Cristóbal Guzmán

Raman Arora, Raef Bassily, Tomás González et al.

We study the problem of approximating stationary points of Lipschitz and smooth functions under $(\varepsilon,δ)$-differential privacy (DP) in both the finite-sum and stochastic settings. A point $\widehat{w}$ is called an $α$-stationary point of a function $F:\mathbb{R}^d\rightarrow\mathbb{R}$ if $\|\nabla F(\widehat{w})\|\leq α$. We provide a new efficient algorithm that finds an $\tilde{O}\big(\big[\frac{\sqrt{d}}{n\varepsilon}\big]^{2/3}\big)$-stationary point in the finite-sum setting, where $n$ is the number of samples. This improves on the previous best rate of $\tilde{O}\big(\big[\frac{\sqrt{d}}{n\varepsilon}\big]^{1/2}\big)$. We also give a new construction that improves over the existing rates in the stochastic optimization setting, where the goal is to find approximate stationary points of the population risk. Our construction finds a $\tilde{O}\big(\frac{1}{n^{1/3}} + \big[\frac{\sqrt{d}}{n\varepsilon}\big]^{1/2}\big)$-stationary point of the population risk in time linear in $n$. Furthermore, under the additional assumption of convexity, we completely characterize the sample complexity of finding stationary points of the population risk (up to polylog factors) and show that the optimal rate on population stationarity is $\tilde Θ\big(\frac{1}{\sqrt{n}}+\frac{\sqrt{d}}{n\varepsilon}\big)$. Finally, we show that our methods can be used to provide dimension-independent rates of $O\big(\frac{1}{\sqrt{n}}+\min\big(\big[\frac{\sqrt{rank}}{n\varepsilon}\big]^{2/3},\frac{1}{(n\varepsilon)^{2/5}}\big)\big)$ on population stationarity for Generalized Linear Models (GLM), where $rank$ is the rank of the design matrix, which improves upon the previous best known rate.

Raman Arora, Raef Bassily, Cristóbal Guzmán et al.

We study the problem of $(ε,δ)$-differentially private learning of linear predictors with convex losses. We provide results for two subclasses of loss functions. The first case is when the loss is smooth and non-negative but not necessarily Lipschitz (such as the squared loss). For this case, we establish an upper bound on the excess population risk of $\tilde{O}\left(\frac{\Vert w^*\Vert}{\sqrt{n}} + \min\left\{\frac{\Vert w^* \Vert^2}{(nε)^{2/3}},\frac{\sqrt{d}\Vert w^*\Vert^2}{nε}\right\}\right)$, where $n$ is the number of samples, $d$ is the dimension of the problem, and $w^*$ is the minimizer of the population risk. Apart from the dependence on $\Vert w^\ast\Vert$, our bound is essentially tight in all parameters. In particular, we show a lower bound of $\tildeΩ\left(\frac{1}{\sqrt{n}} + {\min\left\{\frac{\Vert w^*\Vert^{4/3}}{(nε)^{2/3}}, \frac{\sqrt{d}\Vert w^*\Vert}{nε}\right\}}\right)$. We also revisit the previously studied case of Lipschitz losses [SSTT20]. For this case, we close the gap in the existing work and show that the optimal rate is (up to log factors) $Θ\left(\frac{\Vert w^*\Vert}{\sqrt{n}} + \min\left\{\frac{\Vert w^*\Vert}{\sqrt{nε}},\frac{\sqrt{\text{rank}}\Vert w^*\Vert}{nε}\right\}\right)$, where $\text{rank}$ is the rank of the design matrix. This improves over existing work in the high privacy regime. Finally, our algorithms involve a private model selection approach that we develop to enable attaining the stated rates without a-priori knowledge of $\Vert w^*\Vert$.

