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Kasper Green Larsen

Publications and source records attributed to Kasper Green Larsen.

At least 19 recordsLinked to original sources
arXiv2026

The Binary Tree Mechanism is Optimal for Differentially Private Continual Counting

Private continual counting is a fundamental problem in differential privacy: given a binary stream of length $n$, where each $1$ corresponds to the contribution of one individual, the goal is to release all running counts while protecting the privacy of each individual. For fixed privacy parameters, the standard binary tree mechanism achieves expected $\ell_\infty$ error $O(\log^{3/2} n)$ under approximate differential privacy and $O(\log^2 n)$ under pure differential privacy. Whether these dependences on the stream length are necessary has remained a central open problem. For fixed $\varepsilon\in(0,1)$, we prove a lower bound of $Ω(\log^{3/2} n)$ under approximate DP with sufficiently small fixed $δ>0$, and a lower bound of $Ω(\log^2 n)$ under pure DP. These bounds establish the optimality of the binary tree mechanism in both settings. The bounds hold for arbitrary mechanisms, even when the entire stream is available in advance. Both proofs use the same decomposition and accumulation of residual noise along a tree. As a consequence of the approximate-DP bound, we also obtain a largest-possible separation between hereditary discrepancy and private $\ell_\infty$ error for linear queries, showing that the known general upper bound in terms of hereditary discrepancy has the optimal dependence on the number of queries.

cs.DS↗
arXiv2026

The Interplay Between Interpolation and Aggregation in Regression: Optimal Sample Complexity

This work investigates theoretically the interplay between interpolation and aggregation in regression. We establish that the $γ$-graph dimension characterizes learnability for a broad class of natural aggregation procedures. Furthermore, we prove that an extremely simple aggregation procedure, combining three interpolating hypotheses via the median, is optimal among all these aggregation procedures, and is strictly more powerful than proper learning. Finally, we show that some hypothesis classes are learnable only by aggregating infinitely many hypotheses or by using non-interpolating aggregation rules (which may predict outside the range of their inputs), and any finite interpolating aggregation fails to achieve even trivial performance.

cs.LG↗
arXiv2026

Actively Learning Halfspaces without Synthetic Data

In the classic point location problem, one is given an arbitrary dataset $X \subset \mathbb{R}^d$ of $n$ points with query access to an unknown halfspace $f : \mathbb{R}^d \to \{0,1\}$, and the goal is to learn the label of every point in $X$. This problem is extremely well-studied and a nearly-optimal $\widetilde{O}(d \log n)$ query algorithm is known due to Hopkins-Kane-Lovett-Mahajan (FOCS 2020). However, their algorithm is granted the power to query arbitrary points outside of $X$ (point synthesis), and in fact without this power there is an $Ω(n)$ query lower bound due to Dasgupta (NeurIPS 2004). In this work our goal is to design efficient algorithms for learning halfspaces without point synthesis. To circumvent the $Ω(n)$ lower bound, we consider learning halfspaces whose normal vectors come from a set of size $D$, and show tight bounds of $Θ(D + \log n)$. As a corollary, we obtain an optimal $O(d + \log n)$ query deterministic learner for axis-aligned halfspaces, closing a previous gap of $O(d \log n)$ vs. $Ω(d + \log n)$. In fact, our algorithm solves the more general problem of learning a Boolean function $f$ over $n$ elements which is monotone under at least one of $D$ provided orderings. Our technical insight is to exploit the structure in these orderings to perform a binary search in parallel rather than considering each ordering sequentially, and we believe our approach may be of broader interest. Furthermore, we use our exact learning algorithm to obtain nearly optimal algorithms for PAC-learning. We show that $O(\min(D + \log(1/\varepsilon), 1/\varepsilon) \cdot \log D)$ queries suffice to learn $f$ within error $\varepsilon$, even in a setting when $f$ can be adversarially corrupted on a $c\varepsilon$-fraction of points, for a sufficiently small constant $c$. This bound is optimal up to a $\log D$ factor, including in the realizable setting.

cs.DS↗
arXiv2026

Learning with Monotone Adversarial Corruptions

We study the extent to which standard machine learning algorithms rely on exchangeability and independence of data by introducing a monotone adversarial corruption model. In this model, an adversary, upon looking at a "clean" i.i.d. dataset, inserts additional "corrupted" points of their choice into the dataset. These added points are constrained to be monotone corruptions, in that they get labeled according to the ground-truth target function. Perhaps surprisingly, we demonstrate that in this setting, all known optimal learning algorithms for binary classification can be made to achieve suboptimal expected error on a new independent test point drawn from the same distribution as the clean dataset. On the other hand, we show that uniform convergence-based algorithms do not degrade in their guarantees. Our results showcase how optimal learning algorithms break down in the face of seemingly helpful monotone corruptions, exposing their overreliance on exchangeability.

