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Moses Charikar

Publications and source records attributed to Moses Charikar.

At least 19 recordsLinked to original sources
arXiv2026

Stable Voting Rules on the Edge of Optimal Metric Distortion

We prove the existence of a randomized voting rule with metric distortion at most $2.13713$, within $0.025$ of the lower bound of $2.11264$. Our rule comes from a generalization of stable $k$-lotteries developed in the context of committee selection. In contrast to prior work, our rule samples from a single distribution derived from a zero-sum game, without mixing between voting rules. Our result also gives sharp distortion bounds for stable $k$-lotteries, and in particular shows that stable $2$-lotteries have distortion $7/3$, despite only relying on aggregate preferences over triples of candidates.

cs.GT↗
arXiv2026

Pessimal Elections for Approximately Dominating Sets

Condorcet's paradox is a foundational result in social choice theory, showing that no matter which candidate wins an election, a majority of voters may prefer some losing candidate. Worse still, even if the election can choose a committee of $k$ winners, some loser may beat every winner in a majority vote. Recent work showed that this obstruction can be sidestepped by relaxing the majority threshold. For all $\varepsilon > 0$, any election can select a committee of $O(1/\varepsilon^2)$ winners such that no loser is preferred to every winner by $\frac12 + \varepsilon$ fraction of voters. We present a simple construction, found by GPT-5.6 Sol Ultra, which proves that this result is tight up to a constant factor.

cs.GT↗
arXiv2026

Spectral Dual Fitting for $k$-Means

We give a new dual fitting algorithm which gives improved approximation ratios of $3+\ln 2 + \epsilon\ (\approx 3.694)$ and $4.9+\epsilon$ for $k$-Means in (high-dimensional) Euclidean and general metrics respectively, improving upon the previously known ratios of $4+\epsilon$ [Charikar, Cohen-Addad, Gao, Grandoni, Lee, and van Wijland STOC'26] and $5+\epsilon$ [Byrka, Guo, Hu, Li, Wan, Wang FOCS'26], resp. In particular, our result for Euclidean $k$-Means breaks the hardness barrier of $1+8/e\approx 3.94$ for Metric $k$-Means. Prior to our work, no such separation between general and Euclidean metrics was known for $k$-Median, $k$-Means, or Facility Location in terms of their approximability. Unlike prior dual fitting approaches for $k$-Means, our new dual fitting algorithm tightly accounts for dual payments while still facilitating an effective dual feasibility analysis. We introduce a new framework that uses spectral analysis for determining the approximation factor of our algorithm.

cs.DS↗
arXiv2026

Language Identification with Succinct Machine-Independent Traces

Motivated by the power of large language models, there has been renewed interest in the Gold-Angluin model of language identification in the limit, with an eye toward variants of the model that might overcome the negative results for its original formulation. Recent papers on this question have proposed looking at computational traces and annotations of training strings as a source of additional power for a learner, reflecting empirical regularities such as the way that commented source code is easier to learn from than arbitrary source code, and text annotated with algorithmically generated chain-of-thought tokens can be easier to learn from than the raw text itself. This recent work has shown positive results for language identification in the presence of such computational traces, but the traces in these positive results come from explicit automata-theoretic machine models that generate the language, where the underlying vocabulary of tokens for the traces is very large. In this paper, we address two fundamental issues left open by this line of work: can we achieve positive results with traces that use only a small alphabet, and can we define traces directly from the language itself, without requiring an underlying machine model that generates it? We establish positive results for both of these questions: for an arbitrary collection of languages, we show how to define computational traces that enable identification in the limit, using an alphabet of tokens that is linear in the size of the alphabet that the languages are defined over, and independent of any other properties of the languages.