Raef Bassily, Cristóbal Guzmán, Michael Menart

We show that convex-concave Lipschitz stochastic saddle point problems (also known as stochastic minimax optimization) can be solved under the constraint of $(ε,δ)$-differential privacy with \emph{strong (primal-dual) gap} rate of $\tilde O\big(\frac{1}{\sqrt{n}} + \frac{\sqrt{d}}{nε}\big)$, where $n$ is the dataset size and $d$ is the dimension of the problem. This rate is nearly optimal, based on existing lower bounds in differentially private stochastic optimization. Specifically, we prove a tight upper bound on the strong gap via novel implementation and analysis of the recursive regularization technique repurposed for saddle point problems. We show that this rate can be attained with $O\big(\min\big\{\frac{n^2ε^{1.5}}{\sqrt{d}}, n^{3/2}\big\}\big)$ gradient complexity, and $\tilde{O}(n)$ gradient complexity if the loss function is smooth. As a byproduct of our method, we develop a general algorithm that, given a black-box access to a subroutine satisfying a certain $α$ primal-dual accuracy guarantee with respect to the empirical objective, gives a solution to the stochastic saddle point problem with a strong gap of $\tilde{O}(α+\frac{1}{\sqrt{n}})$. We show that this $α$-accuracy condition is satisfied by standard algorithms for the empirical saddle point problem such as the proximal point method and the stochastic gradient descent ascent algorithm. Further, we show that even for simple problems it is possible for an algorithm to have zero weak gap and suffer from $Ω(1)$ strong gap. We also show that there exists a fundamental tradeoff between stability and accuracy. Specifically, we show that any $Δ$-stable algorithm has empirical gap $Ω\big(\frac{1}{Δn}\big)$, and that this bound is tight. This result also holds also more specifically for empirical risk minimization problems and may be of independent interest.

Alexandre d'Aspremont, Cristóbal Guzmán, Clément Lezane

Inspired by regularization techniques in statistics and machine learning, we study complementary composite minimization in the stochastic setting. This problem corresponds to the minimization of the sum of a (weakly) smooth function endowed with a stochastic first-order oracle, and a structured uniformly convex (possibly nonsmooth and non-Lipschitz) regularization term. Despite intensive work on closely related settings, prior to our work no complexity bounds for this problem were known. We close this gap by providing novel excess risk bounds, both in expectation and with high probability. Our algorithms are nearly optimal, which we prove via novel lower complexity bounds for this class of problems. We conclude by providing numerical results comparing our methods to the state of the art.

Natalia Ponomareva, Zheng Xu, H. Brendan McMahan et al.

High quality data is needed to unlock the full potential of AI for end users. However finding new sources of such data is getting harder: most publicly-available human generated data will soon have been used. Additionally, publicly available data often is not representative of users of a particular system -- for example, a research speech dataset of contractors interacting with an AI assistant will likely be more homogeneous, well articulated and self-censored than real world commands that end users will issue. Therefore unlocking high-quality data grounded in real user interactions is of vital interest. However, the direct use of user data comes with significant privacy risks. Differential Privacy (DP) is a well established framework for reasoning about and limiting information leakage, and is a gold standard for protecting user privacy. The focus of this work, \emph{Differentially Private Synthetic data}, refers to synthetic data that preserves the overall trends of source data,, while providing strong privacy guarantees to individuals that contributed to the source dataset. DP synthetic data can unlock the value of datasets that have previously been inaccessible due to privacy concerns and can replace the use of sensitive datasets that previously have only had rudimentary protections like ad-hoc rule-based anonymization. In this paper we explore the full suite of techniques surrounding DP synthetic data, the types of privacy protections they offer and the state-of-the-art for various modalities (image, tabular, text and decentralized). We outline all the components needed in a system that generates DP synthetic data, from sensitive data handling and preparation, to tracking the use and empirical privacy testing. We hope that work will result in increased adoption of DP synthetic data, spur additional research and increase trust in DP synthetic data approaches.

Xufeng Cai, Chaobing Song, Cristóbal Guzmán et al.

We study stochastic monotone inclusion problems, which widely appear in machine learning applications, including robust regression and adversarial learning. We propose novel variants of stochastic Halpern iteration with recursive variance reduction. In the cocoercive -- and more generally Lipschitz-monotone -- setup, our algorithm attains $ε$ norm of the operator with $\mathcal{O}(\frac{1}{ε^3})$ stochastic operator evaluations, which significantly improves over state of the art $\mathcal{O}(\frac{1}{ε^4})$ stochastic operator evaluations required for existing monotone inclusion solvers applied to the same problem classes. We further show how to couple one of the proposed variants of stochastic Halpern iteration with a scheduled restart scheme to solve stochastic monotone inclusion problems with ${\mathcal{O}}(\frac{\log(1/ε)}{ε^2})$ stochastic operator evaluations under additional sharpness or strong monotonicity assumptions.

