Hilal Asi

Hilal Asi, Jonathan Ullman, Lydia Zakynthinou · berkeley

We study the relationship between two desiderata of algorithms in statistical inference and machine learning: differential privacy and robustness to adversarial data corruptions. Their conceptual similarity was first observed by Dwork and Lei (STOC 2009), who observed that private algorithms satisfy robustness, and gave a general method for converting robust algorithms to private ones. However, all general methods for transforming robust algorithms into private ones lead to suboptimal error rates. Our work gives the first black-box transformation that converts any adversarially robust algorithm into one that satisfies pure differential privacy. Moreover, we show that for any low-dimensional estimation task, applying our transformation to an optimal robust estimator results in an optimal private estimator. Thus, we conclude that for any low-dimensional task, the optimal error rate for $\varepsilon$-differentially private estimators is essentially the same as the optimal error rate for estimators that are robust to adversarially corrupting $1/\varepsilon$ training samples. We apply our transformation to obtain new optimal private estimators for several high-dimensional tasks, including Gaussian (sparse) linear regression and PCA. Finally, we present an extension of our transformation that leads to approximate differentially private algorithms whose error does not depend on the range of the output space, which is impossible under pure differential privacy.

Hilal Asi, Vitaly Feldman, Kunal Talwar

We study the problem of mean estimation of $\ell_2$-bounded vectors under the constraint of local differential privacy. While the literature has a variety of algorithms that achieve the asymptotically optimal rates for this problem, the performance of these algorithms in practice can vary significantly due to varying (and often large) hidden constants. In this work, we investigate the question of designing the protocol with the smallest variance. We show that PrivUnit (Bhowmick et al. 2018) with optimized parameters achieves the optimal variance among a large family of locally private randomizers. To prove this result, we establish some properties of local randomizers, and use symmetrization arguments that allow us to write the optimal randomizer as the optimizer of a certain linear program. These structural results, which should extend to other problems, then allow us to show that the optimal randomizer belongs to the PrivUnit family. We also develop a new variant of PrivUnit based on the Gaussian distribution which is more amenable to mathematical analysis and enjoys the same optimality guarantees. This allows us to establish several useful properties on the exact constants of the optimal error as well as to numerically estimate these constants.

Hilal Asi, Vitaly Feldman, Tomer Koren et al.

Online prediction from experts is a fundamental problem in machine learning and several works have studied this problem under privacy constraints. We propose and analyze new algorithms for this problem that improve over the regret bounds of the best existing algorithms for non-adaptive adversaries. For approximate differential privacy, our algorithms achieve regret bounds of $\tilde{O}(\sqrt{T \log d} + \log d/\varepsilon)$ for the stochastic setting and $\tilde{O}(\sqrt{T \log d} + T^{1/3} \log d/\varepsilon)$ for oblivious adversaries (where $d$ is the number of experts). For pure DP, our algorithms are the first to obtain sub-linear regret for oblivious adversaries in the high-dimensional regime $d \ge T$. Moreover, we prove new lower bounds for adaptive adversaries. Our results imply that unlike the non-private setting, there is a strong separation between the optimal regret for adaptive and non-adaptive adversaries for this problem. Our lower bounds also show a separation between pure and approximate differential privacy for adaptive adversaries where the latter is necessary to achieve the non-private $O(\sqrt{T})$ regret.

Hilal Asi, Vitaly Feldman, Jelani Nelson et al.

We study the problem of locally private mean estimation of high-dimensional vectors in the Euclidean ball. Existing algorithms for this problem either incur sub-optimal error or have high communication and/or run-time complexity. We propose a new algorithmic framework, ProjUnit, for private mean estimation that yields algorithms that are computationally efficient, have low communication complexity, and incur optimal error up to a $1+o(1)$-factor. Our framework is deceptively simple: each randomizer projects its input to a random low-dimensional subspace, normalizes the result, and then runs an optimal algorithm such as PrivUnitG in the lower-dimensional space. In addition, we show that, by appropriately correlating the random projection matrices across devices, we can achieve fast server run-time. We mathematically analyze the error of the algorithm in terms of properties of the random projections, and study two instantiations. Lastly, our experiments for private mean estimation and private federated learning demonstrate that our algorithms empirically obtain nearly the same utility as optimal ones while having significantly lower communication and computational cost.

