Decision Tree Learning on Product Spaces

Decision tree learning has long been a central topic in theoretical computer science, driven by its practical importance. A fundamental and widely used method for decision tree construction is the top-down greedy heuristic, which recursively splits on the most influential variable. Despite its empirical success, theoretical analysis of this heuristic has been limited. A recent breakthrough by Blanc et al. (ITCS, 2020) provided the first rigorous theoretical guarantees for the greedy approach, but only under the uniform distribution. We extend this analysis to the more general and practically relevant setting of arbitrary product distributions. Our main result shows that for any function $f$ computable by an optimal decision tree of size $s$, maximum depth $D_{\text{opt}}$, and average depth $Δ_{\text{opt}}$, the greedy heuristic constructs an $ε$-approximating tree whose size grows at most with $\exp\bigl(Δ_{\text{opt}} D_{\text{opt}} \log(e/ε)\bigr)$. In the special case where the optimal tree is a full binary tree, this bound improves upon the bound of Blanc et al. and holds under a strictly broader class of distributions. Moreover, we present an algorithm based on the top-down greedy heuristic that is entirely parameter-free -- it requires no prior knowledge of the optimal tree's size or depth -- offering a practical advantage over Blanc et al.'s method.

Replicable Composition

Replicability requires that algorithmic conclusions remain consistent when rerun on independently drawn data. A central structural question is composition: given $k$ problems each admitting a $ρ$-replicable algorithm with sample complexity $n$, how many samples are needed to solve all jointly while preserving replicability? The naive analysis yields $\widetilde{O}(nk^2)$ samples, and Bun et al. (STOC'23) observed that reductions through differential privacy give an alternative $\widetilde{O}(n^2k)$ bound, leaving open whether the optimal $\widetilde{O}(nk)$ scaling is achievable. We resolve this open problem and, more generally, show that problems with sample complexities $n_1,\ldots,n_k$ can be jointly solved with $\widetilde{O}(\sum_i n_i)$ samples while preserving constant replicability. Our approach converts each replicable algorithm into a perfectly generalizing one, composes them via a privacy-style analysis, and maps back via correlated sampling. This yields the first advanced composition theorem for replicability. En route, we obtain new bounds for the composition of perfectly generalizing algorithms with heterogeneous parameters. As part of our results, we provide a boosting theorem for the success probability of replicable algorithms. For a broad class of problems, the failure probability appears as a separate additive term independent of $ρ$, immediately yielding improved sample complexity bounds for several problems. Finally, we prove an $Ω(nk^2)$ lower bound for adaptive composition, establishing a quadratic separation from the non-adaptive setting. The key technique, which we call the phantom run, yields structural results of independent interest.

Active Learning for Decision Trees with Provable Guarantees

This paper advances the theoretical understanding of active learning label complexity for decision trees as binary classifiers. We make two main contributions. First, we provide the first analysis of the disagreement coefficient for decision trees-a key parameter governing active learning label complexity. Our analysis holds under two natural assumptions required for achieving polylogarithmic label complexity, (i) each root-to-leaf path queries distinct feature dimensions, and (ii) the input data has a regular, grid-like structure. We show these assumptions are essential, as relaxing them leads to polynomial label complexity. Second, we present the first general active learning algorithm for binary classification that achieves a multiplicative error guarantee, producing a $(1+ε)$-approximate classifier. By combining these results, we design an active learning algorithm for decision trees that uses only a polylogarithmic number of label queries in the dataset size, under the stated assumptions. Finally, we establish a label complexity lower bound, showing our algorithm's dependence on the error tolerance $ε$ is close to optimal.