Michael Menart, Enayat Ullah, Raman Arora et al.

We study private empirical risk minimization (ERM) problem for losses satisfying the $(γ,κ)$-Kurdyka-Łojasiewicz (KL) condition. The Polyak-Łojasiewicz (PL) condition is a special case of this condition when $κ=2$. Specifically, we study this problem under the constraint of $ρ$ zero-concentrated differential privacy (zCDP). When $κ\in[1,2]$ and the loss function is Lipschitz and smooth over a sufficiently large region, we provide a new algorithm based on variance reduced gradient descent that achieves the rate $\tilde{O}\big(\big(\frac{\sqrt{d}}{n\sqrtρ}\big)^κ\big)$ on the excess empirical risk, where $n$ is the dataset size and $d$ is the dimension. We further show that this rate is nearly optimal. When $κ\geq 2$ and the loss is instead Lipschitz and weakly convex, we show it is possible to achieve the rate $\tilde{O}\big(\big(\frac{\sqrt{d}}{n\sqrtρ}\big)^κ\big)$ with a private implementation of the proximal point method. When the KL parameters are unknown, we provide a novel modification and analysis of the noisy gradient descent algorithm and show that this algorithm achieves a rate of $\tilde{O}\big(\big(\frac{\sqrt{d}}{n\sqrtρ}\big)^{\frac{2κ}{4-κ}}\big)$ adaptively, which is nearly optimal when $κ= 2$. We further show that, without assuming the KL condition, the same gradient descent algorithm can achieve fast convergence to a stationary point when the gradient stays sufficiently large during the run of the algorithm. Specifically, we show that this algorithm can approximate stationary points of Lipschitz, smooth (and possibly nonconvex) objectives with rate as fast as $\tilde{O}\big(\frac{\sqrt{d}}{n\sqrtρ}\big)$ and never worse than $\tilde{O}\big(\big(\frac{\sqrt{d}}{n\sqrtρ}\big)^{1/2}\big)$. The latter rate matches the best known rate for methods that do not rely on variance reduction.

Maximilian Fichtl, Cristóbal Guzmán, Nishant A. Mehta

This work introduces a general framework for calibeating based on regret minimization. As compared to Foster and Hart's seminal calibeating work which had specialized treatments of Brier score (squared loss) and log loss, we consider a large family of proper losses that includes $α$-Tsallis losses (for $α\in [1, 2]$) and Lipschitz losses. Our results for Tsallis losses also hold for an unscaled version of Tsallis loss that recovers log loss. Our analysis is oriented around the Bregman divergence view of a proper loss. Technically, our results for the family of Tsallis losses that we consider are U-calibration results, simultaneously obtaining logarithmic regret for all losses in this family while having a weaker dependence on the dimension compared to previous results. Of potential independent interest, we also show a new regret equality for the regret of Be The Regularized Leader. This regret equality holds for general proper losses and itself is based on two results related to online updating formulas for the generalized variance, the latter being a previously introduced generalization of variance based on Bregman divergences.

Tomás González, Cristóbal Guzmán, Courtney Paquette

We study the problem of differentially-private (DP) stochastic (convex-concave) saddle-points in the $\ell_1$ setting. We propose $(\varepsilon, δ)$-DP algorithms based on stochastic mirror descent that attain nearly dimension-independent convergence rates for the expected duality gap, a type of guarantee that was known before only for bilinear objectives. For convex-concave and first-order-smooth stochastic objectives, our algorithms attain a rate of $\sqrt{\log(d)/n} + (\log(d)^{3/2}/[n\varepsilon])^{1/3}$, where $d$ is the dimension of the problem and $n$ the dataset size. Under an additional second-order-smoothness assumption, we show that the duality gap is bounded by $\sqrt{\log(d)/n} + \log(d)/\sqrt{n\varepsilon}$ with high probability, by using bias-reduced gradient estimators. This rate provides evidence of the near-optimality of our approach, since a lower bound of $\sqrt{\log(d)/n} + \log(d)^{3/4}/\sqrt{n\varepsilon}$ exists. Finally, we show that combining our methods with acceleration techniques from online learning leads to the first algorithm for DP Stochastic Convex Optimization in the $\ell_1$ setting that is not based on Frank-Wolfe methods. For convex and first-order-smooth stochastic objectives, our algorithms attain an excess risk of $\sqrt{\log(d)/n} + \log(d)^{7/10}/[n\varepsilon]^{2/5}$, and when additionally assuming second-order-smoothness, we improve the rate to $\sqrt{\log(d)/n} + \log(d)/\sqrt{n\varepsilon}$. Instrumental to all of these results are various extensions of the classical Maurey Sparsification Lemma \cite{Pisier:1980}, which may be of independent interest.

