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Researcher profile

Haim Kaplan

Haim Kaplan contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
Computational GeometryData Structures and AlgorithmsMachine LearningInformation RetrievalArtificial IntelligenceComputer Science and Game TheoryCryptography and Securitymath.OC

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Published work

19 published item(s)

preprint2026arXiv

Cost-Aware Learning

We consider the problem of Cost-Aware Learning, where sampling different component functions of a finite-sum objective incurs different costs. The objective is to reach a target error while minimizing the total cost. First, we propose the Cost-Aware Stochastic Gradient Descent algorithm for convex functions, and derive its cost complexity to attain an error of $ε$. Furthermore, we establish a lower bound for this setting and provide a subset selection algorithm to further reduce the cost of training. We apply our theoretical insights to reinforcement learning with language models, where the computational cost of policy gradients varies with sequence length. To this end, we introduce Cost-Aware GRPO, an algorithm designed to reduce the cost of policy optimization while preserving performance. Empirical results on 1.5B and 8B LLMs demonstrate that our approach reduces the tokens used in policy optimization by up to about 30% while matching or exceeding baseline accuracy.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Locality Sensitive Hashing for Efficient Similar Polygon Retrieval

Locality Sensitive Hashing (LSH) is an effective method of indexing a set of items to support efficient nearest neighbors queries in high-dimensional spaces. The basic idea of LSH is that similar items should produce hash collisions with higher probability than dissimilar items. We study LSH for (not necessarily convex) polygons, and use it to give efficient data structures for similar shape retrieval. Arkin et al. represent polygons by their "turning function" - a function which follows the angle between the polygon's tangent and the $ x $-axis while traversing the perimeter of the polygon. They define the distance between polygons to be variations of the $ L_p $ (for $p=1,2$) distance between their turning functions. This metric is invariant under translation, rotation and scaling (and the selection of the initial point on the perimeter) and therefore models well the intuitive notion of shape resemblance. We develop and analyze LSH near neighbor data structures for several variations of the $ L_p $ distance for functions (for $p=1,2$). By applying our schemes to the turning functions of a collection of polygons we obtain efficient near neighbor LSH-based structures for polygons. To tune our structures to turning functions of polygons, we prove some new properties of these turning functions that may be of independent interest. As part of our analysis, we address the following problem which is of independent interest. Find the vertical translation of a function $ f $ that is closest in $ L_1 $ distance to a function $ g $. We prove tight bounds on the approximation guarantee obtained by the translation which is equal to the difference between the averages of $ g $ and $ f $.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Online Markov Decision Processes with Aggregate Bandit Feedback

We study a novel variant of online finite-horizon Markov Decision Processes with adversarially changing loss functions and initially unknown dynamics. In each episode, the learner suffers the loss accumulated along the trajectory realized by the policy chosen for the episode, and observes aggregate bandit feedback: the trajectory is revealed along with the cumulative loss suffered, rather than the individual losses encountered along the trajectory. Our main result is a computationally efficient algorithm with $O(\sqrt{K})$ regret for this setting, where $K$ is the number of episodes. We establish this result via an efficient reduction to a novel bandit learning setting we call Distorted Linear Bandits (DLB), which is a variant of bandit linear optimization where actions chosen by the learner are adversarially distorted before they are committed. We then develop a computationally-efficient online algorithm for DLB for which we prove an $O(\sqrt{T})$ regret bound, where $T$ is the number of time steps. Our algorithm is based on online mirror descent with a self-concordant barrier regularization that employs a novel increasing learning rate schedule.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Separating Adaptive Streaming from Oblivious Streaming

We present a streaming problem for which every adversarially-robust streaming algorithm must use polynomial space, while there exists a classical (oblivious) streaming algorithm that uses only polylogarithmic space. This is the first separation between oblivious streaming and adversarially-robust streaming, and resolves one of the central open questions in adversarial robust streaming.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Adversarially Robust Streaming Algorithms via Differential Privacy

A streaming algorithm is said to be adversarially robust if its accuracy guarantees are maintained even when the data stream is chosen maliciously, by an adaptive adversary. We establish a connection between adversarial robustness of streaming algorithms and the notion of differential privacy. This connection allows us to design new adversarially robust streaming algorithms that outperform the current state-of-the-art constructions for many interesting regimes of parameters.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Duality-based approximation algorithms for depth queries and maximum depth

