Search for: All records

Estimating frequencies of elements appearing in a data stream is a key task in large-scale data analysis. Popular sketching approaches to this problem (e.g., CountMin and CountSketch) come with worst-case guarantees that probabilistically bound the error of the estimated frequencies for any possible input. The work of Hsu et al.~(2019) introduced the idea of using machine learning to tailor sketching algorithms to the specific data distribution they are being run on. In particular, their learning-augmented frequency estimation algorithm uses a learned heavy-hitter oracle which predicts which elements will appear many times in the stream. We give a novel algorithm, which in some parameter regimes, already theoretically outperforms the learning based algorithm of Hsu et al. without the use of any predictions. Augmenting our algorithm with heavy-hitter predictions further reduces the error and improves upon the state of the art. Empirically, our algorithms achieve superior performance in all experiments compared to prior approaches. 

In this work, we study the convergence \emph{in high probability} of clipped gradient methods when the noise distribution has heavy tails, i.e., with bounded $p$th moments, for some $1 more » « less

Free, publicly-accessible full text available September 21, 2024

In this work, we revisit the generalization error of stochastic mirror descent for quadratically bounded losses studied in Telgarsky (2022). Quadratically bounded losses is a broad class of loss functions, capturing both Lipschitz and smooth functions, for both regression and classification problems. We study the high probability generalization for this class of losses on linear predictors in both realizable and non-realizable cases when the data are sampled IID or from a Markov chain. The prior work relies on an intricate coupling argument between the iterates of the original problem and those projected onto a bounded domain. This approach enables blackbox application of concentration inequalities, but also leads to suboptimal guarantees due in part to the use of a union bound across all iterations. In this work, we depart significantly from the prior work of Telgarsky (2022), and introduce a novel approach for establishing high probability generalization guarantees. In contrast to the prior work, our work directly analyzes the moment generating function of a novel supermartingale sequence and leverages the structure of stochastic mirror descent. As a result, we obtain improved bounds in all aforementioned settings. Specifically, in the realizable case and non-realizable case with light-tailed sub-Gaussian data, we improve the bounds by a $\log T$ factor, matching the correct rates of $1/T$ and $1/\sqrt{T}$, respectively. In the more challenging case of heavy-tailed polynomial data, we improve the existing bound by a $\mathrm{poly}\ T$ factor. 

In this work, we study the problem of privately maximizing a submodular function in the streaming setting. Extensive work has been done on privately maximizing submodular functions in the general case when the function depends upon the private data of individuals. However, when the size of the data stream drawn from the domain of the objective function is large or arrives very fast, one must privately optimize the objective within the constraints of the streaming setting. We establish fundamental differentially private baselines for this problem and then derive better trade-offs between privacy and utility for the special case of decomposable submodular functions. A submodular function is decomposable when it can be written as a sum of submodular functions; this structure arises naturally when each summand function models the utility of an individual and the goal is to study the total utility of the whole population as in the well-known Combinatorial Public Projects Problem. Finally, we complement our theoretical analysis with experimental corroboration. 

In this work, we describe a generic approach to show convergence with high probability for both stochastic convex and non-convex optimization with sub-Gaussian noise. In previous works for convex optimization, either the convergence is only in expectation or the bound depends on the diameter of the domain. Instead, we show high probability convergence with bounds depending on the initial distance to the optimal solution. The algorithms use step sizes analogous to the standard settings and are universal to Lipschitz functions, smooth functions, and their linear combinations. The method can be applied to the non-convex case. We demonstrate an $O((1+\sigma^{2}\log(1/\delta))/T+\sigma/\sqrt{T})$ convergence rate when the number of iterations $T$ is known and an $O((1+\sigma^{2}\log(T/\delta))/\sqrt{T})$ convergence rate when $T$ is unknown for SGD, where $1-\delta$ is the desired success probability. These bounds improve over existing bounds in the literature. We also revisit AdaGrad-Norm \cite{ward2019adagrad} and show a new analysis to obtain a high probability bound that does not require the bounded gradient assumption made in previous works. The full version of our paper contains results for the standard per-coordinate AdaGrad. 