Badih Ghazi, Cristóbal Guzmán, Pritish Kamath et al.

Motivated by applications of large embedding models, we study differentially private (DP) optimization problems under sparsity of individual gradients. We start with new near-optimal bounds for the classic mean estimation problem but with sparse data, improving upon existing algorithms particularly for the high-dimensional regime. Building on this, we obtain pure- and approximate-DP algorithms with almost optimal rates for stochastic convex optimization with sparse gradients; the former represents the first nearly dimension-independent rates for this problem. Finally, we study the approximation of stationary points for the empirical loss in approximate-DP optimization and obtain rates that depend on sparsity instead of dimension, modulo polylogarithmic factors.

Jelena Diakonikolas, Cristóbal Guzmán

We initiate the study of nonsmooth optimization problems under bounded local subgradient variation, which postulates bounded difference between (sub)gradients in small local regions around points, in either average or maximum sense. The resulting class of objective functions encapsulates the classes of objective functions traditionally studied in optimization, which are defined based on either Lipschitz continuity of the objective or Hölder/Lipschitz continuity of its gradient. Further, the defined class contains functions that are neither Lipschitz continuous nor have a Hölder continuous gradient. When restricted to the traditional classes of optimization problems, the parameters defining the studied classes lead to more fine-grained complexity bounds, recovering traditional oracle complexity bounds in the worst case but generally leading to lower oracle complexity for functions that are not ``worst case.'' Some highlights of our results are that: (i) it is possible to obtain complexity results for both convex and nonconvex problems with the (local or global) Lipschitz constant being replaced by a constant of local subgradient variation and (ii) mean width of the subdifferential set around the optima plays a role in the complexity of nonsmooth optimization, particularly in parallel settings. A consequence of (ii) is that for any error parameter $ε> 0$, parallel oracle complexity of nonsmooth Lipschitz convex optimization is lower than its sequential oracle complexity by a factor $\tildeΩ\big(\frac{1}ε\big)$ whenever the objective function is piecewise linear with polynomially many pieces in the input size. This is particularly surprising as existing parallel complexity lower bounds are based on such classes of functions. The seeming contradiction is resolved by considering the region in which the algorithm is allowed to query the objective.

Enayat Ullah, Michael Menart, Raef Bassily et al.

We study the limits and capability of public-data assisted differentially private (PA-DP) algorithms. Specifically, we focus on the problem of stochastic convex optimization (SCO) with either labeled or unlabeled public data. For complete/labeled public data, we show that any $(ε,δ)$-PA-DP has excess risk $\tildeΩ\big(\min\big\{\frac{1}{\sqrt{n_{\text{pub}}}},\frac{1}{\sqrt{n}}+\frac{\sqrt{d}}{nε} \big\} \big)$, where $d$ is the dimension, ${n_{\text{pub}}}$ is the number of public samples, ${n_{\text{priv}}}$ is the number of private samples, and $n={n_{\text{pub}}}+{n_{\text{priv}}}$. These lower bounds are established via our new lower bounds for PA-DP mean estimation, which are of a similar form. Up to constant factors, these lower bounds show that the simple strategy of either treating all data as private or discarding the private data, is optimal. We also study PA-DP supervised learning with \textit{unlabeled} public samples. In contrast to our previous result, we here show novel methods for leveraging public data in private supervised learning. For generalized linear models (GLM) with unlabeled public data, we show an efficient algorithm which, given $\tilde{O}({n_{\text{priv}}}ε)$ unlabeled public samples, achieves the dimension independent rate $\tilde{O}\big(\frac{1}{\sqrt{n_{\text{priv}}}} + \frac{1}{\sqrt{n_{\text{priv}}ε}}\big)$. We develop new lower bounds for this setting which shows that this rate cannot be improved with more public samples, and any fewer public samples leads to a worse rate. Finally, we provide extensions of this result to general hypothesis classes with finite fat-shattering dimension with applications to neural networks and non-Euclidean geometries.

