User-level differentially private (DP) stochastic convex optimization has garnered significant attention due to the paramount importance of safeguarding user privacy in modern large-scale ML applications. Current methods, such as those based on DP stochastic gradient descent (SGD), often struggle with high gradient computation complexity or suboptimal utility due to the need to privatize every intermediate iterate. In this work, we introduce a new nearly linear-time algorithm that resolves this trade-off and achieves the optimal excess rates via an adaptive outlier removal framework. Our key innovation is integrating the sparse vector technique directly into the SGD loop, supported by a novel robust divergence analysis. This approach naturally bounds the sensitivity of gradient estimates without requiring privatization of all intermediate steps. Specifically, our mechanism computes a local concentration score to adaptively filter out users whose updates diverge from the population geometry. Crucially, this approach preserves the unbiasedness of the gradient estimate in well-concentrated regimes while strictly bounding sensitivity in the presence of outliers. We also explore extensions to the $\ell_\infty$ setting demonstrating the generality of our analysis.

Machine unlearning algorithms, designed for selective removal of training data from models, have emerged as a promising approach to growing privacy concerns. In this work, we expose a critical yet underexplored vulnerability in the deployment of unlearning systems: the assumption that the data requested for removal is always part of the original training set. We present a threat model where an attacker can degrade model accuracy by submitting adversarial unlearning requests for data \textit{not} present in the training set. We propose white-box and black-box attack algorithms and evaluate them through a case study on image classification tasks using the CIFAR-10 and ImageNet datasets, targeting a family of widely used unlearning methods. Our results show extremely poor test accuracy following the attack—3.6% on CIFAR-10 and 0.4% on ImageNet for white-box attacks, and 8.5% on CIFAR-10 and 1.3% on ImageNet for black-box attacks. Additionally, we evaluate various verification mechanisms to detect the legitimacy of unlearning requests and reveal the challenges in verification, as most of the mechanisms fail to detect stealthy attacks without severely impairing their ability to process valid requests. These findings underscore the urgent need for research on more robust request verification methods and unlearning protocols, should the deployment of machine unlearning systems become more prevalent in the future.

There is a gap between finding a first-order stationary point (FOSP) and a second-order stationary point (SOSP) under differential privacy constraints, and it remains unclear whether privately finding an SOSP is more challenging than finding an FOSP. Specifically, Ganesh et al. (2023) claimed that an $\alpha$-SOSP can be found with $\alpha=\Tilde{O}(\frac{1}{n^{1/3}}+(\frac{\sqrt{d}}{n\epsilon})^{3/7})$, where $n$ is the dataset size, $d$ is the dimension, and $\epsilon$ is the differential privacy parameter. However, a recent analysis revealed an issue in their saddle point escape procedure, leading to weaker guarantees. Building on the SpiderBoost algorithm framework, we propose a new approach that uses adaptive batch sizes and incorporates the binary tree mechanism. Our method not only corrects this issue but also improves the results for privately finding an SOSP, achieving $\alpha=\Tilde{O}(\frac{1}{n^{1/3}}+(\frac{\sqrt{d}}{n\epsilon})^{1/2})$. This improved bound matches the state-of-the-art for finding a FOSP, suggesting that privately finding an SOSP may be achievable at no additional cost.

Language models (LMs) are trained on vast amounts of text data, which may include private and copyrighted content. Data owners may request the removal of their data from a trained model due to privacy or copyright concerns. However, exactly unlearning only these datapoints (i.e., retraining with the data removed) is intractable in modern-day models. This has led to the development of many approximate unlearning algorithms. The evaluation of the efficacy of these algorithms has traditionally been narrow in scope, failing to precisely quantify the success and practicality of the algorithm from the perspectives of both the model deployers and the data owners. We address this issue by proposing MUSE, a comprehensive machine unlearning evaluation benchmark that enumerates six diverse desirable properties for unlearned models: (1) no verbatim memorization, (2) no knowledge memorization, (3) no privacy leakage, (4) utility preservation on data not intended for removal, (5) scalability with respect to the size of removal requests, and (6) sustainability over sequential unlearning requests. Using these criteria, we benchmark how effectively eight popular unlearning algorithms on 7B-parameter LMs can unlearn Harry Potter books and news articles. Our results demonstrate that most algorithms can prevent verbatim memorization and knowledge memorization to varying degrees, but only one algorithm does not lead to severe privacy leakage. Furthermore, existing algorithms fail to meet deployer's expectations because they often degrade general model utility and also cannot sustainably accommodate successive unlearning requests or large-scale content removal. Our findings identify key issues with the practicality of existing unlearning algorithms on language models.

