We introduce a new notion of stability--which we call stable list decoding--and demonstrate its applicability in designing differentially private density estimators. This definition is weaker than global stability [ABLMM22] and is related to the notions of replicability [ILPS22] and list replicability [CMY23]. We show that if a class of distributions is stable list decodable, then it can be learned privately in the agnostic setting. As the main application of our framework, we prove the first upper bound on the sample complexity of private density estimation for Gaussian Mixture Models in the agnostic setting, extending the realizable result of Afzali et al. [AAL24].

In this work, we investigate the online learning problem of revenue maximization in ad auctions, where the seller needs to learn the click-through rates (CTRs) of each ad candidate and charge the price of the winner through a pay-per-click manner. We focus on two models of the advertisers' strategic behaviors. First, we assume that the advertiser is completely myopic; i.e.~in each round, they aim to maximize their utility only for the current round. In this setting, we develop an online mechanism based on upper-confidence bounds that achieves a tight $O(\sqrt{T})$ regret in the worst-case and negative regret when the values are static across all the auctions and there is a gap between the highest expected value (i.e.~value multiplied by their CTR) and second highest expected value ad. Next, we assume that the advertiser is non-myopic and cares about their long term utility. This setting is much more complex since an advertiser is incentivized to influence the mechanism by bidding strategically in earlier rounds. In this setting, we provide an algorithm to achieve negative regret for the static valuation setting (with a positive gap), which is in sharp contrast with the prior work that shows $O(T^{2/3})$ regret when the valuation is generated by adversary.

Density estimation--also known as distribution learning--is the fundamental task of estimating a distribution given samples generated from it. In the most commonly studied setting, the data points are assumed to be generated independently from an unknown distribution f and the goal is to find a distribution ˆf that is close to f with respect to the total variation (TV) distance. Assuming that f belongs to (or is close to a member of) a class of distributions F, an important question is to characterize the number of samples that is required to guarantee that with high probability ˆf is close to f in TV distance. There is a large body of work on characterizing the optimal sample complexity (or the related minimax error rate) of learning various classes of distributions (see Diakonikolas [Dia16], Devroye and Lugosi [DL01], and Ashtiani and Mehrabian [AM18] for an overview). Nevertheless, determining the sample complexity (or even learnability) of a general class of distributions remains an important open problem (e.g., Open Problem 15.1 in Diakonikolas [Dia16]).

We study the problem of privately estimating the parameters of $d$-dimensional Gaussian Mixture Models (GMMs) with $k$ components. For this, we develop a technique to reduce the problem to its non-private counterpart. This allows us to privatize existing non-private algorithms in a blackbox manner, while incurring only a small overhead in the sample complexity and running time. As the main application of our framework, we develop an $(\varepsilon, \delta)$-differentially private algorithm to learn GMMs using the non-private algorithm of Moitra and Valiant [MV10] as a blackbox. Consequently, this gives the first sample complexity upper bound and first polynomial time algorithm for privately learning GMMs without any boundedness assumptions on the parameters. As part of our analysis, we prove a tight (up to a constant factor) lower bound on the total variation distance of high-dimensional Gaussians which can be of independent interest.

We propose a new Markov Decision Process (MDP) model for ad auctions to capture the user response to the quality of ads, with the objective of maximizing the long-term discounted revenue. By incorporating user response, our model takes into consideration all three parties involved in the auction (advertiser, auctioneer, and user). The state of the user is modeled as a user-specific click-through rate (CTR) with the CTR changing in the next round according to the set of ads shown to the user in the current round. We characterize the optimal mechanism for this MDP as a Myerson's auction with a notion of modified virtual value, which relies on the value distribution of the advertiser, the current user state, and the future impact of showing the ad to the user. Moreover, we propose a simple mechanism built upon second price auctions with personalized reserve prices and show it can achieve a constant-factor approximation to the optimal long term discounted revenue.