Raef Bassily, Cristóbal Guzmán, Michael Menart

In this work, we conduct a systematic study of stochastic saddle point problems (SSP) and stochastic variational inequalities (SVI) under the constraint of $(ε,δ)$-differential privacy (DP) in both Euclidean and non-Euclidean setups. We first consider Lipschitz convex-concave SSPs in the $\ell_p/\ell_q$ setup, $p,q\in[1,2]$. Here, we obtain a bound of $\tilde{O}\big(\frac{1}{\sqrt{n}} + \frac{\sqrt{d}}{nε}\big)$ on the strong SP-gap, where $n$ is the number of samples and $d$ is the dimension. This rate is nearly optimal for any $p,q\in[1,2]$. Without additional assumptions, such as smoothness or linearity requirements, prior work under DP has only obtained this rate when $p=q=2$ (i.e., only in the Euclidean setup). Further, existing algorithms have each only been shown to work for specific settings of $p$ and $q$ and under certain assumptions on the loss and the feasible set, whereas we provide a general algorithm for DP SSPs whenever $p,q\in[1,2]$. Our result is obtained via a novel analysis of the recursive regularization algorithm. In particular, we develop new tools for analyzing generalization, which may be of independent interest. Next, we turn our attention towards SVIs with a monotone, bounded and Lipschitz operator and consider $\ell_p$-setups, $p\in[1,2]$. Here, we provide the first analysis which obtains a bound on the strong VI-gap of $\tilde{O}\big(\frac{1}{\sqrt{n}} + \frac{\sqrt{d}}{nε}\big)$. For $p-1=Ω(1)$, this rate is near optimal due to existing lower bounds. To obtain this result, we develop a modified version of recursive regularization. Our analysis builds on the techniques we develop for SSPs as well as employing additional novel components which handle difficulties arising from adapting the recursive regularization framework to SVIs.

Juan Pablo Contreras, Cristóbal Guzmán, David Martínez-Rubio

We develop algorithms for the optimization of convex objectives that have Hölder continuous $q$-th derivatives by using a $q$-th order oracle, for any $q \geq 1$. Our algorithms work for general norms under mild conditions, including the $\ell_p$-settings for $1\leq p\leq \infty$. We can also optimize structured functions that allow for inexactly implementing a non-Euclidean ball optimization oracle. We do this by developing a non-Euclidean inexact accelerated proximal point method that makes use of an \emph{inexact uniformly convex regularizer}. We show a lower bound for general norms that demonstrates our algorithms are nearly optimal in high-dimensions in the black-box oracle model for $\ell_p$-settings and all $q \geq 1$, even in randomized and parallel settings. This new lower bound, when applied to the first-order smooth case, resolves an open question in parallel convex optimization.

Badih Ghazi, Cristóbal Guzmán, Pritish Kamath et al.

We introduce $\mathsf{PREM}$ (Private Relative Error Multiplicative weight update), a new framework for generating synthetic data that achieves a relative error guarantee for statistical queries under $(\varepsilon, δ)$ differential privacy (DP). Namely, for a domain ${\cal X}$, a family ${\cal F}$ of queries $f : {\cal X} \to \{0, 1\}$, and $ζ> 0$, our framework yields a mechanism that on input dataset $D \in {\cal X}^n$ outputs a synthetic dataset $\widehat{D} \in {\cal X}^n$ such that all statistical queries in ${\cal F}$ on $D$, namely $\sum_{x \in D} f(x)$ for $f \in {\cal F}$, are within a $1 \pm ζ$ multiplicative factor of the corresponding value on $\widehat{D}$ up to an additive error that is polynomial in $\log |{\cal F}|$, $\log |{\cal X}|$, $\log n$, $\log(1/δ)$, $1/\varepsilon$, and $1/ζ$. In contrast, any $(\varepsilon, δ)$-DP mechanism is known to require worst-case additive error that is polynomial in at least one of $n, |{\cal F}|$, or $|{\cal X}|$. We complement our algorithm with nearly matching lower bounds.