cs.CL↗
arXiv2026

Globally Consistent Coloring Schemes for Language Identification

We study how little extra information is needed to make adversarial language learning possible. In Gold's model of language identification in the limit, a learner is given an enumeration of the strings from an unknown language chosen from a countable language collection. The learner guesses the identity of the language over the course of the enumeration, and it succeeds if, eventually, all of its guesses are the correct language. Classical results of Gold and Angluin show that many natural collections cannot be learned in this way. Recent work on trace colorings, motivated by the success of thinking-trace strategies in language learning, overcomes this obstruction by annotating every symbol of every string with a color. We ask whether the learner really needs this whole sequence of colors, or whether one color at the end of each string (a terminal coloring) is enough for language identification. We show that just one terminal bit per string is enough for every countable collection of infinite languages. In fact, the colorings can be chosen collection-independently: there is a single assignment of a two-color terminal coloring to every infinite language such that the same preassigned colorings identify every countable subcollection. Thus, in this model, an entire color trace can be compressed to one bit attached to the end of each example. Our global construction uses transfinite recursion, and we prove that this kind of nonconstructivity is unavoidable for any bounded number of colors. As a notion of constructivity, we use the formalism of Borel maps (a regularity condition satisfied by natural explicit constructions); we show that no global terminal coloring with a finite number of colors defined by a Borel map can identify all countable subcollections. By contrast, known trace-coloring constructions are Borel when encoded as terminal colorings, but require infinitely many colors.

cs.CL↗
arXiv2026

From Non-Convex to Strongly Convex: Curvature-Adaptive FTPL for Online Optimization

Curvature adaptivity is a classical theme in online optimization: for convex Lipschitz losses, adaptive methods interpolate between the optimal $O(\sqrt{T})$ regret for general convex losses and $O(\log T)$ regret under strong convexity. Recent work has shown that Follow-the-Perturbed-Leader (FTPL) achieves optimal $O(\sqrt{T})$ regret even for online non-convex Lipschitz losses, assuming access to an approximate offline-optimization oracle, but these guarantees do not exploit curvature. We show that FTPL can be made curvature-adaptive in the non-convex setting, without knowing in advance how curvature will accumulate over time. Our algorithm replaces the fixed perturbation scale of standard FTPL with a time-varying scale chosen using only past information. We give a simple follow-the-leader tuning rule for this scale and show that it competes, up to constants, with the best choice in hindsight. The resulting method achieves $O(\sqrt{T})$ regret for arbitrary non-convex Lipschitz losses and improves as cumulative curvature grows; with sufficiently accurate oracle calls, it achieves $O(\log T)$ regret when cumulative curvature grows linearly, which includes the classical strongly convex regime. We complement these upper bounds with matching lower bounds for prescribed cumulative-curvature sequences, already for one-dimensional convex losses, showing that the tradeoff between worst-case non-convex regret and curvature-driven fast rates is intrinsic.

cs.LG↗
arXiv2026

An Improved Greedy Approximation for (Metric) $k$-Means

Clustering is a basic task in data analysis and machine learning, and the optimization of clustering objectives are well-studied optimization problems; amongst these, the $k$-Means objective is arguably the most well known. Given a collection of points in a metric space, the goal is to partition them into $k$ clusters, each with an associated center, so as to minimize the sum of squared distances of points to their cluster centers. In this paper, we present a polynomial-time $3+2\sqrt{2}+\epsilon<5.83$-approximation algorithm for $k$-Means in general metrics. This substantially improves on the current-best $(9+\epsilon)$-approximation in [Ahmadian, Norouzi-Fard, Svensson, Ward - FOCS'17, SICOMP'20], and even slightly improves on the $5.92$-approximation in [Cohen-Addad, Esfandiari, Mirrokni, Narayanan - STOC'22] for the Euclidean special case. A natural approach for $k$-Means is to leverage Lagrangian Multiplier Preserving (LMP) approximations for the facility location problem. The previous best results for $k$-Means build upon an adaptation of an LMP $3$-approximation for facility location with metric connection costs in [Jain, Vazirani - J.ACM'01] based on a primal-dual method, rather than on the improved LMP greedy $2$-approximation for the same problem in [Jain, Mahdian, Markakis, Saberi, Vazirani - J.ACM'03]. The barrier to using the improved LMP algorithm was that no adaptation of this algorithm and its analysis to the case of squared metric connection costs was known (since squared distances violate triangle inequality). Our main contribution is overcoming this barrier by providing such an adaptation. This new LMP approximation algorithm is then combined with the framework recently introduced in [Cohen-Addad, Grandoni, Lee, Schwiegelshohn, Svensson - STOC'25] for the related (metric) $k$-Median problem.