We design an efficient data structure for computing a suitably defined approximate depth of any query point in the arrangement $\mathcal{A}(S)$ of a collection $S$ of $n$ halfplanes or triangles in the plane or of halfspaces or simplices in higher dimensions. We then use this structure to find a point of an approximate maximum depth in $\mathcal{A}(S)$. Specifically, given an error parameter $ε>0$, we compute, for any query point $q$, an underestimate $d^-(q)$ of the depth of $q$, that counts only objects containing $q$, but is allowed to exclude objects when $q$ is $ε$-close to their boundary. Similarly, we compute an overestimate $d^+(q)$ that counts all objects containing $q$ but may also count objects that do not contain $q$ but $q$ is $ε$-close to their boundary. Our algorithms for halfplanes and halfspaces are linear in the number of input objects and in the number of queries, and the dependence of their running time on $ε$ is considerably better than that of earlier techniques. Our improvements are particularly substantial for triangles and in higher dimensions.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

How to Find a Point in the Convex Hull Privately

We study the question of how to compute a point in the convex hull of an input set $S$ of $n$ points in ${\mathbb R}^d$ in a differentially private manner. This question, which is trivial non-privately, turns out to be quite deep when imposing differential privacy. In particular, it is known that the input points must reside on a fixed finite subset $G\subseteq{\mathbb R}^d$, and furthermore, the size of $S$ must grow with the size of $G$. Previous works focused on understanding how $n$ needs to grow with $|G|$, and showed that $n=O\left(d^{2.5}\cdot8^{\log^*|G|}\right)$ suffices (so $n$ does not have to grow significantly with $|G|$). However, the available constructions exhibit running time at least $|G|^{d^2}$, where typically $|G|=X^d$ for some (large) discretization parameter $X$, so the running time is in fact $Ω(X^{d^3})$. In this paper we give a differentially private algorithm that runs in $O(n^d)$ time, assuming that $n=Ω(d^4\log X)$. To get this result we study and exploit some structural properties of the Tukey levels (the regions $D_{\ge k}$ consisting of points whose Tukey depth is at least $k$, for $k=0,1,...$). In particular, we derive lower bounds on their volumes for point sets $S$ in general position, and develop a rather subtle mechanism for handling point sets $S$ in degenerate position (where the deep Tukey regions have zero volume). A naive approach to the construction of the Tukey regions requires $n^{O(d^2)}$ time. To reduce the cost to $O(n^d)$, we use an approximation scheme for estimating the volumes of the Tukey regions (within their affine spans in case of degeneracy), and for sampling a point from such a region, a scheme that is based on the volume estimation framework of Lovász and Vempala (FOCS 2003) and of Cousins and Vempala (STOC 2015). Making this framework differentially private raises a set of technical challenges that we address.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Locality Sensitive Hashing for Set-Queries, Motivated by Group Recommendations

Locality Sensitive Hashing (LSH) is an effective method to index a set of points such that we can efficiently find the nearest neighbors of a query point. We extend this method to our novel Set-query LSH (SLSH), such that it can find the nearest neighbors of a set of points, given as a query. Let $ s(x,y) $ be the similarity between two points $ x $ and $ y $. We define a similarity between a set $ Q$ and a point $ x $ by aggregating the similarities $ s(p,x) $ for all $ p\in Q $. For example, we can take $ s(p,x) $ to be the angular similarity between $ p $ and $ x $ (i.e., $1-{\angle (x,p)}/π$), and aggregate by arithmetic or geometric averaging, or taking the lowest similarity. We develop locality sensitive hash families and data structures for a large set of such arithmetic and geometric averaging similarities, and analyze their collision probabilities. We also establish an analogous framework and hash families for distance functions. Specifically, we give a structure for the euclidean distance aggregated by either averaging or taking the maximum. We leverage SLSH to solve a geometric extension of the approximate near neighbors problem. In this version, we consider a metric for which the unit ball is an ellipsoid and its orientation is specified with the query. An important application that motivates our work is group recommendation systems. Such a system embeds movies and users in the same feature space, and the task of recommending a movie for a group to watch together, translates to a set-query $ Q $ using an appropriate similarity.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Near-optimal Regret Bounds for Stochastic Shortest Path

Stochastic shortest path (SSP) is a well-known problem in planning and control, in which an agent has to reach a goal state in minimum total expected cost. In the learning formulation of the problem, the agent is unaware of the environment dynamics (i.e., the transition function) and has to repeatedly play for a given number of episodes while reasoning about the problem's optimal solution. Unlike other well-studied models in reinforcement learning (RL), the length of an episode is not predetermined (or bounded) and is influenced by the agent's actions. Recently, Tarbouriech et al. (2019) studied this problem in the context of regret minimization and provided an algorithm whose regret bound is inversely proportional to the square root of the minimum instantaneous cost. In this work we remove this dependence on the minimum cost---we give an algorithm that guarantees a regret bound of $\widetilde{O}(B_\star |S| \sqrt{|A| K})$, where $B_\star$ is an upper bound on the expected cost of the optimal policy, $S$ is the set of states, $A$ is the set of actions and $K$ is the number of episodes. We additionally show that any learning algorithm must have at least $Ω(B_\star \sqrt{|S| |A| K})$ regret in the worst case.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