We study differentially private (DP) optimization algorithms for stochastic and empirical objectives which are neither smooth nor convex, and propose methods that return a Goldstein-stationary point with sample complexity bounds that improve on existing works. We start by providing a single-pass $(\epsilon,\delta)$-DP algorithm that returns an $(\alpha,\beta)$-stationary point as long as the dataset is of size $\widetilde{\Omega}(\sqrt{d}/\alpha\beta^{3}+d/\epsilon\alpha\beta^{2})$, which is $\Omega(\sqrt{d})$ times smaller than the algorithm of Zhang et al. (2024) for this task, where $d$ is the dimension. We then provide a multi-pass polynomial time algorithm which further improves the sample complexity to $\widetilde{\Omega}\left(d/\beta^2+d^{3/4}/\epsilon\alpha^{1/2}\beta^{3/2}\right)$, by designing a sample efficient ERM algorithm, and proving that Goldstein-stationary points generalize from the empirical loss to the population loss.

Estimating the geometric median of a dataset is a robust counterpart to mean estimation, and is a fundamental problem in computational geometry. Recently, [HSU24] gave an $(\epsilon, \delta)$-differentially private algorithm obtaining an $\alpha$-multiplicative approximation to the geometric median objective, $\frac 1 n \sum_{i \in [n]} \|\cdot - \mathbf{x}_i\|$, given a dataset $D$ of $x_i$ for $i \in [n]$. Their algorithm requires $n \gtrsim \sqrt d \cdot \frac 1 {\alpha\epsilon}$ samples, which they prove is information-theoretically optimal. This result is surprising because its error scales with the effective radius of $D$ (i.e., of a ball capturing most points), rather than the worst-case radius. We give an improved algorithm that obtains the same approximation quality, also using $n \gtrsim \sqrt d \cdot \frac 1 {\alpha\epsilon}$ samples, but in time $\widetilde{O}(nd + \frac d {\alpha^2})$. Our runtime is nearly-linear, plus the cost of the cheapest non-private first-order method due to [CLMPS16]. To achieve our results, we use subsampling and geometric aggregation tools inspired by FriendlyCore [TCKMS22] to speed up the "warm start" component of the [HSU24] algorithm, combined with a careful custom analysis of DP-SGD's sensitivity for the geometric median objective.

Bilevel optimization, in which one optimization problem is nested inside another, underlies many machine learning applications with a hierarchical structure---such as meta-learning and hyperparameter optimization. Such applications often involve sensitive training data, raising pressing concerns about individual privacy. Motivated by this, we study differentially private bilevel optimization. We first focus on settings where the outer-level objective is convex, and provide novel upper and lower bounds on the excess empirical risk for both pure and approximate differential privacy. These bounds are nearly tight and essentially match the optimal rates for standard single-level differentially private ERM, up to additional terms that capture the intrinsic complexity of the nested bilevel structure. We also provide population loss bounds for bilevel stochastic optimization. The bounds are achieved in polynomial time via efficient implementations of the exponential and regularized exponential mechanisms. A key technical contribution is a new method and analysis of log-concave sampling under inexact function evaluations, which may be of independent interest. In the non-convex setting, we develop novel algorithms with state-of-the-art rates for privately finding approximate stationary points. Notably, our bounds do not depend on the dimension of the inner problem.