Recently a multi-agent variant of the classical multi-armed bandit was proposed to tackle fairness issues in online learning. Inspired by a long line of work in social choice and economics, the goal is to optimize the Nash social welfare instead of the total utility. Unfortunately previous algorithms either are not efficient or achieve sub-optimal regret in terms of the number of rounds. We propose a new efficient algorithm with lower regret than even previous inefficient ones. We also complement our efficient algorithm with an inefficient approach with regret that matches the lower bound for one agent. The experimental findings confirm the effectiveness of our efficient algorithm compared to the previous approaches.

 

Existing analysis of AdaGrad and other adaptive methods for smooth convex optimization is typically for functions with bounded domain diameter. In unconstrained problems, previous works guarantee an asymptotic convergence rate without an explicit constant factor that holds true for the entire function class. Furthermore, in the stochastic setting, only a modified version of AdaGrad, different from the one commonly used in practice, in which the latest gradient is not used to update the stepsize, has been analyzed. Our paper aims at bridging these gaps and developing a deeper understanding of AdaGrad and its variants in the standard setting of smooth convex functions as well as the more general setting of quasar convex functions. First, we demonstrate new techniques to explicitly bound the convergence rate of the vanilla AdaGrad for unconstrained problems in both deterministic and stochastic settings. Second, we propose a variant of AdaGrad for which we can show the convergence of the last iterate, instead of the average iterate. Finally, we give new accelerated adaptive algorithms and their convergence guarantee in the deterministic setting with explicit dependency on the problem parameters, improving upon the asymptotic rate shown in previous works. 

We consider the problem of clustering in the learning-augmented setting. We are given a data set in $d$-dimensional Euclidean space, and a label for each data point given by a predictor indicating what subsets of points should be clustered together. This setting captures situations where we have access to some auxiliary information about the data set relevant for our clustering objective, for instance the labels output by a neural network. Following prior work, we assume that there are at most an $\alpha \in (0,c)$ for some $c<1$ fraction of false positives and false negatives in each predicted cluster, in the absence of which the labels would attain the optimal clustering cost $\mathrm{OPT}$. For a dataset of size $m$, we propose a deterministic $k$-means algorithm that produces centers with an improved bound on the clustering cost compared to the previous randomized state-of-the-art algorithm while preserving the $O( d m \log m)$ runtime. Furthermore, our algorithm works even when the predictions are not very accurate, i.e., our cost bound holds for $\alpha$ up to $1/2$, an improvement from $\alpha$ being at most $1/7$ in previous work. For the $k$-medians problem we again improve upon prior work by achieving a biquadratic improvement in the dependence of the approximation factor on the accuracy parameter $\alpha$ to get a cost of $(1+O(\alpha))\mathrm{OPT}$, while requiring essentially just $O(md \log^3 m/\alpha)$ runtime. 

We develop new adaptive algorithms for variational inequalities with monotone operators, which capture many problems of interest, notably convex optimization and convex-concave saddle point problems. Our algorithms automatically adapt to unknown problem parameters such as the smoothness and the norm of the operator, and the variance of the stochastic evaluation oracle. We show that our algorithms are universal and simultaneously achieve the optimal convergence rates in the non-smooth, smooth, and stochastic settings. The convergence guarantees of our algorithms improve over existing adaptive methods and match the optimal non-adaptive algorithms. Additionally, prior works require that the optimization domain is bounded. In this work, we remove this restriction and give algorithms for unbounded domains that are adaptive and universal. Our general proof techniques can be used for many variants of the algorithm using one or two operator evaluations per iteration. The classical methods based on the ExtraGradient/MirrorProx algorithm require two operator evaluations per iteration, which is the dominant factor in the running time in many settings.