On Radial Isotropic Position: Theory and Algorithms

We review the theory of, and develop algorithms for transforming a finite point set in ${\bf R}^d$ into a set in \emph{radial isotropic position} by a nonsingular linear transformation followed by rescaling each image point to the unit sphere. This problem arises in a wide spectrum of applications in computer science and mathematics. Our algorithms use gradient descent methods for a particular convex function $f$ whose minimum defines the transformation, and our main focus is on analyzing their performance. Although the minimum can be computed exactly, by expensive symbolic algebra techniques, gradient descent only approximates the desired minimum to any desired level of accuracy. We show that computing the gradient of $f$ amounts to computing the Singular Value Decomposition (SVD) of a certain matrix associated with the input set, making it simple to implement. We believe it to be superior to other approximate techniques (mainly the ellipsoid algorithm) used previously to find this transformation, and it should run much faster in practice. We prove that $f$ is smooth, which yields convergence rate proportional to $1/ε$, where $ε$ is the desired approximation accuracy. To complete the analysis, we provide upper bounds on the norm of the optimal solution which depend on new parameters measuring "the degeneracy" in our input. We believe that our parameters capture degeneracy better than other, seemingly weaker, parameters used in previous works. We next analyze the strong convexity of $f$, and present two worst-case lower bounds on the smallest eigenvalue of its Hessian. This gives another worst-case bound on the convergence rate of another variant of gradient decent that depends only logarithmically on $1/ε$.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Output sensitive algorithms for approximate incidences and their applications

An $ε$-approximate incidence between a point and some geometric object (line, circle, plane, sphere) occurs when the point and the object lie at distance at most $ε$ from each other. Given a set of points and a set of objects, computing the approximate incidences between them is a major step in many database and web-based applications in computer vision and graphics, including robust model fitting, approximate point pattern matching, and estimating the fundamental matrix in epipolar (stereo) geometry. In a typical approximate incidence problem of this sort, we are given a set $P$ of $m$ points in two or three dimensions, a set $S$ of $n$ objects (lines, circles, planes, spheres), and an error parameter $ε>0$, and our goal is to report all pairs $(p,s)\in P\times S$ that lie at distance at most $ε$ from one another. We present efficient output-sensitive approximation algorithms for quite a few cases, including points and lines or circles in the plane, and points and planes, spheres, lines, or circles in three dimensions. Several of these cases arise in the applications mentioned above.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Planning in Hierarchical Reinforcement Learning: Guarantees for Using Local Policies

We consider a settings of hierarchical reinforcement learning, in which the reward is a sum of components. For each component we are given a policy that maximizes it and our goal is to assemble a policy from the individual policies that maximizes the sum of the components. We provide theoretical guarantees for assembling such policies in deterministic MDPs with collectible rewards. Our approach builds on formulating this problem as a traveling salesman problem with discounted reward. We focus on local solutions, i.e., policies that only use information from the current state; thus, they are easy to implement and do not require substantial computational resources. We propose three local stochastic policies and prove that they guarantee better performance than any deterministic local policy in the worst case; experimental results suggest that they also perform better on average.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Unknown mixing times in apprenticeship and reinforcement learning

We derive and analyze learning algorithms for apprenticeship learning, policy evaluation, and policy gradient for average reward criteria. Existing algorithms explicitly require an upper bound on the mixing time. In contrast, we build on ideas from Markov chain theory and derive sampling algorithms that do not require such an upper bound. For these algorithms, we provide theoretical bounds on their sample-complexity and running time.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Voronoi diagrams on planar graphs, and computing the diameter in deterministic $\tilde{O}(n^{5/3})$ time

We present an explicit and efficient construction of additively weighted Voronoi diagrams on planar graphs. Let $G$ be a planar graph with $n$ vertices and $b$ sites that lie on a constant number of faces. We show how to preprocess $G$ in $\tilde O(nb^2)$ time (footnote: The $\tilde O$ notation hides polylogarithmic factors.) so that one can compute any additively weighted Voronoi diagram for these sites in $\tilde O(b)$ time. We use this construction to compute the diameter of a directed planar graph with real arc lengths in $\tilde{O}(n^{5/3})$ time. This improves the recent breakthrough result of Cabello (SODA'17), both by improving the running time (from $\tilde{O}(n^{11/6})$), and by providing a deterministic algorithm. It is in fact the first truly subquadratic {\em deterministic} algorithm for this problem. Our use of Voronoi diagrams to compute the diameter follows that of Cabello, but he used abstract Voronoi diagrams, which makes his diameter algorithm more involved, more expensive, and randomized. As in Cabello's work, our algorithm can compute, for every vertex $v$, both the farthest vertex from $v$ (i.e., the eccentricity of $v$), and the sum of distances from $v$ to all other vertices. Hence, our algorithm can also compute the radius, median, and Wiener index (sum of all pairwise distances) of a planar graph within the same time bounds. Our construction of Voronoi diagrams for planar graphs is of independent interest.