Hilal Asi, Vitaly Feldman, Tomer Koren et al.

We consider online learning problems in the realizable setting, where there is a zero-loss solution, and propose new Differentially Private (DP) algorithms that obtain near-optimal regret bounds. For the problem of online prediction from experts, we design new algorithms that obtain near-optimal regret ${O} \big( \varepsilon^{-1} \log^{1.5}{d} \big)$ where $d$ is the number of experts. This significantly improves over the best existing regret bounds for the DP non-realizable setting which are ${O} \big( \varepsilon^{-1} \min\big\{d, T^{1/3}\log d\big\} \big)$. We also develop an adaptive algorithm for the small-loss setting with regret $O(L^\star\log d + \varepsilon^{-1} \log^{1.5}{d})$ where $L^\star$ is the total loss of the best expert. Additionally, we consider DP online convex optimization in the realizable setting and propose an algorithm with near-optimal regret $O \big(\varepsilon^{-1} d^{1.5} \big)$, as well as an algorithm for the smooth case with regret $O \big( \varepsilon^{-2/3} (dT)^{1/3} \big)$, both significantly improving over existing bounds in the non-realizable regime.

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.

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.

Hilal Asi, Daogao Liu

We study differentially private stochastic convex optimization (DP-SCO) under user-level privacy, where each user may hold multiple data items. Existing work for user-level DP-SCO either requires super-polynomial runtime [Ghazi et al. (2023)] or requires the number of users to grow polynomially with the dimensionality of the problem with additional strict assumptions [Bassily et al. (2023)]. We develop new algorithms for user-level DP-SCO that obtain optimal rates for both convex and strongly convex functions in polynomial time and require the number of users to grow only logarithmically in the dimension. Moreover, our algorithms are the first to obtain optimal rates for non-smooth functions in polynomial time. These algorithms are based on multiple-pass DP-SGD, combined with a novel private mean estimation procedure for concentrated data, which applies an outlier removal step before estimating the mean of the gradients.

Hilal Asi, Vitaly Feldman, Jelani Nelson et al. · apple-ml

We study the problem of private vector mean estimation in the shuffle model of privacy where $n$ users each have a unit vector $v^{(i)} \in\mathbb{R}^d$. We propose a new multi-message protocol that achieves the optimal error using $\tilde{\mathcal{O}}\left(\min(n\varepsilon^2,d)\right)$ messages per user. Moreover, we show that any (unbiased) protocol that achieves optimal error requires each user to send $Ω(\min(n\varepsilon^2,d)/\log(n))$ messages, demonstrating the optimality of our message complexity up to logarithmic factors. Additionally, we study the single-message setting and design a protocol that achieves mean squared error $\mathcal{O}(dn^{d/(d+2)}\varepsilon^{-4/(d+2)})$. Moreover, we show that any single-message protocol must incur mean squared error $Ω(dn^{d/(d+2)})$, showing that our protocol is optimal in the standard setting where $\varepsilon = Θ(1)$. Finally, we study robustness to malicious users and show that malicious users can incur large additive error with a single shuffler.