Mario Bravo, Juan P. Flores-Mella, Cristóbal Guzmán

We study the mixing time of the projected Langevin algorithm (LA) and the privacy curve of noisy Stochastic Gradient Descent (SGD), beyond nonexpansive iterations. Specifically, we derive new mixing time bounds for the projected LA which are, in some important cases, dimension-free and poly-logarithmic on the accuracy, closely matching the existing results in the smooth convex case. Additionally, we establish new upper bounds for the privacy curve of the subsampled noisy SGD algorithm. These bounds show a crucial dependency on the regularity of gradients, and are useful for a wide range of convex losses beyond the smooth case. Our analysis relies on a suitable extension of the Privacy Amplification by Iteration (PABI) framework (Feldman et al., 2018; Altschuler and Talwar, 2022, 2023) to noisy iterations whose gradient map is not necessarily nonexpansive. This extension is achieved by designing an optimization problem which accounts for the best possible Rényi divergence bound obtained by an application of PABI, where the tractability of the problem is crucially related to the modulus of continuity of the associated gradient mapping. We show that, in several interesting cases -- namely the nonsmooth convex, weakly smooth and (strongly) dissipative -- such optimization problem can be solved exactly and explicitly, yielding the tightest possible PABI-based bounds.

Hédi Hadiji, Sarah Sachs, Cristóbal Guzmán

Tracking the solution of time-varying variational inequalities is an important problem with applications in game theory, optimization, and machine learning. Existing work considers time-varying games or time-varying optimization problems. For strongly convex optimization problems or strongly monotone games, these results provide tracking guarantees under the assumption that the variation of the time-varying problem is restrained, that is, problems with a sublinear solution path. In this work we extend existing results in two ways: In our first result, we provide tracking bounds for (1) variational inequalities with a sublinear solution path but not necessarily monotone functions, and (2) for periodic time-varying variational inequalities that do not necessarily have a sublinear solution path-length. Our second main contribution is an extensive study of the convergence behavior and trajectory of discrete dynamical systems of periodic time-varying VI. We show that these systems can exhibit provably chaotic behavior or can converge to the solution. Finally, we illustrate our theoretical results with experiments.

Sarah Sachs, Hédi Hadiji, Tim van Erven et al.

Stochastic and adversarial data are two widely studied settings in online learning. But many optimization tasks are neither i.i.d. nor fully adversarial, which makes it of fundamental interest to get a better theoretical understanding of the world between these extremes. In this work we establish novel regret bounds for online convex optimization in a setting that interpolates between stochastic i.i.d. and fully adversarial losses. By exploiting smoothness of the expected losses, these bounds replace a dependence on the maximum gradient length by the variance of the gradients, which was previously known only for linear losses. In addition, they weaken the i.i.d. assumption by allowing, for example, adversarially poisoned rounds, which were previously considered in the expert and bandit setting. Our results extend this to the online convex optimization framework. In the fully i.i.d. case, our bounds match the rates one would expect from results in stochastic acceleration, and in the fully adversarial case they gracefully deteriorate to match the minimax regret. We further provide lower bounds showing that our regret upper bounds are tight for all intermediate regimes in terms of the stochastic variance and the adversarial variation of the loss gradients.

Raef Bassily, Cristóbal Guzmán, Michael Menart

We study differentially private stochastic optimization in convex and non-convex settings. For the convex case, we focus on the family of non-smooth generalized linear losses (GLLs). Our algorithm for the $\ell_2$ setting achieves optimal excess population risk in near-linear time, while the best known differentially private algorithms for general convex losses run in super-linear time. Our algorithm for the $\ell_1$ setting has nearly-optimal excess population risk $\tilde{O}\big(\sqrt{\frac{\log{d}}{n\varepsilon}}\big)$, and circumvents the dimension dependent lower bound of \cite{Asi:2021} for general non-smooth convex losses. In the differentially private non-convex setting, we provide several new algorithms for approximating stationary points of the population risk. For the $\ell_1$-case with smooth losses and polyhedral constraint, we provide the first nearly dimension independent rate, $\tilde O\big(\frac{\log^{2/3}{d}}{(n\varepsilon)^{1/3}}\big)$ in linear time. For the constrained $\ell_2$-case with smooth losses, we obtain a linear-time algorithm with rate $\tilde O\big(\frac{1}{n^{1/3}}+\frac{d^{1/5}}{(n\varepsilon)^{2/5}}\big)$. Finally, for the $\ell_2$-case we provide the first method for {\em non-smooth weakly convex} stochastic optimization with rate $\tilde O\big(\frac{1}{n^{1/4}}+\frac{d^{1/6}}{(n\varepsilon)^{1/3}}\big)$ which matches the best existing non-private algorithm when $d= O(\sqrt{n})$. We also extend all our results above for the non-convex $\ell_2$ setting to the $\ell_p$ setting, where $1 < p \leq 2$, with only polylogarithmic (in the dimension) overhead in the rates.