We introduce FlexOLMo, a new class of language models (LMs) that supports (1) distributed training without data sharing, where different model parameters are independently trained on private datasets, and (2) data-flexible inference, where these parameters along with their associated data can be easily included or excluded from model inferences with no further training. FlexOLMo employs a mixture-of-experts (MoE) architecture where each expert is trained independently on private datasets and later integrated through a new nonparametric routing without any joint training across datasets. FlexOLMo is trained on FLEXMIX, a corpus we curate comprising seven restricted sets, either real or realistic approximations, alongside publicly available datasets. We evaluate models with up to 37 billion parameters (20 billion active) on 31 diverse downstream tasks. We show that a general expert trained on public data can be effectively combined with independently trained experts from other data owners significantly benefiting from these restricted sets (an average 41% relative improvement) while allowing flexible opt-out at inference time (e.g., for users without appropriate licenses or permissions). Our approach also outperforms prior model merging methods by 10.1% on average and surpasses the standard MoE trained without data restrictions using the same training FLOPs. Altogether, FlexOLMo enables training on restricted data while keeping data local and supports fine-grained control of data access at inference.

We propose SCONE (**S**calable, **C**ontextualized, **O**ffloaded, **N**-gram **E**mbedding), a new method for extending input embedding layers to enhance language model performance. To avoid increased decoding costs, SCONE retains the original vocabulary while introducing embeddings for a set of frequent $n$-grams. These embeddings provide contextualized representation for each input token and are learned with a separate model during training. After training, embeddings are precomputed and stored in off-accelerator memory; during inference, querying them has minimal impact on latency due to the low complexity of embedding lookups. SCONE enables two new scaling strategies: increasing the number of $n$-gram embeddings and scaling the model used to learn them, both while maintaining fixed accelerator usage during inference (in terms of FLOPS and memory). We show that scaling both aspects enables a model with 1B accelerator-resident parameters to outperform a 1.9B-parameter baseline across diverse corpora, while using only about half the FLOPS and accelerator memory during inference.

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.

Although large language models (LLMs) are widely deployed, the data used to train them is rarely disclosed. Given the incredible scale of this data, up to trillions of tokens, it is all but certain that it includes potentially problematic text such as copyrighted materials, personally identifiable information, and test data for widely reported reference benchmarks. However, we currently have no way to know which data of these types is included or in what proportions. In this paper, we study the pretraining data detection problem: given a piece of text and black-box access to an LLM without knowing the pretraining data, can we determine if the model was trained on the provided text? To facilitate this study, we introduce a dynamic benchmark WIKIMIA that uses data created before and after model training to support gold truth detection. We also introduce a new detection method MIN-K PROB based on a simple hypothesis: an unseen example is likely to contain a few outlier words with low probabilities under the LLM, while a seen example is less likely to have words with such low probabilities. MIN-K PROB can be applied without any knowledge about the pretrainig corpus or any additional training, departing from previous detection methods that require training a reference model on data that is similar to the pretraining data. Moreover, our experiments demonstrate that MIN-K PROB achieves a 7.4% improvement on WIKIMIA over these previous methods. We apply MIN-K PROB to two real-world scenarios, copyrighted book detection and contaminated downstream example detection, and find that it to be a consistently effective solution.

We provide an improved analysis of standard differentially private gradient descent for linear regression under the squared error loss. Under modest assumptions on the input, we characterize the distribution of the iterate at each time step. Our analysis leads to new results on the algorithm’s accuracy: for a proper fixed choice of hyperparameters, the sample complexity depends only linearly on the dimension of the data. This matches the dimension-dependence of the (non-private) ordinary least squares estimator as well as that of recent private algorithms that rely on sophisticated adaptive gradient-clipping schemes (Varshney et al., 2022; Liu et al., 2023). Our analysis of the iterates’ distribution also allows us to construct confidence intervals for the empirical optimizer which adapt automatically to the variance of the algorithm on a particular data set. We validate our theorems through experiments on synthetic data.

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.

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 $G_2 \cdot \frac 1 {\sqrt n} + G_k \cdot (\frac{\sqrt d}{n\epsilon})^{1 - \frac 1 k}$ under $(\epsilon, \delta)$-approximate differential privacy, up to a mild $\textup{polylog}(\frac{1}{\delta})$ 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 [LR23]. We then give a suite of private algorithms for DP-SCO with heavy-tailed gradients improving 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.

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.