cs.DS↗
arXiv2025

A Characterization of List Language Identification in the Limit

We study the problem of language identification in the limit, where given a sequence of examples from a target language, the goal of the learner is to output a sequence of guesses for the target language such that all the guesses beyond some finite time are correct. Classical results of Gold showed that language identification in the limit is impossible for essentially any interesting collection of languages. Later, Angluin gave a precise characterization of language collections for which this task is possible. Motivated by recent positive results for the related problem of language generation, we revisit the classic language identification problem in the setting where the learner is given the additional power of producing a list of $k$ guesses at each time step. The goal is to ensure that beyond some finite time, one of the guesses is correct at each time step. We give an exact characterization of collections of languages that can be $k$-list identified in the limit, based on a recursive version of Angluin's characterization (for language identification with a list of size $1$). This further leads to a conceptually appealing characterization: A language collection can be $k$-list identified in the limit if and only if the collection can be decomposed into $k$ collections of languages, each of which can be identified in the limit (with a list of size $1$). We also use our characterization to establish rates for list identification in the statistical setting where the input is drawn as an i.i.d. stream from a distribution supported on some language in the collection. Our results show that if a collection is $k$-list identifiable in the limit, then the collection can be $k$-list identified at an exponential rate, and this is best possible. On the other hand, if a collection is not $k$-list identifiable in the limit, then it cannot be $k$-list identified at any rate that goes to zero.

cs.CL↗
arXiv2025

Pareto-optimal Non-uniform Language Generation

Kleinberg and Mullainathan (2024) recently proposed an interesting model for language generation in the limit: Given a countable collection of languages, and an adversary enumerating the strings of some language $L$ from the collection, the objective is to generate new strings from the target language, such that all strings generated beyond some finite time are valid. Li, Raman and Tewari (2024) and Charikar and Pabbaraju (2024) showed strong non-uniform generation guarantees in this model, giving algorithms that generate new valid strings from $L$ after seeing a number of distinct input strings $t(L)$ that depends only on $L$ (and the collection), but not the enumeration order. However, for both these works, the language-wise generation times $t(L)$ of the algorithm can be strictly sub-optimal. In this work, we study Pareto-optimality of non-uniform language generation in the limit. We propose an algorithm, whose generation times $t^\star(L)$ are (almost) Pareto-optimal: any other algorithm whose generation time for some language $L$ is strictly smaller than $t^\star(L)$, must satisfy that its generation time for some other language $L'$ is strictly worse than $t^\star(L')$. Pareto-optimality is essentially the best that one can achieve for non-uniform generation. Our algorithmic framework conveniently adapts to further give Pareto-optimal non-uniform generation algorithms in the practically motivated settings of noisy as well as representative generation.

cs.DS↗
arXiv2025

Metric Distortion for Tournament Voting and Beyond

In the well-studied metric distortion problem in social choice, we have voters and candidates located in a shared metric space, and the objective is to design a voting rule that selects a candidate with minimal total distance to the voters. However, the voting rule has limited information about the distances in the metric, such as each voter's ordinal rankings of the candidates in order of distances. The central question is whether we can design rules that, for any election and underlying metric space, select a candidate whose total cost deviates from the optimal by only a small factor, referred to as the distortion. A long line of work resolved the optimal distortion of deterministic rules, and recent work resolved the optimal distortion of randomized (weighted) tournament rules, which only use the aggregate preferences between pairs of candidates. In both cases, simple rules achieve the optimal distortion of $3$. Can we achieve the best of both worlds: a deterministic tournament rule matching the lower bound of $3$? Prior to our work, the best rules have distortion $2 + \sqrt{5} \approx 4.2361$. In this work, we establish a lower bound of $3.1128$ on the distortion of any deterministic tournament rule, even when there are only 5 candidates, and improve the upper bound with a novel rule guaranteeing distortion $3.9312$. We then generalize tournament rules to the class of $k$-tournament rules which obtain the aggregate preferences between $k$-tuples of candidates. We show that there is a family of deterministic $k$-tournament rules that achieves distortion approaching $3$ as $k$ grows. Finally, we show that even with $k = 3$, a randomized $k$-tournament rule can achieve distortion less than $3$, which had been a longstanding barrier even for the larger class of ranked voting rules.

