Adaptive Frontier Exploration on Graphs with Applications to Network-Based Disease Testing

We study a sequential decision-making problem on a $n$-node graph $G$ where each node has an unknown label from a finite set $\mathbf{\Sigma}$, drawn from a joint distribution $P$ that is Markov with respect to $G$. At each step, selecting a node reveals its label and yields a label-dependent reward. The goal is to adaptively choose nodes to maximize expected accumulated discounted rewards. We impose a frontier exploration constraint, where actions are limited to neighbors of previously selected nodes, reflecting practical constraints in settings such as contact tracing and robotic exploration. We design a Gittins index-based policy that applies to general graphs and is provably optimal when $G$ is a forest. Our implementation runs in $O(n^2 \cdot |\mathbf{\Sigma}|^2)$ time while using $O(n \cdot |\mathbf{\Sigma}|^2)$ oracle calls to $P$ and $O(n^2 \cdot |\mathbf{\Sigma}|)$ space. Experiments on synthetic and real-world graphs show that our method consistently outperforms natural baselines, including in non-tree, budget-limited, and undiscounted settings. For example, in HIV testing simulations on real-world sexual interaction networks, our policy detects nearly all positive cases with only half the population tested, substantially outperforming other baselines.

Learning-Augmented Online Bipartite Fractional Matching

Online bipartite matching is a fundamental problem in online optimization, extensively studied both in its integral and fractional forms due to its theoretical significance and practical applications, such as online advertising and resource allocation. Motivated by recent progress in learning-augmented algorithms, we study online bipartite fractional matching when the algorithm is given advice in the form of a suggested matching in each iteration. We develop algorithms for both the vertex-weighted and unweighted variants that provably dominate the naive "coin flip" strategy of randomly choosing between the advice-following and advice-free algorithms. Moreover, our algorithm for the vertex-weighted setting extends to the AdWords problem under the small bids assumption, yielding a significant improvement over the seminal work of Mahdian, Nazerzadeh, and Saberi (EC 2007, TALG 2012). Complementing our positive results, we establish a hardness bound on the robustness-consistency tradeoff that is attainable by any algorithm. We empirically validate our algorithms through experiments on synthetic and real-world data.

On Sequential Fault-Intolerant Process Planning

We propose and study a planning problem we call Sequential Fault-Intolerant Process Planning (SFIPP). SFIPP captures a reward structure common in many sequential multi-stage decision problems where the planning is deemed successful only if all stages succeed. Such reward structures are different from classic additive reward structures and arise in important applications such as drug/material discovery, security, and quality-critical product design. We design provably tight online algorithms for settings in which we need to pick between different actions with unknown success chances at each stage. We do so both for the foundational case in which the behavior of actions is deterministic, and the case of probabilistic action outcomes, where we effectively balance exploration for learning and exploitation for planning through the usage of multi-armed bandit algorithms. In our empirical evaluations, we demonstrate that the specialized algorithms we develop, which leverage additional information about the structure of the SFIPP instance, outperform our more general algorithm.

A partition cover approach to tokenization

Tokenization is the process of encoding strings into tokens from a fixed vocabulary of size $k$ and is widely utilized in Natural Language Processing applications. The leading tokenization algorithm today is Byte Pair Encoding (BPE), which formulates the tokenization problem as a compression problem and tackles it by performing sequences of merges. In this work, we formulate tokenization as an optimization objective, show that it is NP-hard via a simple reduction from vertex cover, and propose a polynomial-time greedy algorithm GreedTok. Our formulation naturally relaxes to the well-studied weighted maximum coverage problem which has a simple $(1 - 1/e)$-approximation algorithm GreedWMC. Through empirical evaluations on real-world corpora, we show that GreedTok outperforms BPE, while achieving a comparable objective score as GreedWMC (which could have achieved a higher score due to relaxation).

Learning multivariate Gaussians with imperfect advice

We revisit the problem of distribution learning within the framework of learning-augmented algorithms. In this setting, we explore the scenario where a probability distribution is provided as potentially inaccurate advice on the true, unknown distribution. Our objective is to develop learning algorithms whose sample complexity decreases as the quality of the advice improves, thereby surpassing standard learning lower bounds when the advice is sufficiently accurate. Specifically, we demonstrate that this outcome is achievable for the problem of learning a multivariate Gaussian distribution $N(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ in the PAC learning setting. Classically, in the advice-free setting, $\tilde{\Theta}(d^2/\varepsilon^2)$ samples are sufficient and worst case necessary to learn $d$-dimensional Gaussians up to TV distance $\varepsilon$ with constant probability. When we are additionally given a parameter $\tilde{\boldsymbol{\Sigma}}$ as advice, we show that $\tilde{O}(d^{2-\beta}/\varepsilon^2)$ samples suffices whenever $\| \tilde{\boldsymbol{\Sigma}}^{-1/2} \boldsymbol{\Sigma} \tilde{\boldsymbol{\Sigma}}^{-1/2} - \boldsymbol{I_d} \|_1 \leq \varepsilon d^{1-\beta}$ (where $\|\cdot\|_1$ denotes the entrywise $\ell_1$ norm) for any $\beta > 0$, yielding a polynomial improvement over the advice-free setting.