The development of efficient sampling algorithms catering to non-Euclidean geometries has been a challenging endeavor, as discretization techniques which succeed in the Euclidean setting do not readily carry over to more general settings. We develop a non-Euclidean analog of the recent proximal sampler of [LST21], which naturally induces regularization by an object known as the log-Laplace transform (LLT) of a density. We prove new mathematical properties (with an algorithmic flavor) of the LLT, such as strong convexity-smoothness duality and an isoperimetric inequality, which are used to prove a mixing time on our proximal sampler matching [LST21] under a warm start. As our main application, we show our warm-started sampler improves the value oracle complexity of differentially private convex optimization in $\ell_p$ and Schatten-$p$ norms for $p \in [1, 2]$ to match the Euclidean setting [GLL22], while retaining state-of-the-art excess risk bounds [GLLST23]. We find our investigation of the LLT to be a promising proof-of-concept of its utility as a tool for designing samplers, and outline directions for future exploration.

We introduce a new tool for stochastic convex optimization (SCO): a Reweighted Stochastic Query (ReSQue) estimator for the gradient of a function convolved with a (Gaussian) probability density. Combining ReSQue with recent advances in ball oracle acceleration [CJJ+20], [ACJ+21], we develop algorithms achieving state-of-the-art complexities for SCO in parallel and private settings. For a SCO objective constrained to the unit ball in $\mathbb{R}^{d}$, we obtain the following results (up to polylogarithmic factors).1)We give a parallel algorithm obtaining optimization error $\epsilon_{\text {opt} }$ with $d^{1 / 3} \epsilon_{\text {opt} }^{-2 / 3}$ gradient oracle query depth and $d^{1 / 3} \epsilon_{\text {opt} }^{-2 / 3}+\epsilon_{\text {opt} }^{-2}$ gradient queries in total, assuming access to a bounded-variance stochastic gradient estimator. For $\epsilon_{\text {opt} } \in\left[d^{-1}, d^{-1 / 4}\right]$, our algorithm matches the state-of-the-art oracle depth of [BJL+19] while maintaining the optimal total work of stochastic gradient descent.2)Given n samples of Lipschitz loss functions, prior works [BFTT19], [BFGT20], [AFKT21], [KLL21] established that if $n \gt rsim d \epsilon_{\mathrm{dp}}^{-2},\left(\epsilon_{\mathrm{dp}}, \delta\right)$-differential privacy is attained at no asymptotic cost to the SCO utility. However, these prior works all required a superlinear number of gradient queries. We close this gap for sufficiently large $n \gt rsim d^{2} \epsilon_{{\bf d p}}^{-3}$, by using ReSQue to design an algorithm with near-linear gradient query complexity in this regime.

We reconsider the challenge of non-convex optimization under differential privacy constraint. Building upon the previous variance-reduced algorithm SpiderBoost, we propose a novel framework that employs two types of gradient oracles: one that estimates the gradient at a single point and a more cost-effective option that calculates the gradient difference between two points. Our framework can ensure continuous accuracy of gradient estimations and subsequently enhances the rates of identifying second-order stationary points. Additionally, we consider a more challenging task by attempting to locate the global minima of a non-convex objective via the exponential mechanism without almost any assumptions. Our preliminary results suggest that the regularized exponential mechanism can effectively emulate previous empirical and population risk bounds, negating the need for smoothness assumptions for algorithms with polynomial running time. Furthermore, with running time factors excluded, the exponential mechanism demonstrates promising population risk bound performance, and we provide a nearly matching lower bound.

We propose a new framework for differentially private optimization of convex functions which are Lipschitz in an arbitrary norm ||·||x. Our algorithms are based on a regularized exponential mechanism which samples from the density ∞ exp(-k(F + μr)) where F is the empirical loss and τ is a regularizer which is strongly convex with respect to ||·||x, generalizing a recent work of [GLL22] to non-Euclidean settings. We show that this mechanism satisfies Gaussian differential privacy and solves both DP-ERM (empirical risk minimization) and DP-SCO (stochastic convex optimization), by using localization tools from convex geometry. Our framework is the first to apply to private convex optimization in general normed spaces, and directly recovers non-private SCO rates achieved by mirror descent, as the privacy parameter ε → ∞. As applications, for Lipschitz optimization in ℓp norms for all p ∈ (1, 2), we obtain the first optimal privacy-utility tradeoffs; for p = 1, we improve tradeoffs obtained by the recent works [AFKT21, BGN21] by at least a logarithmic factor. Our ℓp norm and Schatten-p norm optimization frameworks are complemented with polynomial-time samplers whose query complexity we explicitly bound.