Cristóbal Guzmán, Nishant A. Mehta, Ali Mortazavi

Much of the work in online learning focuses on the study of sublinear upper bounds on the regret. In this work, we initiate the study of best-case lower bounds in online convex optimization, wherein we bound the largest improvement an algorithm can obtain relative to the single best action in hindsight. This problem is motivated by the goal of better understanding the adaptivity of a learning algorithm. Another motivation comes from fairness: it is known that best-case lower bounds are instrumental in obtaining algorithms for decision-theoretic online learning (DTOL) that satisfy a notion of group fairness. Our contributions are a general method to provide best-case lower bounds in Follow The Regularized Leader (FTRL) algorithms with time-varying regularizers, which we use to show that best-case lower bounds are of the same order as existing upper regret bounds: this includes situations with a fixed learning rate, decreasing learning rates, timeless methods, and adaptive gradient methods. In stark contrast, we show that the linearized version of FTRL can attain negative linear regret. Finally, in DTOL with two experts and binary predictions, we fully characterize the best-case sequences, which provides a finer understanding of the best-case lower bounds.

Digvijay Boob, Cristóbal Guzmán

In this work, we conduct the first systematic study of stochastic variational inequality (SVI) and stochastic saddle point (SSP) problems under the constraint of differential privacy (DP). We propose two algorithms: Noisy Stochastic Extragradient (NSEG) and Noisy Inexact Stochastic Proximal Point (NISPP). We show that a stochastic approximation variant of these algorithms attains risk bounds vanishing as a function of the dataset size, with respect to the strong gap function; and a sampling with replacement variant achieves optimal risk bounds with respect to a weak gap function. We also show lower bounds of the same order on weak gap function. Hence, our algorithms are optimal. Key to our analysis is the investigation of algorithmic stability bounds, both of which are new even in the nonprivate case. The dependence of the running time of the sampling with replacement algorithms, with respect to the dataset size $n$, is $n^2$ for NSEG and $\tilde{O}(n^{3/2})$ for NISPP.

Siqi Zhang, Junchi Yang, Cristóbal Guzmán et al.

This paper studies the complexity for finding approximate stationary points of nonconvex-strongly-concave (NC-SC) smooth minimax problems, in both general and averaged smooth finite-sum settings. We establish nontrivial lower complexity bounds of $Ω(\sqrtκΔLε^{-2})$ and $Ω(n+\sqrt{nκ}ΔLε^{-2})$ for the two settings, respectively, where $κ$ is the condition number, $L$ is the smoothness constant, and $Δ$ is the initial gap. Our result reveals substantial gaps between these limits and best-known upper bounds in the literature. To close these gaps, we introduce a generic acceleration scheme that deploys existing gradient-based methods to solve a sequence of crafted strongly-convex-strongly-concave subproblems. In the general setting, the complexity of our proposed algorithm nearly matches the lower bound; in particular, it removes an additional poly-logarithmic dependence on accuracy present in previous works. In the averaged smooth finite-sum setting, our proposed algorithm improves over previous algorithms by providing a nearly-tight dependence on the condition number.