Vinod Raman, Hilal Asi, Satyen Kale

Recent advances in test-time alignment methods, such as Best-of-N sampling, offer a simple and effective way to steer language models (LMs) toward preferred behaviors using reward models (RM). However, these approaches can be computationally expensive, especially when applied uniformly across prompts without accounting for differences in alignment difficulty. In this work, we propose a prompt-adaptive strategy for Best-of-N alignment that allocates inference-time compute more efficiently. Motivated by latency concerns, we develop a two-stage algorithm: an initial exploratory phase estimates the reward distribution for each prompt using a small exploration budget, and a second stage adaptively allocates the remaining budget using these estimates. Our method is simple, practical, and compatible with any LM-RM combination. Empirical results on prompts from the AlpacaEval, HH-RLHF, and PKU-SafeRLHF datasets for 12 LM/RM pairs and 50 different batches of prompts show that our adaptive strategy outperforms the uniform allocation with the same inference budget. Moreover, we show that our adaptive strategy remains competitive against uniform allocations with 20 percent larger inference budgets and improves in performance as the batch size grows.

Hilal Asi, Vitaly Feldman, Hannah Keller et al. · apple-ml

We revisit the problem of secure aggregation of high-dimensional vectors in a two-server system such as Prio. These systems are typically used to aggregate vectors such as gradients in private federated learning, where the aggregate itself is protected via noise addition to ensure differential privacy. Existing approaches require communication scaling with the dimensionality, and thus limit the dimensionality of vectors one can efficiently process in this setup. We propose PREAMBLE: {\bf Pr}ivate {\bf E}fficient {\bf A}ggregation {\bf M}echanism via {\bf BL}ock-sparse {\bf E}uclidean Vectors. PREAMBLE builds on an extension of distributed point functions that enables communication- and computation-efficient aggregation of {\em block-sparse vectors}, which are sparse vectors where the non-zero entries occur in a small number of clusters of consecutive coordinates. We show that these block-sparse DPFs can be combined with random sampling and privacy amplification by sampling results, to allow asymptotically optimal privacy-utility trade-offs for vector aggregation, at a fraction of the communication cost. When coupled with recent advances in numerical privacy accounting, our approach incurs a negligible overhead in noise variance, compared to the Gaussian mechanism used with Prio.

Andrew Lowy, Daogao Liu, Hilal Asi

We study private stochastic convex optimization (SCO) under user-level differential privacy (DP) constraints. In this setting, there are $n$ users (e.g., cell phones), each possessing $m$ data items (e.g., text messages), and we need to protect the privacy of each user's entire collection of data items. Existing algorithms for user-level DP SCO are impractical in many large-scale machine learning scenarios because: (i) they make restrictive assumptions on the smoothness parameter of the loss function and require the number of users to grow polynomially with the dimension of the parameter space; or (ii) they are prohibitively slow, requiring at least $(mn)^{3/2}$ gradient computations for smooth losses and $(mn)^3$ computations for non-smooth losses. To address these limitations, we provide novel user-level DP algorithms with state-of-the-art excess risk and runtime guarantees, without stringent assumptions. First, we develop a linear-time algorithm with state-of-the-art excess risk (for a non-trivial linear-time algorithm) under a mild smoothness assumption. Our second algorithm applies to arbitrary smooth losses and achieves optimal excess risk in $\approx (mn)^{9/8}$ gradient computations. Third, for non-smooth loss functions, we obtain optimal excess risk in $n^{11/8} m^{5/4}$ gradient computations. Moreover, our algorithms do not require the number of users to grow polynomially with the dimension.

Aadirupa Saha, Hilal Asi

We consider the well-studied dueling bandit problem, where a learner aims to identify near-optimal actions using pairwise comparisons, under the constraint of differential privacy. We consider a general class of utility-based preference matrices for large (potentially unbounded) decision spaces and give the first differentially private dueling bandit algorithm for active learning with user preferences. Our proposed algorithms are computationally efficient with near-optimal performance, both in terms of the private and non-private regret bound. More precisely, we show that when the decision space is of finite size $K$, our proposed algorithm yields order optimal $O\Big(\sum_{i = 2}^K\log\frac{KT}{Δ_i} + \frac{K}ε\Big)$ regret bound for pure $ε$-DP, where $Δ_i$ denotes the suboptimality gap of the $i$-th arm. We also present a matching lower bound analysis which proves the optimality of our algorithms. Finally, we extend our results to any general decision space in $d$-dimensions with potentially infinite arms and design an $ε$-DP algorithm with regret $\tilde{O} \left( \frac{d^6}{κε} + \frac{ d\sqrt{T }}κ \right)$, providing privacy for free when $T \gg d$.