The ability to resolve detail in the object that is being imaged, named by resolution, is the core parameter of an imaging system. Super-resolution is a class of techniques that can enhance the resolution of an imaging system and even transcend the diffraction limit of systems. Despite huge success in the application, super-resolution is not well understood on the theoretical side, especially for any dimension d ≥ 2. In particular, in order to recover a k-sparse signal, all previous results suffer from either/both poly(k) samples or running time. We design robust algorithms for any (constant) dimension under a strong noise model based on developing some new techniques in Sparse Fourier transform (Sparse FT), such as inverting a robust linear system, “eggshell” sampling schemes, and partition and voting methods in high dimension. These algorithms are the first to achieve running time and sample complexity (nearly) linear in the number of source points and logarithmic in bandwidth for any constant dimension, and we believe the techniques developed in the work can find their further applications on the Super-resolution and Sparse FT problem. * The full version of the paper can be accessed at https://arxiv.org/abs/2005.06156

Weitzman (1979) introduced the Pandora Box problem as a model for sequential search with inspection costs, and gave an elegant index-based policy that attains provably optimal expected payoff. In various scenarios, the searching agent may select an option without making a costly inspection. The variant of the Pandora box problem with non-obligatory inspection has attracted interest from both economics and algorithms researchers. Various simple algorithms have proved suboptimal, with the best known 0.8-approximation algorithm due to Guha et al. (2008). No hardness result for the problem was known.

In this paper, we study the private optimization problems for non-smooth convex functions $F(x)=\mathbb{E}_i f_i(x)$ on $\mathbb{R}^d$. We show that modifying the exponential mechanism by adding an $\ell_2^2$ regularizer to $F(x)$ and sampling from $\pi(x)\propto \exp(-k(F(x)+\mu\|x\|_2^2/2))$ recovers both the known optimal empirical risk and population loss under $(\eps,\delta)$-DP. Furthermore, we show how to implement this mechanism using $\widetilde{O}(n \min(d, n))$ queries to $f_i(x)$ where $n$ is the number of samples/users in the DP-SCO. We also give a (nearly) matching lower bound $\widetilde{\Omega}(n \min(d, n))$ on the number of evaluation queries. Our results utilize the following tools that are of independent interests: \begin{itemize} \item We prove Gaussian Differential Privacy (GDP) of the exponential mechanism if the loss function is strongly convex and the perturbation is Lipschitz. Our privacy bound is \emph{optimal} as it includes the privacy of Gaussian mechanism as a special case. \item We show how to sample from $\exp(-F(x)-\mu \|x\|^2_2/2)$ for $G$-Lipschitz $F$ with $\eta$ error in TV distance using $\widetilde{O}((G^2/\mu) \log^2(d/\eta))$ unbiased queries to $F(x)$. This is the first sampler whose query complexity has \emph{polylogarithmic dependence} on both dimension $d$ and accuracy $\eta$. \end{itemize}

In machine learning, correlation clustering is an important problem whose goal is to partition the individuals into groups that correlate with their pairwise similarities as much as possible. In this work, we revisit the correlation clustering under the differential privacy constraints. Particularly, we improve previous results and achieve an $\Tilde{O}(n^{1.5})$ additive error compared to the optimal cost in expectation on general graphs. As for unweighted complete graphs, we improve the results further and propose a more involved algorithm which achieves $\Tilde{O}(n \sqrt{\Delta^*})$ additive error, where $\Delta^*$ is the maximum degrees of positive edges among all nodes.

Large pretrained models can be fine-tuned with differential privacy to achieve performance approaching that of non-private models. A common theme in these results is the surprising observation that high-dimensional models can achieve favorable privacy-utility trade-offs. This seemingly contradicts known results on the model-size dependence of differentially private convex learning and raises the following research question: When does the performance of differentially private learning not degrade with increasing model size? We identify that the magnitudes of gradients projected onto subspaces is a key factor that determines performance. To precisely characterize this for private convex learning, we introduce a condition on the objective that we term restricted Lipschitz continuity and derive improved bounds for the excess empirical and population risks that are dimension- independent under additional conditions. We empirically show that in private fine-tuning of large language models, gradients obtained during fine-tuning are mostly controlled by a few principal components. This behavior is similar to conditions under which we obtain dimension-independent bounds in convex settings. Our theoretical and empirical results together provide a possible explanation for the recent success of large-scale private fine-tuning. Code to reproduce our results can be found at https://github.com/lxuechen/private-transformers/tree/main/examples/classification/spectral_analysis.