We present a fairly general framework for reducing $(\varepsilon, \delta)$ differentially private (DP) statistical estimation to its non-private counterpart. As the main application of this framework, we give a polynomial time and $(\varepsilon,\delta)$-DP algorithm for learning (unrestricted) Gaussian distributions in $\mathbb{R}^d$. The sample complexity of our approach for learning the Gaussian up to total variation distance $\alpha$ is $\widetilde{O}\left(\frac{d^2}{\alpha^2}+\frac{d^2 \sqrt{\ln{1/\delta}}}{\alpha\varepsilon} \right)$, matching (up to logarithmic factors) the best known information-theoretic (non-efficient) sample complexity upper bound of Aden-Ali, Ashtiani, Kamath~(ALT'21). In an independent work, Kamath, Mouzakis, Singhal, Steinke, and Ullman~(arXiv:2111.04609) proved a similar result using a different approach and with $O(d^{5/2})$ sample complexity dependence on $d$. As another application of our framework, we provide the first polynomial time $(\varepsilon, \delta)$-DP algorithm for robust learning of (unrestricted) Gaussians.

We consider stochastic gradient descent algorithms for minimizing a non-smooth, strongly-convex function. Several forms of this algorithm, including suffix averaging, are known to achieve the optimal $O(1/T)$ convergence rate in expectation. We consider a simple, non-uniform averaging strategy of Lacoste-Julien et al. (2011) and prove that it achieves the optimal $O(1/T)$ convergence rate with high probability. Our proof uses a recently developed generalization of Freedman's inequality. Finally, we compare several of these algorithms experimentally and show that this non-uniform averaging strategy outperforms many standard techniques, and with smaller variance.

We prove that ϴ(k d^2 / ε^2) samples are necessary and sufficient for learning a mixture of k Gaussians in R^d, up to error ε in total variation distance. This improves both the known upper bounds and lower bounds for this problem. For mixtures of axis-aligned Gaussians, we show that O(k d / ε^2) samples suffice, matching a known lower bound. The upper bound is based on a novel technique for distribution learning based on a notion of sample compression. Any class of distributions that allows such a sample compression scheme can also be learned with few samples. Moreover, if a class of distributions has such a compression scheme, then so do the classes of products and mixtures of those distributions. The core of our main result is showing that the class of Gaussians in R^d has an efficient sample compression.

We prove that ϴ(k d^2 / ε^2) samples are necessary and sufficient for learning a mixture of k Gaussians in R^d, up to error ε in total variation distance. This improves both the known upper bounds and lower bounds for this problem. For mixtures of axis-aligned Gaussians, we show that O(k d / ε^2) samples suffice, matching a known lower bound. The upper bound is based on a novel technique for distribution learning based on a notion of sample compression. Any class of distributions that allows such a sample compression scheme can also be learned with few samples. Moreover, if a class of distributions has such a compression scheme, then so do the classes of products and mixtures of those distributions. The core of our main result is showing that the class of Gaussians in R^d has an efficient sample compression.

Consider the problem of minimizing functions that are Lipschitz and strongly convex, but not necessarily differentiable. We prove that after $T$ steps of stochastic gradient descent, the error of the final iterate is $O(\log(T)/T)$ with high probability. We also construct a function from this class for which the error of the final iterate of deterministic gradient descent is $\Omega(\log(T)/T)$. This shows that the upper bound is tight and that, in this setting, the last iterate of stochastic gradient descent has the same general error rate (with high probability) as deterministic gradient descent. This resolves both open questions posed by Shamir (2012). An intermediate step of our analysis proves that the suffix averaging method achieves error $O(1/T)$ with high probability, which is optimal (for any first-order optimization method). This improves results of Rakhlin (2012) and Hazan and Kale (2014), both of which achieved error $O(1/T)$, but only in expectation, and achieved a high probability error bound of $O(\log \log(T)/T)$, which is suboptimal. We prove analogous results for functions that are Lipschitz and convex, but not necessarily strongly convex or differentiable. After $T$ steps of stochastic gradient descent, the error of the final iterate is $O(\log(T)/\sqrt{T})$ with high probability, and there exists a function for which the error of the final iterate of deterministic gradient descent is $\Omega(\log(T)/\sqrt{T})$.