Raef Bassily, Cristóbal Guzmán, Anupama Nandi

Differentially private (DP) stochastic convex optimization (SCO) is a fundamental problem, where the goal is to approximately minimize the population risk with respect to a convex loss function, given a dataset of $n$ i.i.d. samples from a distribution, while satisfying differential privacy with respect to the dataset. Most of the existing works in the literature of private convex optimization focus on the Euclidean (i.e., $\ell_2$) setting, where the loss is assumed to be Lipschitz (and possibly smooth) w.r.t. the $\ell_2$ norm over a constraint set with bounded $\ell_2$ diameter. Algorithms based on noisy stochastic gradient descent (SGD) are known to attain the optimal excess risk in this setting. In this work, we conduct a systematic study of DP-SCO for $\ell_p$-setups under a standard smoothness assumption on the loss. For $1< p\leq 2$, under a standard smoothness assumption, we give a new, linear-time DP-SCO algorithm with optimal excess risk. Previously known constructions with optimal excess risk for $1< p <2$ run in super-linear time in $n$. For $p=1$, we give an algorithm with nearly optimal excess risk. Our result for the $\ell_1$-setup also extends to general polyhedral norms and feasible sets. Moreover, we show that the excess risk bounds resulting from our algorithms for $1\leq p \leq 2$ are attained with high probability. For $2 < p \leq \infty$, we show that existing linear-time constructions for the Euclidean setup attain a nearly optimal excess risk in the low-dimensional regime. As a consequence, we show that such constructions attain a nearly optimal excess risk for $p=\infty$. Our work draws upon concepts from the geometry of normed spaces, such as the notions of regularity, uniform convexity, and uniform smoothness.

Jelena Diakonikolas, Cristóbal Guzmán

Composite minimization is a powerful framework in large-scale convex optimization, based on decoupling of the objective function into terms with structurally different properties and allowing for more flexible algorithmic design. We introduce a new algorithmic framework for complementary composite minimization, where the objective function decouples into a (weakly) smooth and a uniformly convex term. This particular form of decoupling is pervasive in statistics and machine learning, due to its link to regularization. The main contributions of our work are summarized as follows. First, we introduce the problem of complementary composite minimization in general normed spaces; second, we provide a unified accelerated algorithmic framework to address broad classes of complementary composite minimization problems; and third, we prove that the algorithms resulting from our framework are near-optimal in most of the standard optimization settings. Additionally, we show that our algorithmic framework can be used to address the problem of making the gradients small in general normed spaces. As a concrete example, we obtain a nearly-optimal method for the standard $\ell_1$ setup (small gradients in the $\ell_{\infty}$ norm), essentially matching the bound of Nesterov (2012) that was previously known only for the Euclidean setup. Finally, we show that our composite methods are broadly applicable to a number of regression and other classes of optimization problems, where regularization plays a key role. Our methods lead to complexity bounds that are either new or match the best existing ones.

Raef Bassily, Vitaly Feldman, Cristóbal Guzmán et al.

Uniform stability is a notion of algorithmic stability that bounds the worst case change in the model output by the algorithm when a single data point in the dataset is replaced. An influential work of Hardt et al. (2016) provides strong upper bounds on the uniform stability of the stochastic gradient descent (SGD) algorithm on sufficiently smooth convex losses. These results led to important progress in understanding of the generalization properties of SGD and several applications to differentially private convex optimization for smooth losses. Our work is the first to address uniform stability of SGD on {\em nonsmooth} convex losses. Specifically, we provide sharp upper and lower bounds for several forms of SGD and full-batch GD on arbitrary Lipschitz nonsmooth convex losses. Our lower bounds show that, in the nonsmooth case, (S)GD can be inherently less stable than in the smooth case. On the other hand, our upper bounds show that (S)GD is sufficiently stable for deriving new and useful bounds on generalization error. Most notably, we obtain the first dimension-independent generalization bounds for multi-pass SGD in the nonsmooth case. In addition, our bounds allow us to derive a new algorithm for differentially private nonsmooth stochastic convex optimization with optimal excess population risk. Our algorithm is simpler and more efficient than the best known algorithm for the nonsmooth case Feldman et al. (2020).

Jelena Diakonikolas, Cristóbal Guzmán

We study the question of whether parallelization in the exploration of the feasible set can be used to speed up convex optimization, in the local oracle model of computation. We show that the answer is negative for both deterministic and randomized algorithms applied to essentially any of the interesting geometries and nonsmooth, weakly-smooth, or smooth objective functions. In particular, we show that it is not possible to obtain a polylogarithmic (in the sequential complexity of the problem) number of parallel rounds with a polynomial (in the dimension) number of queries per round. In the majority of these settings and when the dimension of the space is polynomial in the inverse target accuracy, our lower bounds match the oracle complexity of sequential convex optimization, up to at most a logarithmic factor in the dimension, which makes them (nearly) tight. Prior to our work, lower bounds for parallel convex optimization algorithms were only known in a small fraction of the settings considered in this paper, mainly applying to Euclidean ($\ell_2$) and $\ell_\infty$ spaces. Our work provides a more general approach for proving lower bounds in the setting of parallel convex optimization.