We study the problem of generating synthetic data under differential privacy. We establish fixed-parameter tractability (FPT) for this problem where the parameter is the treewidth of the query family’s incidence graph. Our algorithms attain optimal error rates across all regimes and are realized by two different approaches: the first is based on linear programming (LP) and the FPT of the separation problem for the LP dual; the second is based on a subsampled private multiplicative weights method, where we obtain FPT for sampling from Gibbs distributions. Both approaches are unified by a dynamic programming framework over a tree decomposition.

High quality data is of vital importance for unlocking the full potential of AI for end users. Villalobos et al. stated in 2024 that finding new sources of such data is getting harder as 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 that real world commands that end users will issue. Therefore unlocking high-quality data grounded in real user interactions is of vital interest to both system creators and end users themselves. However, the direct use of user data comes with significant privacy risks, which must be addressed before the data can be used. 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, Differentially Private Synthetic data, refers to synthetic data that preserves the overall trends of source data (often user-generated), 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. Additionally, DP synthetic data can replace the use of sensitive datasets that previously have only had rudimentary protections like ad-hoc rule-based anonymization. In this survey we explore the full suite of techniques surrounding DP synthetic data, the types of privacy protections different generation approaches can offer, and the state-of-the-art for various modalities including image, tabular, text and federated (decentralized) data. We outline all the components needed in a system that generates DP synthetic data, from sensitive data handling and preparation, to tracking the use of synthetic data and empirical privacy testing. We hope that work will result in increased adoption of DP synthetic data, spur additional research in still underexplored domains, and additionally increase trust in DP synthetic data approaches.

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 \textit{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.

We introduce $\mathsf{PREM}$ (Private Relative Error Multiplicative weight update), a new framework for generating synthetic data that achieves a {\em relative} error guarantee for statistical queries under $(\varepsilon, \delta)$-differential privacy (DP). Namely, for a domain ${\cal X}$, a family ${\cal F}$ of queries $f : {\cal X} \to \{0, 1\}$, and $\zeta > 0$, 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 $1 \pm \zeta$ {\em multiplicative} factor of the corresponding value on $\widehat{D}$ up to an {\em additive} error that is polynomial in $\log |{\cal F}|$, $\log |{\cal X}|$, $\log n$, $\log(1/\delta)$, $1/\varepsilon$, and $1/\zeta$. In contrast, any $(\varepsilon, \delta)$-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.

The minimax sample complexity of group distributionally robust optimization (GDRO) has been determined up to a $\log(K)$ factor, where $K$ is the number of groups. In this work, we venture beyond the minimax perspective via a novel notion of sparsity that we call $(\lambda, \beta)$-sparsity. In short, this condition means that at any parameter $\theta$, there is a set of at most $\beta$ groups whose risks at $\theta$ are all at least $\lambda$ larger than the risks of the other groups. To find an $\epsilon$-optimal $\theta$, we show via a novel algorithm and analysis that the $\epsilon$-dependent term in the sample complexity can swap a linear dependence on $K$ for a linear dependence on the potentially much smaller $\beta$. This improvement leverages recent progress in sleeping bandits, showing a fundamental connection between the two-player zero-sum game optimization framework for GDRO and per-action regret bounds in sleeping bandits. We next show an adaptive algorithm which, up to logarithmic factors, obtains a sample complexity bound that adapts to the best $(\lambda, \beta)$-sparsity condition that holds. We also show how to obtain a dimension-free semi-adaptive sample complexity bound with a computationally efficient method. Finally, we demonstrate the practicality of the $(\lambda, \beta)$-sparsity condition and the improved sample efficiency of our algorithms on both synthetic and real-life datasets.

We study the mixing time of the projected Langevin algorithm and the privacy curve of noisy stochastic gradient descent, beyond nonexpansive iterations. Specifically, we derive new mixing time bounds for the projected Langevin algorithm which are, in some important cases, dimension-free and poly-logarithmic in 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 stochastic gradient descent 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 framework 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 privacy amplification by iteration, 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 cases, such an optimization problem can be solved exactly and explicitly, yielding the tightest possible privacy amplification by iteration based bounds.