A Dynamic Algorithm for Weighted Submodular Cover Problem

We initiate the study of the submodular cover problem in dynamic setting where the elements of the ground set are inserted and deleted. In the classical submodular cover problem, we are given a monotone submodular function $f : 2^{V} \to \mathbb{R}^{\ge 0}$ and the goal is to obtain a set $S \subseteq V$ that minimizes the cost subject to the constraint $f(S) = f(V)$. This is a classical problem in computer science and generalizes the Set Cover problem, 2-Set Cover, and dominating set problem among others. We consider this problem in a dynamic setting where there are updates to our set $V$, in the form of insertions and deletions of elements from a ground set $\mathcal{V}$, and the goal is to maintain an approximately optimal solution with low query complexity per update. For this problem, we propose a randomized algorithm that, in expectation, obtains a $(1-O(\epsilon), O(\epsilon^{-1}))$-bicriteria approximation using polylogarithmic query complexity per update.

Dynamic Non-monotone Submodular Maximization

Maximizing submodular functions has been increasingly used in many applications of machine learning, such as data summarization, recommendation systems, and feature selection. Moreover, there has been a growing interest in both submodular maximization and dynamic algorithms. In 2020, Monemizadeh and Lattanzi, Mitrovic, Norouzi{-}Fard, Tarnawski, and Zadimoghaddam initiated developing dynamic algorithms for the monotone submodular maximization problem under the cardinality constraint $k$. Recently, there have been some improvements on the topic made by Banihashem, Biabani, Goudarzi, Hajiaghayi, Jabbarzade, and Monemizadeh. In 2022, Chen and Peng studied the complexity of this problem and raised an important open question: "Can we extend [fully dynamic] results (algorithm or hardness) to non-monotone submodular maximization?". We affirmatively answer their question by demonstrating a reduction from maximizing a non-monotone submodular function under the cardinality constraint $k$ to maximizing a monotone submodular function under the same constraint. Through this reduction, we obtain the first dynamic algorithms to solve the non-monotone submodular maximization problem under the cardinality constraint $k$. Our algorithms maintain an $(8+\epsilon)$-approximate of the solution and use expected amortized $O(\epsilon^{-3}k^3\log^3(n)\log(k))$ or $O(\epsilon^{-1}k^2\log^3(k))$ oracle queries per update, respectively. Furthermore, we showcase the benefits of our dynamic algorithm for video summarization and max-cut problems on several real-world data sets.

An Improved Relaxation for Oracle-Efficient Adversarial Contextual Bandits

We present an oracle-efficient relaxation for the adversarial contextual bandits problem, where the contexts are sequentially drawn i.i.d from a known distribution and the cost sequence is chosen by an online adversary. Our algorithm has a regret bound of $O(T^{\frac{2}{3}}(K\log(|\Pi|))^{\frac{1}{3}})$ and makes at most $O(K)$ calls per round to an offline optimization oracle, where $K$ denotes the number of actions, $T$ denotes the number of rounds and $\Pi$ denotes the set of policies. This is the first result to improve the prior best bound of $O((TK)^{\frac{2}{3}}(\log(|\Pi|))^{\frac{1}{3}})$ as obtained by Syrgkanis et al. at NeurIPS 2016, and the first to match the original bound of Langford and Zhang at NeurIPS 2007 which was obtained for the stochastic case.

Dynamic Algorithms for Matroid Submodular Maximization

Submodular maximization under matroid and cardinality constraints are classical problems with a wide range of applications in machine learning, auction theory, and combinatorial optimization. In this paper, we consider these problems in the dynamic setting where (1) we have oracle access to a monotone submodular function $f: 2^{V} \rightarrow \mathbb{R}^+$ and (2) we are given a sequence $\mathcal{S}$ of insertions and deletions of elements of an underlying ground set $V$. We develop the first parameterized (by the rank $k$ of a matroid $\mathcal{M}$) dynamic $(4+\epsilon)$-approximation algorithm for the submodular maximization problem under the matroid constraint using an expected worst-case $O(k\log(k)\log^3{(k/\epsilon)})$ query complexity where $0 < \epsilon \le 1$. Chen and Peng at STOC'22 studied the complexity of this problem in the insertion-only dynamic model (a restricted version of the fully dynamic model where deletion is not allowed), and they raised the following important open question: *"for fully dynamic streams [sequences of insertions and deletions of elements], there is no known constant-factor approximation algorithm with poly(k) amortized queries for matroid constraints."* Our dynamic algorithm answers this question as well as an open problem of Lattanzi et al. (NeurIPS'20) affirmatively. As a byproduct, for the submodular maximization under the cardinality constraint $k$, we propose a parameterized (by the cardinality constraint $k$) dynamic algorithm that maintains a $(2+\epsilon)$-approximate solution of the sequence $\mathcal{S}$ at any time $t$ using the expected amortized worst-case complexity $O(k\epsilon^{-1}\log^2(k))$. This is the first dynamic algorithm for the problem that has a query complexity independent of the size of ground set $V$.

