a reasonable set of conditions, the combined parameter achieves mean-squared error that decays as $\order(N^{-1}+(N/m)^{-2})$. Whenever $m \le \sqrt{N}$, this guarantee matches the best possible rate achievable by a centralized algorithm having access to all $\totalnumobs$ samples. The second algorithm is a novel method, based on an appropriate form of bootstrap subsampling. Requiring only a single round of communication, it has mean-squared error that decays as $\order(N^{-1} + (N/m)^{-3})$, and so is more robust to the amount of parallelization. In addition, we show that a stochastic gradient-based method attains mean-squared error decaying as $O(N^{-1} + (N/ m)^{-3/2})$, easing computation at the expense of penalties in the rate of convergence. We also provide experimental evaluation of our methods, investigating their performance both on simulated data and on a large-scale regression problem from the internet search domain. In particular, we show that our methods can be used to efficiently solve an advertisement prediction problem from the Chinese SoSo Search Engine, which involves logistic regression with $N \approx 2.4 \times 10^8$ samples and $d \approx 740,000$ covariates.

Alekh Agarwal, Peter L. Bartlett, John C. Duchi

We analyze general model selection procedures using penalized empirical loss minimization under computational constraints. While classical model selection approaches do not consider computational aspects of performing model selection, we argue that any practical model selection procedure must not only trade off estimation and approximation error, but also the computational effort required to compute empirical minimizers for different function classes. We provide a framework for analyzing such problems, and we give algorithms for model selection under a computational budget. These algorithms satisfy oracle inequalities that show that the risk of the selected model is not much worse than if we had devoted all of our omputational budget to the optimal function class.

John C. Duchi, Lester Mackey, Michael I. Jordan

We consider the predictive problem of supervised ranking, where the task is to rank sets of candidate items returned in response to queries. Although there exist statistical procedures that come with guarantees of consistency in this setting, these procedures require that individuals provide a complete ranking of all items, which is rarely feasible in practice. Instead, individuals routinely provide partial preference information, such as pairwise comparisons of items, and more practical approaches to ranking have aimed at modeling this partial preference data directly. As we show, however, such an approach raises serious theoretical challenges. Indeed, we demonstrate that many commonly used surrogate losses for pairwise comparison data do not yield consistency; surprisingly, we show inconsistency even in low-noise settings. With these negative results as motivation, we present a new approach to supervised ranking based on aggregation of partial preferences, and we develop $U$-statistic-based empirical risk minimization procedures. We present an asymptotic analysis of these new procedures, showing that they yield consistency results that parallel those available for classification. We complement our theoretical results with an experiment studying the new procedures in a large-scale web-ranking task.