We study private empirical risk minimization (ERM) problem for losses satisfying the $(\gamma,\kappa)$-Kurdyka-{Ł}ojasiewicz (KL) condition, that is, the empirical loss $F$ satisfies $F(w)-\min_{w}F(w) \leq \gamma^\kappa \|\nabla F(w)\|^\kappa$. The Polyak-{Ł}ojasiewicz (PL) condition is a special case of this condition when $\kappa=2$. Specifically, we study this problem under the constraint of $\rho$ zero-concentrated differential privacy (zCDP). When $\kappa\in[1,2]$ 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 $\tilde{O}\big(\big(\frac{\sqrt{d}}{n\sqrt{\rho}}\big)^\kappa\big)$ 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 $\kappa \geq 2$ and the loss is instead Lipschitz and weakly convex, we show it is possible to achieve the rate $\tilde{O}\big(\big(\frac{\sqrt{d}}{n\sqrt{\rho}}\big)^\kappa\big)$ 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 $\tilde{O}\big(\big(\frac{\sqrt{d}}{n\sqrt{\rho}}\big)^{\frac{2\kappa}{4-\kappa}}\big)$ adaptively, which is nearly optimal when $\kappa = 2$. 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 $\tilde{O}\big(\frac{\sqrt{d}}{n\sqrt{\rho}}\big)$ and never worse than $\tilde{O}\big(\big(\frac{\sqrt{d}}{n\sqrt{\rho}}\big)^{1/2}\big)$. The latter rate matches the best known rate for methods that do not rely on variance reduction.

We study the problem of differentially-private (DP) stochastic (convex-concave) saddle-points in the polyhedral setting. We propose $(\varepsilon, \delta)$-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 improve the rate on the expected gap to $\sqrt{\log(d)/n} + (\log(d)^{3/2}/[n\varepsilon])^{2/5}$. Under this additional assumption, we also show, by using bias-reduced gradient estimators, that the duality gap is bounded by $\log(d)/\sqrt{n} + \log(d)/[n\varepsilon]^{1/2}$ with constant success probability. This result provides evidence of the near-optimality of the approach. 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 polyhedral 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, which may be of independent interest.

In this work, we conduct a systematic study of stochastic saddle point problems (SSP) and stochastic variational inequalities (SVI) under the constraint of $(\epsilon,\delta)$-differential privacy (DP) in both Euclidean and non-Euclidean setups. We first consider Lipschitz convex-concave SSPs in the $\ell_p/\ell_q$ setup, $p,q\in[1,2]$. That is, we consider the case where the primal problem has an $\ell_p$-setup (i.e., the primal parameter is constrained to an $\ell_p$ bounded domain and the loss is $\ell_p$-Lipschitz with respect to the primal parameter) and the dual problem has an $\ell_q$ setup. Here, we obtain a bound of $\tilde{O}\big(\frac{1}{\sqrt{n}} + \frac{\sqrt{d}}{n\epsilon}\big)$ 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 $\tilde{O}\big(\frac{1}{\sqrt{n}} + \frac{\sqrt{d}}{n\epsilon}\big)$. For $p-1=\Omega(1)$, 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.

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. The corresponding lower bounds are based on a novel block-diagonal construction that is combined with existing DP mean estimation lower bounds. Next, 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. Furthermore, by introducing novel analyses of bias reduction in mean estimation and randomly-stopped biased SGD we obtain nearly dimension-independent rates for near-stationary points for the empirical risk in nonconvex settings under approximate-DP.

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 $(\epsilon,\delta)$-PA-DP has excess risk $\tilde{\Omega}\big(\min(\frac{1}{\sqrt{n_{\text{pub}}}},\frac{1}{\sqrt{n}}+\frac{\sqrt{d}}{n\epsilon} ) \big)$, 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 $\tilde{O}({n_{\text{priv}}}\epsilon)$ unlabeled public samples, achieves the dimension independent rate $\tilde{O}\big(\frac{1}{\sqrt{{n_{\text{priv}}}}} + \frac{1}{\sqrt{{n_{\text{priv}}}\epsilon}}\big)$. 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 \textit{fat-shattering dimension} with applications to neural networks and non-Euclidean geometries.

This note closes a gap in the proof of Theorem VI.3 from the article “Lower Bounds on the Oracle Complexity of Nonsmooth Convex Optimization via Information Theory” (2017).

We show that convex-concave Lipschitz stochastic saddle point problems (also known as stochastic minimax optimization) can be solved under the constraint of $(\epsilon,\delta)$-differential privacy with \emph{strong (primal-dual) gap} rate of $\tilde O\big(\frac{1}{\sqrt{n}} + \frac{\sqrt{d}}{n\epsilon}\big)$, 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 $O\big(\min\big\{\frac{n^2\epsilon^{1.5}}{\sqrt{d}}, n^{3/2}\big\}\big)$ 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 $\alpha$ primal-dual accuracy guarantee with respect to the empirical objective, gives a solution to the stochastic saddle point problem with a strong gap of $\tilde{O}(\alpha+\frac{1}{\sqrt{n}})$. We show that this $\alpha$-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. Finally, to emphasize the importance of the strong gap as a convergence criterion compared to the weaker notion of primal-dual gap, commonly known as the \emph{weak gap}, we show that even for simple problems it is possible for an algorithm to have zero weak gap and suffer from $\Omega(1)$ strong gap. We also show that there exists a fundamental tradeoff between stability and accuracy. Specifically, we show that any $\Delta$-stable algorithm has empirical gap $\Omega\big(\frac{1}{\Delta n}\big)$, and that this bound is tight. This result also holds also more specifically for empirical risk minimization problems and may be of independent interest.

