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Large Language Models (LLMs) often falter in complex reasoning tasks due to their static, parametric knowledge, leading to hallucinations and poor performance in specialized domains like mathematics. This work explores a fundamental…

Modern ML systems increasingly augment input instances with additional relevant information to enhance final prediction. Despite growing interest in such retrieval-augmented models, their fundamental properties and training are not well…

In this paper, we study arbitrary infinite binary information systems each of which consists of an infinite set called universe and an infinite set of two-valued functions (attributes) defined on the universe. We consider the notion of a…

In 2010, P\v{a}tra\c{s}cu proposed the following three-phase dynamic problem, as a candidate for proving polynomial lower bounds on the operational time of dynamic data structures: I: Preprocess a collection of sets $\vec{S} = S_1, \ldots ,…

Dynamic diversification---finding a set of data points with maximum diversity from a time-dependent sample pool---is an important task in recommender systems, web search, database search, and notification services, to avoid showing users…

The representation of a dynamic ordered set of $n$ integer keys drawn from a universe of size $m$ is a fundamental data structuring problem. Many solutions to this problem achieve optimal time but take polynomial space, therefore preserving…

Policy evaluation in reinforcement learning is often conducted using two-timescale stochastic approximation, which results in various gradient temporal difference methods such as GTD(0), GTD2, and TDC. Here, we provide convergence rate…

Finite element methods usually construct basis functions and quadrature rules for multidimensional domains via tensor products of one-dimensional counterparts. While straightforward, this approach results in integration spaces larger than…

A dynamic dictionary is a data structure that maintains sets of cardinality at most $n$ from a given universe and supports insertions, deletions, and membership queries. A filter approximates membership queries with a one-sided error that…

We study the point location problem in incremental (possibly disconnected) planar subdivisions, that is, dynamic subdivisions allowing insertions of edges and vertices only. Specifically, we present an $O(n\log n)$-space data structure for…

We study the incremental knapsack problem, where one wishes to sequentially pack items into a knapsack whose capacity expands over a finite planning horizon, with the objective of maximizing time-averaged profits. While various…

Stochastic resetting, where a dynamical process is intermittently returned to a fixed reference state, has emerged as a powerful mechanism for optimizing first-passage properties. Existing theory largely treats static, non-learning…

In a dynamic data structure problem we wish to maintain an encoding of some data in memory, in such a way that we may efficiently carry out a sequence of queries and updates to the data. A long-standing open problem in this area is to prove…

We study sequential search without priors. Our interest lies in decision rules that are close to being optimal under each prior and after each history. We call these rules dynamically robust. The search literature employs optimal rules…

The strategy of stochastic resetting is known to expedite the first passage to a target, in diffusive systems. Consequently, the mean first passage time is minimized at an optimal resetting parameter. With Poisson resetting, vanishing…

Stable states in complex systems correspond to local minima on the associated potential energy surface. Transitions between these local minima govern the dynamics of such systems. Precisely determining the transition pathways in complex and…

Data structures that realize a dictionary are characterized by three basic instructions: (1) Insert (a new entry <key,value>). (2) Search by a key, returning the associated value. (3) Delete an entry. Known realizations are hashing schemes…

For any $\epsilon \in (0,1)$, a $(1+\epsilon)$-approximate range mode query asks for the position of an element whose frequency in the query range is at most a factor $(1+\epsilon)$ smaller than the true mode. For this problem, we design an…

Designing an embedding retrieval system requires navigating a complex design space of conflicting trade-offs between efficiency and effectiveness. This work structures these decisions as a vertical traversal of the system design stack. We…

Given a query and dataset, the optimal way of answering the query is to make use all the information available. Modern LLMs exhibit impressive ability to memorize training data, but data not deemed important during training is forgotten,…