cs.GT↗
arXiv2025

Approximately Dominating Sets in Elections

Condorcet's paradox is a fundamental result in social choice theory which states that there exist elections in which, no matter which candidate wins, a majority of voters prefer a different candidate. In fact, even if we can select any $k$ winners, there still may exist another candidate that would beat each of the winners in a majority vote. That is, elections may require arbitrarily large dominating sets. We show that approximately dominating sets of constant size always exist. In particular, for every $\varepsilon > 0$, every election (irrespective of the number of voters or candidates) can select $O(\frac{1}{\varepsilon ^2})$ winners such that no other candidate beats each of the winners by a margin of more than $\varepsilon$ fraction of voters. Our proof uses a simple probabilistic construction using samples from a maximal lottery, a well-studied distribution over candidates derived from the Nash equilibrium of a two-player game. In stark contrast to general approximate equilibria, which may require support logarithmic in the number of pure strategies, we show that maximal lotteries can be approximated with constant support size. These approximate maximal lotteries may be of independent interest.

cs.GT↗
arXiv2024

Exploring Facets of Language Generation in the Limit

The recent work of Kleinberg & Mullainathan [KM24] provides a concrete model for language generation in the limit: given a sequence of examples from an unknown target language, the goal is to generate new examples from the target language such that no incorrect examples are generated beyond some point. In sharp contrast to strong negative results for the closely related problem of language identification, they establish positive results for language generation in the limit for all countable collections of languages. Follow-up work by Raman & Tewari [RT24] studies bounds on the number of distinct inputs required by an algorithm before correct language generation is achieved -- namely, whether this is a constant for all languages in the collection (uniform generation) or a language-dependent constant (non-uniform generation). We show that every countable language collection has a generator which has the stronger property of non-uniform generation in the limit. However, while the generation algorithm of [KM24] can be implemented using membership queries, we show that any algorithm cannot non-uniformly generate even for collections of just two languages, using only membership queries. We also formalize the tension between validity and breadth in the generation algorithm of [KM24] by introducing a definition of exhaustive generation, and show a strong negative result for exhaustive generation. Our result shows that a tradeoff between validity and breadth is inherent for generation in the limit. We also provide a precise characterization of the language collections for which exhaustive generation is possible. Finally, inspired by algorithms that can choose to obtain feedback, we consider a model of uniform generation with feedback, completely characterizing language collections for which such uniform generation with feedback is possible in terms of a complexity measure of the collection.

cs.DS↗
arXiv2024

Six Candidates Suffice to Win a Voter Majority

A cornerstone of social choice theory is Condorcet's paradox which says that in an election where $n$ voters rank $m$ candidates it is possible that, no matter which candidate is declared the winner, a majority of voters would have preferred an alternative candidate. Instead, can we always choose a small committee of winning candidates that is preferred to any alternative candidate by a majority of voters? Elkind, Lang, and Saffidine raised this question and called such a committee a Condorcet winning set. They showed that winning sets of size $2$ may not exist, but sets of size logarithmic in the number of candidates always do. In this work, we show that Condorcet winning sets of size $6$ always exist, regardless of the number of candidates or the number of voters. More generally, we show that if $\frac{\alpha}{1 - \ln \alpha} \geq \frac{2}{k + 1}$, then there always exists a committee of size $k$ such that less than an $\alpha$ fraction of the voters prefer an alternate candidate. These are the first nontrivial positive results that apply for all $k \geq 2$. Our proof uses the probabilistic method and the minimax theorem, inspired by recent work on approximately stable committee selection. We construct a distribution over committees that performs sufficiently well (when compared against any candidate on any small subset of the voters) so that this distribution must contain a committee with the desired property in its support.