We study the problem of approximating stationary points of Lipschitz and smooth functions under $(\varepsilon,\delta)$-differential privacy (DP) in both the finite-sum and stochastic settings. A point $\widehat{w}$ is called an $\alpha$-stationary point of a function $F:\mathbb{R}^d\rightarrow\mathbb{R}$ if $\|\nabla F(\widehat{w})\|\leq \alpha$. We 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 given $n$ samples. 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 $n$. We also provide an 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. This improves on the previous best rate of $\tilde{O}\big(\big[\frac{\sqrt{d}}{n\varepsilon}\big]^{1/2}\big)$. 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 $\tilde \Theta\big(\frac{1}{\sqrt{n}}+\frac{\sqrt{d}}{n\varepsilon}\big)$. 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.

We study the problem of $(\epsilon,\delta)$-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 $\tilde{O}\left(\frac{\Vert w^*\Vert}{\sqrt{n}} + \min\left\{\frac{\Vert w^* \Vert^2}{(n\epsilon)^{2/3}},\frac{\sqrt{d}\Vert w^*\Vert^2}{n\epsilon}\right\}\right)$, 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 $\tilde{\Omega}\left(\frac{1}{\sqrt{n}} + {\min\left\{\frac{\Vert w^*\Vert^{4/3}}{(n\epsilon)^{2/3}}, \frac{\sqrt{d}\Vert w^*\Vert}{n\epsilon}\right\}}\right)$. We also revisit the previously studied case of Lipschitz losses \cite{SSTT21}. For this case, we close the gap in the existing work and show that the optimal rate is (up to log factors) $\Theta\left(\frac{\Vert w^*\Vert}{\sqrt{n}} + \min\left\{\frac{\Vert w^*\Vert}{\sqrt{n\epsilon}},\frac{\sqrt{\text{rank}}\Vert w^*\Vert}{n\epsilon}\right\}\right)$, 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 $\Vert w^*\Vert$.

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 $\epsilon$ norm of the operator with $\mathcal{O}(\frac{1}{\epsilon^3})$ stochastic operator evaluations, which significantly improves over state of the art $\mathcal{O}(\frac{1}{\epsilon^4})$ 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 ${\mathcal{O}}(\frac{\log(1/\epsilon)}{\epsilon^2})$ stochastic operator evaluations under additional sharpness or strong monotonicity assumptions.

Stochastic and adversarial data are two widely studied settings in online learning. But many optimizationtasks 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 aretight for all intermediate regimes in terms of the stochastic variance and theadversarial variation of the loss gradients.

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 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. For $p=1$, under a standard smoothness assumption, we give a new algorithm with nearly optimal excess risk. This result also extends to general polyhedral norms and feasible sets. For $p\in(1, 2)$, we give two new algorithms, whose central building block is a novel privacy mechanism, which generalizes the Gaussian mechanism. Moreover, we establish a lower bound on the excess risk for this range of $p$, showing a necessary dependence on $\sqrt{d}$, where $d$ is the dimension of the space. Our lower bound implies a sudden transition of the excess risk at $p=1$, where the dependence on $d$ changes from logarithmic to polynomial, resolving an open question in prior work \citep{TTZ15a}. For $p\in (2, \infty)$, noisy SGD attains optimal excess risk in the low-dimensional regime; in particular, this proves the optimality of noisy SGD 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.

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 $\tilde{O}\big(\sqrt{\frac{\log{d}}{n}}\big)$, 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, $\tilde O\big(\frac{\log^{2/3}{d}}{{n^{1/3}}}\big)$ in linear time. For the constrained $\ell_2$-case, with smooth losses, we obtain a linear-time algorithm with rate $\tilde O\big(\frac{1}{n^{3/10}d^{1/10}}+\big(\frac{d}{n^2}\big)^{1/5}\big)$. Finally, for the $\ell_2$-case we provide the first method for {\em non-smooth weakly convex} stochastic optimization with rate $\tilde O\big(\frac{1}{n^{1/4}}+\big(\frac{d}{n^2}\big)^{1/6}\big)$ 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.

