An influential paper of Hsu et al. (ICLR'19) introduced the study of learning-augmented streaming algorithms in the context of frequency estimation. A fundamental problem in the streaming literature, the goal of frequency estimation is to approximate the number of occurrences of items appearing in a long stream of data using only a small amount of memory. Hsu et al. develop a natural framework to combine the worst-case guarantees of popular solutions such as CountMin and CountSketch with learned predictions of high frequency elements. They demonstrate that learning the underlying structure of data can be used to yield better streaming algorithms, both in theory and practice. We simplify and generalize past work on learning-augmented frequency estimation. Our first contribution is a learning-augmented variant of the Misra-Gries algorithm which improves upon the error of learned CountMin and learned CountSketch and achieves the state-of-the-art performance of randomized algorithms (Aamand et al., NeurIPS'23) with a simpler, deterministic algorithm. Our second contribution is to adapt learning-augmentation to a high-dimensional generalization of frequency estimation corresponding to finding important directions (top singular vectors) of a matrix given its rows one-by-one in a stream. We analyze a learning-augmented variant of the Frequent Directions algorithm, extending the theoretical and empirical understanding of learned predictions to matrix streaming.

We design improved approximation algorithms for NP-hard graph problems by incorporating predictions (e.g., learned from past data). Our prediction model builds upon and extends the $\varepsilon$-prediction framework by Cohen-Addad, d’Orsi, Gupta, Lee, and Panigrahi (NeurIPS 2024). We consider an edge-based version of this model, where each edge provides two bits of information, corresponding to predictions about whether each of its endpoints belong to an optimal solution. Even with weak predictions where each bit is only $\varepsilon$-correlated with the true solution, this information allows us to break approximation barriers in the standard setting. We develop algorithms with improved approximation ratios for MaxCut, Vertex Cover, Set Cover, and Maximum Independent Set problems (among others). Across these problems, our algorithms share a unifying theme, where we separately satisfy constraints related to high degree vertices (using predictions) and low-degree vertices (without using predictions) and carefully combine the answers.

Graph-based data structures have become powerful and ubiquitous tools for scalable approximate nearest-neighbor (ANN) search over the past decade. In spite of their apparent practical performance, there has only recently been progress on the **worst-case** performance of these data structures. Indeed, the influential work of Indyx and Xu (2023) introduced the key concept of $\alpha$-reachable graphs, showing that graphs constructed by the DiskANN algorithm (Subramanya, et. al. 2023) produce an $\left(\frac{\alpha+1}{\alpha-1}\right)$-approximate solution with a simple best-first search that runs in poly-logarithmic query time. In our work, we improve and generalize this analysis as follows: - We introduce **sorted** $\alpha$-reachable graphs, and use this notion to obtain a stronger approximation factor of $\frac{\alpha}{\alpha-1}$ for the DiskANN algorithm on Euclidean metrics. - We present the **first** worst-case theoretical analysis for the popular **beam-search** algorithm, which is used in practice to search these graphs for $k > 1$ candidate nearest neighbors. We also present empirical results validating the significance of sorted $\alpha$-reachable graphs, which aligns with our theoretical findings.

The joint replenishment problem (JRP) is a classical inventory management problem. We consider a natural generalization with outliers, where we are allowed to reject (that is, not service) a subset of demand points. In this paper, we are motivated by issues of fairness - if we do not serve all of the demands, we wish to “spread out the pain” in a balanced way among customers, communities, or any specified market segmentation. One approach is to constrain the rejections allowed, and to have separate bounds for each given customer. In our most general setting, we consider a set of C features, where each demand point has an associated rejection cost for each feature, and we have a given bound on the allowed rejection cost incurred in total for each feature. This generalizes an extensively studied model of fairness introduced in earlier work on the Colorful k—Center problem in which (analogously) each demand point has a given color, and we bound the number of rejections of each color class. In the JRP, we seek to balance the cost incurred by a fixed ordering overhead with the cost of maintaining on-hand inventory over a longer period in advance of when it is needed. More precisely, there is a given set of item types, for which there is specified demand over a finite, discrete-time horizon, and placing any order at a given time incurs a general ordering cost and item-specific ordering costs (independent of the total demand serviced); in addition, for each unit of demand held in inventory for an interval of time, there is a corresponding item-specific holding cost incurred; the aim is to minimize the total cost.

Given a set of $n$ vectors in $\mathbb{R}^d$, the goal of the \emph{determinant maximization} problem is to pick $k$ vectors with the maximum volume. Determinant maximization is the MAP-inference task for determinantal point processes (DPP) and has recently received considerable attention for modeling diversity. As most applications for the problem use large amounts of data, this problem has been studied in the relevant \textit{composable coreset} setting. In particular, [Indyk-Mahabadi-OveisGharan-Rezaei--SODA'20, ICML'19] showed that one can get composable coresets with optimal approximation factor of $\tilde O(k)^k$ for the problem, and that a local search algorithm achieves an almost optimal approximation guarantee of $O(k)^{2k}$. In this work, we show that the widely-used Greedy algorithm also provides composable coresets with an almost optimal approximation factor of $O(k)^{3k}$, which improves over the previously known guarantee of $C^{k^2}$, and supports the prior experimental results showing the practicality of the greedy algorithm as a coreset. Our main result follows by showing a local optimality property for Greedy: swapping a single point from the greedy solution with a vector that was not picked by the greedy algorithm can increase the volume by a factor of at most $(1+\sqrt{k})$. This is tight up to the additive constant $1$. Finally, our experiments show that the local optimality of the greedy algorithm is even lower than the theoretical bound on real data sets.

As Approximate Nearest Neighbor Search (ANNS)-based dense retrieval becomes ubiquitous for search and recommendation scenarios, efficiently answering filtered ANNS queries has become a critical requirement. Filtered ANNS queries ask for the nearest neighbors of a query’s embedding from the points in the index that match the query’s labels such as date, price range, language. There has been little prior work on algorithms that use label metadata associated with vector data to build efficient indices for filtered ANNS queries. Consequently, current indices have high search latency or low recall which is not practical in interactive web-scenarios. We present two algorithms with native support for faster and more accurate filtered ANNS queries: one with streaming support, and another based on batch construction. Central to our algorithms is the construction of a graph-structured index which forms connections not only based on the geometry of the vector data, but also the associated label set. On real-world data with natural labels, both algorithms are an order of magnitude or more efficient for filtered queries than the current state of the art algorithms. The generated indices also be queried from an SSD and support thousands of queries per second at over recall@10.