Hilal Asi, Vinod Raman, Kunal Talwar · apple-ml

We design new differentially private algorithms for the problems of adversarial bandits and bandits with expert advice. For adversarial bandits, we give a simple and efficient conversion of any non-private bandit algorithm to a private bandit algorithm. Instantiating our conversion with existing non-private bandit algorithms gives a regret upper bound of $O\left(\frac{\sqrt{KT}}{\sqrtε}\right)$, improving upon the existing upper bound $O\left(\frac{\sqrt{KT \log(KT)}}ε\right)$ for all $ε\leq 1$. In particular, our algorithms allow for sublinear expected regret even when $ε\leq \frac{1}{\sqrt{T}}$, establishing the first known separation between central and local differential privacy for this problem. For bandits with expert advice, we give the first differentially private algorithms, with expected regret $O\left(\frac{\sqrt{NT}}{\sqrtε}\right), O\left(\frac{\sqrt{KT\log(N)}\log(KT)}ε\right)$, and $\tilde{O}\left(\frac{N^{1/6}K^{1/2}T^{2/3}\log(NT)}{ε^{1/3}} + \frac{N^{1/2}\log(NT)}ε\right)$, where $K$ and $N$ are the number of actions and experts respectively. These rates allow us to get sublinear regret for different combinations of small and large $K, N$ and $ε.$

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.

Aadirupa Saha, Vinod Raman, Hilal Asi

We design differentially private algorithms for the problem of prediction with expert advice under dynamic regret, also known as tracking the best expert. Our work addresses three natural types of adversaries, stochastic with shifting distributions, oblivious, and adaptive, and designs algorithms with sub-linear regret for all three cases. In particular, under a shifting stochastic adversary where the distribution may shift $S$ times, we provide an $ε$-differentially private algorithm whose expected dynamic regret is at most $O\left( \sqrt{S T \log (NT)} + \frac{S \log (NT)}ε\right)$, where $T$ and $N$ are the epsilon horizon and number of experts, respectively. For oblivious adversaries, we give a reduction from dynamic regret minimization to static regret minimization, resulting in an upper bound of $O\left(\sqrt{S T \log(NT)} + \frac{S T^{1/3}\log(T/δ) \log(NT)}{ε^{2/3}}\right)$ on the expected dynamic regret, where $S$ now denotes the allowable number of switches of the best expert. Finally, similar to static regret, we establish a fundamental separation between oblivious and adaptive adversaries for the dynamic setting: while our algorithms show that sub-linear regret is achievable for oblivious adversaries in the high-privacy regime $ε\le \sqrt{S/T}$, we show that any $(ε, δ)$-differentially private algorithm must suffer linear dynamic regret under adaptive adversaries for $ε\le \sqrt{S/T}$. Finally, to complement this lower bound, we give an $ε$-differentially private algorithm that attains sub-linear dynamic regret under adaptive adversaries whenever $ε\gg \sqrt{S/T}$.

Hilal Asi, Tomer Koren, Daogao Liu et al.

We study the problem of private online learning, specifically, online prediction from experts (OPE) and online convex optimization (OCO). We propose a new transformation that transforms lazy online learning algorithms into private algorithms. We apply our transformation for differentially private OPE and OCO using existing lazy algorithms for these problems. Our final algorithms obtain regret, which significantly improves the regret in the high privacy regime $\varepsilon \ll 1$, obtaining $\sqrt{T \log d} + T^{1/3} \log(d)/\varepsilon^{2/3}$ for DP-OPE and $\sqrt{T} + T^{1/3} \sqrt{d}/\varepsilon^{2/3}$ for DP-OCO. We also complement our results with a lower bound for DP-OPE, showing that these rates are optimal for a natural family of low-switching private algorithms.