0 citations0 reviewsNo code linkedOpen access
preprint2019arXiv

Reachability Oracles for Directed Transmission Graphs

Let $P \subset \mathbb{R}^d$ be a set of $n$ points in $d$ dimensions such that each point $p \in P$ has an associated radius $r_p > 0$. The transmission graph $G$ for $P$ is the directed graph with vertex set $P$ such that there is an edge from $p$ to $q$ if and only if $|pq| \leq r_p$, for any $p, q \in P$. A reachability oracle is a data structure that decides for any two vertices $p, q \in G$ whether $G$ has a path from $p$ to $q$. The quality of the oracle is measured by the space requirement $S(n)$, the query time $Q(n)$, and the preprocessing time. For transmission graphs of one-dimensional point sets, we can construct in $O(n \log n)$ time an oracle with $Q(n) = O(1)$ and $S(n) = O(n)$. For planar point sets, the ratio $Ψ$ between the largest and the smallest associated radius turns out to be an important parameter. We present three data structures whose quality depends on $Ψ$: the first works only for $Ψ< \sqrt{3}$ and achieves $Q(n) = O(1)$ with $S(n) = O(n)$ and preprocessing time $O(n\log n)$; the second data structure gives $Q(n) = O(Ψ^3 \sqrt{n})$ and $S(n) = O(Ψ^3 n^{3/2})$; the third data structure is randomized with $Q(n) = O(n^{2/3}\log^{1/3} Ψ\log^{2/3} n)$ and $S(n) = O(n^{5/3}\log^{1/3} Ψ\log^{2/3} n)$ and answers queries correctly with high probability.

0 citations0 reviewsNo code linkedOpen access
preprint2019arXiv

Thompson Sampling for Adversarial Bit Prediction

We study the Thompson sampling algorithm in an adversarial setting, specifically, for adversarial bit prediction. We characterize the bit sequences with the smallest and largest expected regret. Among sequences of length $T$ with $k < \frac{T}{2}$ zeros, the sequences of largest regret consist of alternating zeros and ones followed by the remaining ones, and the sequence of smallest regret consists of ones followed by zeros. We also bound the regret of those sequences, the worse case sequences have regret $O(\sqrt{T})$ and the best case sequence have regret $O(1)$. We extend our results to a model where false positive and false negative errors have different weights. We characterize the sequences with largest expected regret in this generalized setting, and derive their regret bounds. We also show that there are sequences with $O(1)$ regret.

0 citations0 reviewsNo code linkedOpen access
preprint2010arXiv

A Kinetic Triangulation Scheme for Moving Points in The Plane

We present a simple randomized scheme for triangulating a set $P$ of $n$ points in the plane, and construct a kinetic data structure which maintains the triangulation as the points of $P$ move continuously along piecewise algebraic trajectories of constant description complexity. Our triangulation scheme experiences an expected number of $O(n^2β_{s+2}(n)\log^2n)$ discrete changes, and handles them in a manner that satisfies all the standard requirements from a kinetic data structure: compactness, efficiency, locality and responsiveness. Here $s$ is the maximum number of times where any specific triple of points of $P$ can become collinear, $β_{s+2}(q)=λ_{s+2}(q)/q$, and $λ_{s+2}(q)$ is the maximum length of Davenport-Schinzel sequences of order $s+2$ on $n$ symbols. Thus, compared to the previous solution of Agarwal et al.~\cite{AWY}, we achieve a (slightly) improved bound on the number of discrete changes in the triangulation. In addition, we believe that our scheme is simpler to implement and analyze.

0 citations0 reviewsNo code linkedOpen access
preprint2010arXiv

On the Interplay between Incentive Compatibility and Envy Freeness

We study mechanisms for an allocation of goods among agents, where agents have no incentive to lie about their true values (incentive compatible) and for which no agent will seek to exchange outcomes with another (envy-free). Mechanisms satisfying each requirement separately have been studied extensively, but there are few results on mechanisms achieving both. We are interested in those allocations for which there exist payments such that the resulting mechanism is simultaneously incentive compatible and envy-free. Cyclic monotonicity is a characterization of incentive compatible allocations, local efficiency is a characterization for envy-free allocations. We combine the above to give a characterization for allocations which are both incentive compatible and envy free. We show that even for allocations that allow payments leading to incentive compatible mechanisms, and other payments leading to envy free mechanisms, there may not exist any payments for which the mechanism is simultaneously incentive compatible and envy-free. The characterization that we give lets us compute the set of Pareto-optimal mechanisms that trade off envy freeness for incentive compatibility.

0 citations0 reviewsNo code linkedOpen access