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 \emph{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 losses, we fully characterize the best-case sequences, which provides a finer understanding of the best-case lower bounds.

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 for the two settings, respectively. 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.

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 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 bound 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, due to Feldman et al. [2020].

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 and in the high-dimensional regime. 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. Another conceptual contribution of our work is in providing a general and streamlined framework for proving lower bounds in the setting of parallel convex optimization. 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.

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 and streamlined approach for proving lower bounds in the setting of parallel convex optimization.

Network pricing games provide a framework for modeling real-world settings with two types of strategic agents: owners (operators) of the network and users of the network. Owners of the network post a price for usage of the link they own so as to attract users and maximize profit; users of the network select routes based on price and level of use by other users. We point out that an equilibrium in these games may not exist, may not be unique and may induce an arbitrarily inefficient network performance. Our main result is to observe that a simple regulation on the network owners market solves all three issues above. Specifically, if an authority could set appropriate caps (upper bounds) on the tolls (prices) operators can charge, then: the game among the link operators has a unique and strong Nash equilibrium and the users' game results in a Wardrop equilibrium that achieves the optimal total delay. We call any price vector with these properties a great set of tolls. As a secondary objective, we want to compute great tolls that minimize total users' payments and we provide a linear program that does this. We obtain multiplicative approximation results compared to the optimal total users' payments for arbitrary networks with polynomial latencies of bounded degree, while in the single-commodity case we obtain a bound that only depends on the topology of the network. Lastly, we show how the same mechanism of setting appropriate caps on the allowable prices extends to the model of elastic demands.

We provide a deterministic construction of the sparse Johnson-Lindenstrauss transform of Kane & Nelson (J.ACM 2014) which runs, under a mild restriction, in the time necessary to apply the sparse embedding matrix to the input vectors.Specifically, given a set of n vectors in R d and target error ε, we give a deterministic algorithm to compute a {-1, 0, 1} embedding matrix of rank O((ln n)/ε 2 ) with O((ln n)/ε) entries per column which preserves the norms of the vectors to within 1±ε.If NNZ, the number of non-zero entries in the input set of vectors, is Ω(d 2 ), our algorithm runs in time O(NNZ • ln n/ε).One ingredient in our construction is an extremely simple proof of the Hanson-Wright inequality for subgaussian random variables, which is more amenable to derandomization.As an interesting byproduct, we are able to derive the essentially optimal form of the inequality in terms of its functional dependence on the parameters.

Stochastic convex optimization, where the objective is the expectation of a random convex function, is an important and widely used method with numerous applications in machine learning, statistics, operations research and other areas.We study the complexity of stochastic convex optimization given only statistical query (SQ) access to the objective function.We show that well-known and popular first-order iterative methods can be implemented using only statistical queries.For many cases of interest we derive nearly matching upper and lower bounds on the estimation (sample) complexity including linear optimization in the most general setting.We then present several consequences for machine learning, differential privacy and proving concrete lower bounds on the power of convex optimization based methods.The key ingredient of our work is SQ algorithms and lower bounds for estimating the mean vector of a distribution over vectors supported on a convex body in R d .This natural problem has not been previously studied and we show that our solutions can be used to get substantially improved SQ versions of Perceptron and other online algorithms for learning halfspaces.

We present an information-theoretic approach to lower bound the oracle complexity of nonsmooth black box convex optimization, unifying previous lower bounding techniques by identifying a combinatorial problem, namely string guessing, as a single source of hardness. As a measure of complexity, we use distributional oracle complexity, which subsumes randomized oracle complexity as well as worst case oracle complexity. We obtain strong lower bounds on distributional oracle complexity for the box [-1, 1]n, as well as for the L p-ball for p ≥ 1 (for both low-scale and large-scale regimes), matching worst case upper bounds, and hence we close the gap between distributional complexity, and in particular, randomized complexity and worst case complexity. Furthermore, the bounds remain essentially the same for high-probability and bounded-error oracle complexity, and even for combination of the two, i.e., bounded-error highprobability oracle complexity. This considerably extends the applicability of known bounds.

First-order convex minimization algorithms are currently the methods of choice for large-scale sparse – and more generally parsimonious – regression models. We pose the question on the limits of performance of black-box oriented methods for convex minimization in \em non-standard settings, where the regularity of the objective is measured in a norm not necessarily induced by the feasible domain. This question is studied for \ell_p/\ell_q-settings, and their matrix analogues (Schatten norms), where we find surprising gaps on lower bounds compared to state of the art methods. We propose a conjecture on the optimal convergence rates for these settings, for which a positive answer would lead to significant improvements on minimization algorithms for parsimonious regression models.