Probably approximately correct high-dimensional causal effect estimation given a valid adjustment set

Accurate estimates of causal effects play a key role in decision-making across applications such as healthcare, economics, and operations. In the absence of randomized experiments, a common approach to estimating causal effects uses \textit{covariate adjustment}. In this paper, we study covariate adjustment for discrete distributions from the PAC learning perspective, assuming knowledge of a valid adjustment set $\bZ$, which might be high-dimensional. Our first main result PAC-bounds the estimation error of covariate adjustment by a term that is exponential in the size of the adjustment set; it is known that such a dependency is unavoidable even if one only aims to minimize the mean squared error. Motivated by this result, we introduce the notion of an \emph{$\eps$-Markov blanket}, give bounds on the misspecification error of using such a set for covariate adjustment, and provide an algorithm for $\eps$-Markov blanket discovery; our second main result upper bounds the sample complexity of this algorithm. Furthermore, we provide a misspecification error bound and a constraint-based algorithm that allow us to go beyond $\eps$-Markov blankets to even smaller adjustment sets. Our third main result upper bounds the sample complexity of this algorithm, and our final result combines the first three into an overall PAC bound. Altogether, our results highlight that one does not need to perfectly recover causal structure in order to ensure accurate estimates of causal effects.

Learnability of Parameter-Bounded Bayes Nets

Bayes nets are extensively used in practice to efficiently represent joint probability distributions over a set of random variables and capture dependency relations. In a seminal paper, Chickering et al. (JMLR 2004) showed that given a distribution $P$, that is defined as the marginal distribution of a Bayes net, it is $\mathsf{NP}$-hard to decide whether there is a parameter-bounded Bayes net that represents $P$. They called this problem LEARN. In this work, we extend the $\mathsf{NP}$-hardness result of LEARN and prove the $\mathsf{NP}$-hardness of a promise search variant of LEARN, whereby the Bayes net in question is guaranteed to exist and one is asked to find such a Bayes net. We complement our hardness result with a positive result about the sample complexity that is sufficient to recover a parameter-bounded Bayes net that is close (in TV distance) to a given distribution $P$, that is represented by some parameter-bounded Bayes net, generalizing a degree-bounded sample complexity result of Brustle et al. (EC 2020).

Online bipartite matching with imperfect advice

We study the problem of online unweighted bipartite matching with $n$ offline vertices and $n$ online vertices where one wishes to be competitive against the optimal offline algorithm. While the classic RANKING algorithm of Karp et al. [1990] provably attains competitive ratio of $1-1/e > 1/2$, we show that no learning-augmented method can be both 1-consistent and strictly better than $1/2$-robust under the adversarial arrival model. Meanwhile, under the random arrival model, we show how one can utilize methods from distribution testing to design an algorithm that takes in external advice about the online vertices and provably achieves competitive ratio interpolating between any ratio attainable by advice-free methods and the optimal ratio of 1, depending on the advice quality.

Causal Discovery under Off-Target Interventions

Causal graph discovery is a significant problem with applications across various disciplines. However, with observational data alone, the underlying causal graph can only be recovered up to its Markov equivalence class, and further assumptions or interventions are necessary to narrow down the true graph. This work addresses the causal discovery problem under the setting of stochastic interventions with the natural goal of minimizing the number of interventions performed. We propose the following stochastic intervention model which subsumes existing adaptive noiseless interventions in the literature while capturing scenarios such as fat-hand interventions and CRISPR gene knockouts: any intervention attempt results in an actual intervention on a random subset of vertices, drawn from a distribution dependent on attempted action. Under this model, we study the two fundamental problems in causal discovery of verification and search and provide approximation algorithms with polylogarithmic competitive ratios and provide some preliminary experimental results.

Learning bounded-degree polytrees with known skeleton

We establish finite-sample guarantees for efficient proper learning of bounded-degree polytrees, a rich class of high-dimensional probability distributions and a subclass of Bayesian networks, a widely-studied type of graphical model. Recently, Bhattacharyya et al. (2021) obtained finite-sample guarantees for recovering tree-structured Bayesian networks, i.e., 1-polytrees. We extend their results by providing an efficient algorithm which learns $d$-polytrees in polynomial time and sample complexity for any bounded $d$ when the underlying undirected graph (skeleton) is known. We complement our algorithm with an information-theoretic sample complexity lower bound, showing that the dependence on the dimension and target accuracy parameters are nearly tight.