Optimal Sparse Recovery with Decision Stumps

Decision trees are widely used for their low computational cost, good predictive performance, and ability to assess the importance of features. Though often used in practice for feature selection, the theoretical guarantees of these methods are not well understood. We here obtain a tight finite sample bound for the feature selection problem in linear regression using single-depth decision trees. We examine the statistical properties of these "decision stumps" for the recovery of the $s$ active features from $p$ total features, where $s \ll p$. Our analysis provides tight sample performance guarantees on high-dimensional sparse systems which align with the finite sample bound of $O(s \log p)$ as obtained by Lasso, improving upon previous bounds for both the median and optimal splitting criteria. Our results extend to the non-linear regime as well as arbitrary sub-Gaussian distributions, demonstrating that tree based methods attain strong feature selection properties under a wide variety of settings and further shedding light on the success of these methods in practice. As a byproduct of our analysis, we show that we can provably guarantee recovery even when the number of active features $s$ is unknown. We further validate our theoretical results and proof methodology using computational experiments.

Bandit Social Learning: Exploration under Myopic Behavior

We study social learning dynamics where the agents collectively follow a simple multi-armed bandit protocol. Agents arrive sequentially, choose arms and receive associated rewards. Each agent observes the full history (arms and rewards) of the previous agents, and there are no private signals. While collectively the agents face exploration-exploitation tradeoff, each agent acts myopically, without regards to exploration. Motivating scenarios concern reviews and ratings on online platforms. We allow a wide range of myopic behaviors that are consistent with (parameterized) confidence intervals, including the "unbiased" behavior as well as various behaviorial biases. While extreme versions of these behaviors correspond to well-known bandit algorithms, we prove that more moderate versions lead to stark exploration failures, and consequently to regret rates that are linear in the number of agents. We provide matching upper bounds on regret by analyzing "moderately optimistic" agents. As a special case of independent interest, we obtain a general result on failure of the greedy algorithm in multi-armed bandits. This is the first such result in the literature, to the best of our knowledge

Run-Off Election: Improved Provable Defense against Data Poisoning Attacks

In data poisoning attacks, an adversary tries to change a model's prediction by adding, modifying, or removing samples in the training data. Recently, ensemble-based approaches for obtaining provable defenses against data poisoning have been proposed where predictions are done by taking a majority vote across multiple base models. In this work, we show that merely considering the majority vote in ensemble defenses is wasteful as it does not effectively utilize available information in the logits layers of the base models. Instead, we propose Run-Off Election (ROE), a novel aggregation method based on a two-round election across the base models: In the first round, models vote for their preferred class and then a second, Run-Off election is held between the top two classes in the first round. Based on this approach, we propose DPA+ROE and FA+ROE defense methods based on Deep Partition Aggregation (DPA) and Finite Aggregation (FA) approaches from prior work. We show how to obtain robustness for these methods using ideas inspired by dynamic programming and duality. We evaluate our methods on MNIST, CIFAR-10, and GTSRB and obtain improvements in certified accuracy by up to 4.73%, 3.63%, and 3.54%, respectively, establishing a new state-of-the-art in (pointwise) certified robustness against data poisoning. In many cases, our approach outperforms the state-of-the-art, even when using 32 times less computational power.