cs.LG↗
arXiv2026

A Fine-Grained Understanding of Uniform Convergence for Halfspaces

We study the fine-grained uniform convergence behavior of halfspaces beyond worst-case VC bounds. For inhomogeneous halfspaces in $\mathbb{R}^d$ with $d\ge 2$, we show that standard first-order VC bounds are essentially tight: even consistent hypotheses can incur population error $Θ(d\ln(n/d)/n)$, and in the agnostic setting the deviation scales as $\sqrt{τ\ln(1/τ)}$ at true error $τ$. In contrast, homogeneous halfspaces in $\mathbb{R}^2$ exhibit a markedly different behavior. In the realizable case, every hypothesis consistent with the sample has error $O(1/n)$. In the agnostic case, we prove a bandwise, log-free deviation bound on each dyadic risk band via a critical-wedge localization argument. Unioning over bands incurs only a $\ln\ln n$ overhead, and we establish a matching lower bound showing this overhead is unavoidable. Together, these results give a fine-grained and nearly complete picture of uniform convergence for halfspaces, revealing sharp dimensional and structural thresholds.

cs.LG↗
arXiv2026

The Sample Complexity of Replicable Realizable PAC Learning

In this paper, we consider the problem of replicable realizable PAC learning. We construct a particularly hard learning problem and show a sample complexity lower bound with a close to $(\log|H|)^{3/2}$ dependence on the size of the hypothesis class $H$. Our proof uses several novel techniques and works by defining a particular Cayley graph associated with $H$ and analyzing a suitable random walk on this graph by examining the spectral properties of its adjacency matrix. Furthermore, we show an almost matching upper bound for the lower bound instance, meaning if a stronger lower bound exists, one would have to consider a different instance of the problem.

cs.LG↗
arXiv2025

An Exponential Separation Between Quantum and Quantum-Inspired Classical Algorithms for Linear Systems

Achieving a provable exponential quantum speedup for an important machine learning task has been a central research goal since the seminal HHL quantum algorithm for solving linear systems and the subsequent quantum recommender systems algorithm by Kerenidis and Prakash. These algorithms were initially believed to be strong candidates for exponential speedups, but a lower bound ruling out similar classical improvements remained absent. In breakthrough work by Tang, it was demonstrated that this lack of progress in classical lower bounds was for good reasons. Concretely, she gave a classical counterpart of the quantum recommender systems algorithm, reducing the quantum advantage to a mere polynomial. Her approach is quite general and was named quantum-inspired classical algorithms. Since then, almost all the initially exponential quantum machine learning speedups have been reduced to polynomial via new quantum-inspired classical algorithms. From the current state-of-affairs, it is unclear whether we can hope for exponential quantum speedups for any natural machine learning task. In this work, we present the first such provable exponential separation between quantum and quantum-inspired classical algorithms for the basic problem of solving a linear system when the input matrix is well-conditioned and has sparse rows and columns.

quant-ph↗
arXiv2025

Optimal Parallelization of Boosting

Recent works on the parallel complexity of Boosting have established strong lower bounds on the tradeoff between the number of training rounds $p$ and the total parallel work per round $t$. These works have also presented highly non-trivial parallel algorithms that shed light on different regions of this tradeoff. Despite these advancements, a significant gap persists between the theoretical lower bounds and the performance of these algorithms across much of the tradeoff space. In this work, we essentially close this gap by providing both improved lower bounds on the parallel complexity of weak-to-strong learners, and a parallel Boosting algorithm whose performance matches these bounds across the entire $p$ vs.~$t$ compromise spectrum, up to logarithmic factors. Ultimately, this work settles the true parallel complexity of Boosting algorithms that are nearly sample-optimal.

cs.LG↗
arXiv2025

AdaBoost is not an Optimal Weak to Strong Learner

AdaBoost is a classic boosting algorithm for combining multiple inaccurate classifiers produced by a weak learner, to produce a strong learner with arbitrarily high accuracy when given enough training data. Determining the optimal number of samples necessary to obtain a given accuracy of the strong learner, is a basic learning theoretic question. Larsen and Ritzert (NeurIPS'22) recently presented the first provably optimal weak-to-strong learner. However, their algorithm is somewhat complicated and it remains an intriguing question whether the prototypical boosting algorithm AdaBoost also makes optimal use of training samples. In this work, we answer this question in the negative. Concretely, we show that the sample complexity of AdaBoost, and other classic variations thereof, are sub-optimal by at least one logarithmic factor in the desired accuracy of the strong learner.