We study a general Markov game with metric switching costs: in each round, the player adaptively chooses one of several Markov chains to advance with the objective of minimizing the expected cost for at least k chains to reach their target states. If the player decides to play a different chain, an additional switching cost is incurred. The special case in which there is no switching cost was solved optimally by Dumitriu, Tetali and Winkler [DTW03] by a variant of the celebrated Gittins Index for the classical multi-armed bandit (MAB) problem with Markovian rewards [Git74, Git79]. However, for Markovian multi-armed bandit with nontrivial switching cost, even if the switching cost is a constant, the classic paper by Banks and Sundaram [BS94] showed that no index strategy can be optimal.1 In this paper, we complement their result and show there is a simple index strategy that achieves a constant approximation factor if the switching cost is constant and k = 1. To the best of our knowledge, this index strategy is the first strategy that achieves a constant approximation factor for a general Markovian MAB variant with switching costs. For the general metric, we propose a more involved constant-factor approximation algorithm, via a nontrivial reduction to the stochastic k-TSP problem, in which a Markov chain is approximated by a random variable. Our analysis makes extensive use of various interesting properties of the Gittins index.

We study the differentially private Empirical Risk Minimization (ERM) and Stochastic Convex Optimization (SCO) problems for non-smooth convex functions. We get a (nearly) optimal bound on the excess empirical risk for ERM with $O(\frac{N^{3/2}}{d^{1/8}}+ \frac{N^2}{d})$ gradient queries, which is achieved with the help of subsampling and smoothing the function via convolution. Combining this result with the iterative localization technique of Feldman et al. \cite{fkt20}, we achieve the optimal excess population loss for the SCO problem with $O(\min\{N^{5/4}d^{1/8},\frac{ N^{3/2}}{d^{1/8}}\})$ gradient queries.Our work makes progress towards resolving a question raised by Bassily et al. \cite{bfgt20}, giving first algorithms for private SCO with subquadratic steps. In a concurrent work, Asi et al. \cite{afkt21} gave other algorithms for private ERM and SCO with subquadratic steps.

Given a metric (V,d) and a root ∈ V, the classic k-TSP problem is to find a tour originating at the root of minimum length that visits at least k nodes in V. In this work, motivated by applications where the input to an optimization problem is uncertain, we study two stochastic versions of k-TSP. In Stoch-Reward k-TSP, originally defined by Ene-Nagarajan-Saket [Ene et al., 2018], each vertex v in the given metric (V,d) contains a stochastic reward R_v. The goal is to adaptively find a tour of minimum expected length that collects at least reward k; here "adaptively" means our next decision may depend on previous outcomes. Ene et al. give an O(log k)-approximation adaptive algorithm for this problem, and left open if there is an O(1)-approximation algorithm. We totally resolve their open question, and even give an O(1)-approximation non-adaptive algorithm for Stoch-Reward k-TSP. We also introduce and obtain similar results for the Stoch-Cost k-TSP problem. In this problem each vertex v has a stochastic cost C_v, and the goal is to visit and select at least k vertices to minimize the expected sum of tour length and cost of selected vertices. Besides being a natural stochastic generalization of k-TSP, this problem is also interesting because it generalizes the Price of Information framework [Singla, 2018] from deterministic probing costs to metric probing costs. Our techniques are based on two crucial ideas: "repetitions" and "critical scaling". In general, replacing a random variable with its expectation leads to very poor results. We show that for our problems, if we truncate the random variables at an ideal threshold, then their expected values form a good surrogate. Here, we rely on running several repetitions of our algorithm with the same threshold, and then argue concentration using Freedman’s and Jogdeo-Samuels' inequalities. Unfortunately, this ideal threshold depends on how far we are from achieving our target k, which a non-adaptive algorithm does not know. To overcome this barrier, we truncate the random variables at various different scales and identify a "critical" scale.