Prashant Gokhale

Kiarash Banihashem, Jeff Giliberti, Prashant Gokhale et al.

We work in the adaptive query model, where one is given a point set $P \subset \mathbb{R}^d$ and seeks to construct a data structure that can answer correctly and efficiently a sequence of adaptive queries. In this model, an adversary observes the answers returned by the data structure to previous queries $q_1, \ldots, q_{i-1}$ and, based on this information, chooses the next query point $q_i$. This setting captures strong forms of adaptivity that naturally arise in modern machine learning pipelines, and rules out many classical randomized techniques that assume oblivious queries. Our focus is the problem of furthest neighbor search in this adaptive setting, a fundamental problem in several learning tasks, including diversity maximization, outlier and anomaly detection, adversarial example generation, and more. We present the first adversarially robust data structure for $c$-approximate furthest neighbor queries that achieves query time $\tilde{O}( \min( d n^{1/c^2}, n^{2/c^2} + d))$. This matches the $n$ dependency in the query time of the seminal result by Indyk~[SODA'03] for $c$-approximate furthest neighbor in the oblivious setting, and improves upon the $\tilde{O}(n + d)$ query time achieved via the adaptive distance estimation framework of Cherapanamjeri and Nelson~[NeurIPS'20] for a wide range of natural parameters. To complement this result, we present an adversarial attack against oblivious approximate furthest neighbor algorithms. Specifically, we show that the data structure from the algorithm by Indyk fails to maintain its guarantees against adaptive queries.

Prashant Gokhale, Mikhail Khodak, Sandeep Silwal

We consider the problem of sequentially approximating functions of each element in a slowly-varying sequence, i.e. one where the magnitude $α_i$ of the difference between the elements at positions $i$ and $i-1$ is small. Recent work on implicit trace estimation shows that when $α_t$ is small, reusing queries to past sequence elements can reduce the overall cost [Dharangutte \& Musco, NeurIPS~2021; Woodruff et al., NeurIPS~2022]. We introduce a framework generalizing this to a variety of linear and nonlinear functions on diverse vector spaces, obtaining novel sequential estimation results for matrix powers, spectral densities, Monte Carlo integration, and a boundary value problem from partial differential equations~(PDEs). Furthermore, we develop a novel algorithm for use with this framework that locally scales the estimation budget with $α_t$, obtaining sharper path-length-style variation bounds of form $\mathcal O(\sum_{i=1}^mα_i)$ on the cost of estimating a sequence of length $m$. This improves upon the previous implicit trace estimation bound of $\mathcal O(m\cdot\max_iα_i)$ [Dharangutte \& Musco, NeurIPS~2021], which is achieved by fixing the query budget using the worst-case $α_i$ and is thus inefficient for stable sequences with rare bursts. Lastly, while all past work assumes a known bound on $α_i$, we show in certain cases how the changes can be estimated on-the-fly with (nearly) no added cost. In summary, our framework makes the sequential approximation toolkit general-purpose and adaptive while improving upon state-of-the-art-guarantees for dynamic trace estimation.

Prashant Gokhale, Piotr Indyk, Yuhao Liu et al.

Computing geometric representations of data is a cornerstone of modern machine learning, typically achieved by training dual encoders which map queries and documents into a shared embedding space. Recent work of You et al. [NeurIPS '25] has extended this approach to hierarchical retrieval, where relevance is determined by the ancestor-descendant relationships in a Directed Acyclic Graph (DAG). While previous work has shown that valid embeddings exist when the number of descendants is small, these bounds degrade significantly for deep hierarchies, requiring dimensions as large as the total number of nodes. In this paper, we investigate compact reachability embeddings for more general graph classes and provide theoretical guarantees for representing hierarchies using embeddings whose dimension depends on structural graph parameters. We prove that for any directed tree, there exists a reachability embedding in constant dimension 3, independent of the tree's size or depth. We generalize this result to graphs characterized by treewidth $t$, constructing embeddings of dimension $O(t \log n)$, where $n$ is the number of nodes. Complementing these upper bounds, we provide matching or near-matching lower bounds, showing that dimension $Ω(n)$ is necessary for general DAGs and $Ω(t/\log(n/t))$ is required for graphs of treewidth $t$. We also obtain upper and lower bounds parameterized by the number of cross-edges in the DAG. We additionally show that our embeddings can be constructed on real world datasets, and that they give much smaller dimensions in high recall regimes compared to prior embeddings with theoretical guarantees.