Hilal Asi, Daogao Liu, Kevin Tian

We study the problem of differentially private stochastic convex optimization (DP-SCO) with heavy-tailed gradients, where we assume a $k^{\text{th}}$-moment bound on the Lipschitz constants of sample functions rather than a uniform bound. We propose a new reduction-based approach that enables us to obtain the first optimal rates (up to logarithmic factors) in the heavy-tailed setting, achieving error $G_2 \cdot \frac 1 {\sqrt n} + G_k \cdot (\frac{\sqrt d}{nε})^{1 - \frac 1 k}$ under $(ε, δ)$-approximate differential privacy, up to a mild $\textup{polylog}(\frac{1}δ)$ factor, where $G_2^2$ and $G_k^k$ are the $2^{\text{nd}}$ and $k^{\text{th}}$ moment bounds on sample Lipschitz constants, nearly-matching a lower bound of [Lowy and Razaviyayn 2023]. We further give a suite of private algorithms in the heavy-tailed setting which improve upon our basic result under additional assumptions, including an optimal algorithm under a known-Lipschitz constant assumption, a near-linear time algorithm for smooth functions, and an optimal linear time algorithm for smooth generalized linear models.

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.

Hilal Asi, Yair Carmon, Arun Jambulapati et al.

We develop a new primitive for stochastic optimization: a low-bias, low-cost estimator of the minimizer $x_\star$ of any Lipschitz strongly-convex function. In particular, we use a multilevel Monte-Carlo approach due to Blanchet and Glynn to turn any optimal stochastic gradient method into an estimator of $x_\star$ with bias $δ$, variance $O(\log(1/δ))$, and an expected sampling cost of $O(\log(1/δ))$ stochastic gradient evaluations. As an immediate consequence, we obtain cheap and nearly unbiased gradient estimators for the Moreau-Yoshida envelope of any Lipschitz convex function, allowing us to perform dimension-free randomized smoothing. We demonstrate the potential of our estimator through four applications. First, we develop a method for minimizing the maximum of $N$ functions, improving on recent results and matching a lower bound up to logarithmic factors. Second and third, we recover state-of-the-art rates for projection-efficient and gradient-efficient optimization using simple algorithms with a transparent analysis. Finally, we show that an improved version of our estimator would yield a nearly linear-time, optimal-utility, differentially-private non-smooth stochastic optimization method.

Hilal Asi, Vitaly Feldman, Tomer Koren et al.

Stochastic convex optimization over an $\ell_1$-bounded domain is ubiquitous in machine learning applications such as LASSO but remains poorly understood when learning with differential privacy. We show that, up to logarithmic factors the optimal excess population loss of any $(\varepsilon,δ)$-differentially private optimizer is $\sqrt{\log(d)/n} + \sqrt{d}/\varepsilon n.$ The upper bound is based on a new algorithm that combines the iterative localization approach of~\citet{FeldmanKoTa20} with a new analysis of private regularized mirror descent. It applies to $\ell_p$ bounded domains for $p\in [1,2]$ and queries at most $n^{3/2}$ gradients improving over the best previously known algorithm for the $\ell_2$ case which needs $n^2$ gradients. Further, we show that when the loss functions satisfy additional smoothness assumptions, the excess loss is upper bounded (up to logarithmic factors) by $\sqrt{\log(d)/n} + (\log(d)/\varepsilon n)^{2/3}.$ This bound is achieved by a new variance-reduced version of the Frank-Wolfe algorithm that requires just a single pass over the data. We also show that the lower bound in this case is the minimum of the two rates mentioned above.

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.

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.

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.

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.