cs.LG↗
arXiv2025

Improved Margin Generalization Bounds for Voting Classifiers

In this paper we establish a new margin-based generalization bound for voting classifiers, refining existing results and yielding tighter generalization guarantees for widely used boosting algorithms such as AdaBoost (Freund and Schapire, 1997). Furthermore, the new margin-based generalization bound enables the derivation of an optimal weak-to-strong learner: a Majority-of-3 large-margin classifiers with an expected error matching the theoretical lower bound. This result provides a more natural alternative to the Majority-of-5 algorithm by (Høgsgaard et al., 2024), and matches the Majority-of-3 result by (Aden-Ali et al., 2024) for the realizable prediction model.

cs.LG↗
arXiv2025

Approximating Klee's Measure Problem and a Lower Bound for Union Volume Estimation

Union volume estimation is a classical algorithmic problem. Given a family of objects $O_1,\ldots,O_n \subseteq \mathbb{R}^d$, we want to approximate the volume of their union. In the special case where all objects are boxes (also known as hyperrectangles) this is known as Klee's measure problem. The state-of-the-art algorithm [Karp, Luby, Madras '89] for union volume estimation and Klee's measure problem in constant dimension $d$ computes a $(1+\varepsilon)$-approximation with constant success probability by using a total of $O(n/\varepsilon^2)$ queries of the form (i) ask for the volume of $O_i$, (ii) sample a point uniformly at random from $O_i$, and (iii) query whether a given point is contained in $O_i$. We show that if one can only interact with the objects via the aforementioned three queries, the query complexity of [Karp, Luby, Madras '89] is indeed optimal, i.e., $Ω(n/\varepsilon^2)$ queries are necessary. Our lower bound already holds for estimating the union of equiponderous axis-aligned polygons in $\mathbb{R}^2$, and even if the algorithm is allowed to inspect the coordinates of the points sampled from the polygons, and still holds when a containment query can ask containment of an arbitrary (not sampled) point. Guided by the insights of the lower bound, we provide a more efficient approximation algorithm for Klee's measure problem improving the $O(n/\varepsilon^2)$ time to $O((n+\frac{1}{\varepsilon^2}) \cdot \log^{O(d)}n)$. We achieve this improvement by exploiting the geometry of Klee's measure problem in various ways: (1) Since we have access to the boxes' coordinates, we can split the boxes into classes of boxes of similar shape. (2) Within each class, we show how to sample from the union of all boxes, by using orthogonal range searching. And (3) we exploit that boxes of different classes have small intersection, for most pairs of classes.

cs.CG↗
arXiv2025

Tight Generalization Bounds for Large-Margin Halfspaces

We prove the first generalization bound for large-margin halfspaces that is asymptotically tight in the tradeoff between the margin, the fraction of training points with the given margin, the failure probability and the number of training points.

cs.LG↗
arXiv2025

A new lower bound for multi-color discrepancy with applications to fair division

A classical problem in combinatorics seeks colorings of low discrepancy. More concretely, the goal is to color the elements of a set system so that the number of appearances of any color among the elements in each set is as balanced as possible. We present a new lower bound for multi-color discrepancy, showing that there is a set system with $n$ subsets over a set of elements in which any $k$-coloring of the elements has discrepancy at least $Ω\left(\sqrt{\frac{n}{\ln{k}}}\right)$. This result improves the previously best-known lower bound of $Ω\left(\sqrt{\frac{n}{k}}\right)$ of Doerr and Srivastav [2003] and may have several applications. Here, we explore its implications on the feasibility of fair division concepts for instances with $n$ agents having valuations for a set of indivisible items. The first such concept is known as consensus $1/k$-division up to $d$ items (\cd$d$) and aims to allocate the items into $k$ bundles so that no matter which bundle each agent is assigned to, the allocation is envy-free up to $d$ items. The above lower bound implies that \cd$d$ can be infeasible for $d\in Ω\left(\sqrt{\frac{n}{\ln{k}}}\right)$. We furthermore extend our proof technique to show that there exist instances of the problem of allocating indivisible items to $k$ groups of $n$ agents in total so that envy-freeness and proportionality up to $d$ items are infeasible for $d\in Ω\left(\sqrt{\frac{n}{k\ln{k}}}\right)$ and $d\in Ω\left(\sqrt{\frac{n}{k^3\ln{k}}}\right)$, respectively. The lower bounds for fair division improve the currently best-known ones by Manurangsi and Suksompong [2022].