cs.GT↗
arXiv2024

Towards SMT Solver Stability via Input Normalization

In many applications, SMT solvers are utilized to solve similar or identical tasks over time. Significant variations in performance due to small changes in the input are not uncommon and lead to frustration for users. This sort of stability problem represents an important usability challenge for SMT solvers. We introduce an approach for mitigating the stability problem based on normalizing solver inputs. We show that a perfect normalizing algorithm exists but is computationally expensive. We then describe an approximate algorithm and evaluate it on a set of benchmarks from related work, as well as a large set of benchmarks sampled from SMT-LIB. Our evaluation shows that our approximate normalizer reduces runtime variability with minimal overhead and is able to normalize a large class of mutated benchmarks to a unique normal form.

cs.LO↗
arXiv2024

Quantifying the Gain in Weak-to-Strong Generalization

Recent advances in large language models have shown capabilities that are extraordinary and near-superhuman. These models operate with such complexity that reliably evaluating and aligning them proves challenging for humans. This leads to the natural question: can guidance from weak models (like humans) adequately direct the capabilities of strong models? In a recent and somewhat surprising work, Burns et al. (2023) empirically demonstrated that when strong models (like GPT-4) are finetuned using labels generated by weak supervisors (like GPT-2), the strong models outperform their weaker counterparts -- a phenomenon they term weak-to-strong generalization. In this work, we present a theoretical framework for understanding weak-to-strong generalization. Specifically, we show that the improvement in performance achieved by strong models over their weaker counterparts is quantified by the misfit error incurred by the strong model on labels generated by the weaker model. Our theory reveals several curious algorithmic insights. For instance, we can predict the amount by which the strong model will improve over the weak model, and also choose among different weak models to train the strong model, based on its misfit error. We validate our theoretical findings through various empirical assessments.

cs.LG↗
arXiv2024

Dynamic Data Layout Optimization with Worst-case Guarantees

Many data analytics systems store and process large datasets in partitions containing millions of rows. By mapping rows to partitions in an optimized way, it is possible to improve query performance by skipping over large numbers of irrelevant partitions during query processing. This mapping is referred to as a data layout. Recent works have shown that customizing the data layout to the anticipated query workload greatly improves query performance, but the performance benefits may disappear if the workload changes. Reorganizing data layouts to accommodate workload drift can resolve this issue, but reorganization costs could exceed query savings if not done carefully. In this paper, we present an algorithmic framework OReO that makes online reorganization decisions to balance the benefits of improved query performance with the costs of reorganization. Our framework extends results from Metrical Task Systems to provide a tight bound on the worst-case performance guarantee for online reorganization, without prior knowledge of the query workload. Through evaluation on real-world datasets and query workloads, our experiments demonstrate that online reorganization with OReO can lead to an up to 32% improvement in combined query and reorganization time compared to using a single, optimized data layout for the entire workload.

cs.DB↗
arXiv2024

Correlation Clustering Beyond the Pivot Algorithm

We study the classic correlation clustering in the dynamic setting. Given $n$ objects and a complete labeling of the object-pairs as either similar or dissimilar, the goal is to partition the objects into arbitrarily many clusters while minimizing disagreements with the labels. In the dynamic setting, an update consists of a flip of a label of an edge. In a breakthrough result, [BDHSS, FOCS'19] showed how to maintain a 3-approximation with polylogarithmic update time by providing a dynamic implementation of the Pivot algorithm of [ACN, STOC'05]. Since then, it has been a major open problem to determine whether the 3-approximation barrier can be broken in the fully dynamic setting. In this paper, we resolve this problem. Our algorithm, Modified Pivot, locally improves the output of Pivot by moving some vertices to other existing clusters or new singleton clusters. We present an analysis showing that this modification does indeed improve the approximation to below 3. We also show that its output can be maintained in polylogarithmic time per update.

cs.DS↗