cs.GT↗
arXiv2025

Improved Replicable Boosting with Majority-of-Majorities

We introduce a new replicable boosting algorithm which significantly improves the sample complexity compared to previous algorithms. The algorithm works by doing two layers of majority voting, using an improved version of the replicable boosting algorithm introduced by Impagliazzo et al. [2022] in the bottom layer.

cs.LG↗
arXiv2024

Boosting, Voting Classifiers and Randomized Sample Compression Schemes

In boosting, we aim to leverage multiple weak learners to produce a strong learner. At the center of this paradigm lies the concept of building the strong learner as a voting classifier, which outputs a weighted majority vote of the weak learners. While many successful boosting algorithms, such as the iconic AdaBoost, produce voting classifiers, their theoretical performance has long remained sub-optimal: The best known bounds on the number of training examples necessary for a voting classifier to obtain a given accuracy has so far always contained at least two logarithmic factors above what is known to be achievable by general weak-to-strong learners. In this work, we break this barrier by proposing a randomized boosting algorithm that outputs voting classifiers whose generalization error contains a single logarithmic dependency on the sample size. We obtain this result by building a general framework that extends sample compression methods to support randomized learning algorithms based on sub-sampling.

cs.LG↗
arXiv2024

Invertible Bloom Lookup Tables with Less Memory and Randomness

In this work we study Invertible Bloom Lookup Tables (IBLTs) with small failure probabilities. IBLTs are highly versatile data structures that have found applications in set reconciliation protocols, error-correcting codes, and even the design of advanced cryptographic primitives. For storing $n$ elements and ensuring correctness with probability at least $1 - δ$, existing IBLT constructions require $Ω(n(\frac{\log(1/δ)}{\log(n)}+1))$ space and they crucially rely on fully random hash functions. We present new constructions of IBLTs that are simultaneously more space efficient and require less randomness. For storing $n$ elements with a failure probability of at most $δ$, our data structure only requires $\mathcal{O}\left(n + \log(1/δ)\log\log(1/δ)\right)$ space and $\mathcal{O}\left(\log(\log(n)/δ)\right)$-wise independent hash functions. As a key technical ingredient we show that hashing $n$ keys with any $k$-wise independent hash function $h:U \to [Cn]$ for some sufficiently large constant $C$ guarantees with probability $1 - 2^{-Ω(k)}$ that at least $n/2$ keys will have a unique hash value. Proving this is non-trivial as $k$ approaches $n$. We believe that the techniques used to prove this statement may be of independent interest. We apply our new IBLTs to the encrypted compression problem, recently studied by Fleischhacker, Larsen, Simkin (Eurocrypt 2023). We extend their approach to work for a more general class of encryption schemes and using our new IBLT we achieve an asymptotically better compression rate.

cs.DS↗
arXiv2024

The NFA Acceptance Hypothesis: Non-Combinatorial and Dynamic Lower Bounds

We pose the fine-grained hardness hypothesis that the textbook algorithm for the NFA Acceptance problem is optimal up to subpolynomial factors, even for dense NFAs and fixed alphabets. We show that this barrier appears in many variations throughout the algorithmic literature by introducing a framework of Colored Walk problems. These yield fine-grained equivalent formulations of the NFA Acceptance problem as problems concerning detection of an $s$-$t$-walk with a prescribed color sequence in a given edge- or node-colored graph. For NFA Acceptance on sparse NFAs (or equivalently, Colored Walk in sparse graphs), a tight lower bound under the Strong Exponential Time Hypothesis has been rediscovered several times in recent years. We show that our hardness hypothesis, which concerns dense NFAs, has several interesting implications: - It gives a tight lower bound for Context-Free Language Reachability. This proves conditional optimality for the class of 2NPDA-complete problems, explaining the cubic bottleneck of interprocedural program analysis. - It gives a tight $(n+nm^{1/3})^{1-o(1)}$ lower bound for the Word Break problem on strings of length $n$ and dictionaries of total size $m$. - It implies the popular OMv hypothesis. Since the NFA acceptance problem is a static (i.e., non-dynamic) problem, this provides a static reason for the hardness of many dynamic problems. Thus, a proof of the NFA Acceptance hypothesis would resolve several interesting barriers. Conversely, a refutation of the NFA Acceptance hypothesis may lead the way to attacking the current barriers observed for Context-Free Language Reachability, the Word Break problem and the growing list of dynamic problems proven hard under the OMv hypothesis.

cs.CC↗