Pasin Manurangsi
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We prove the following asymptotically tight lower bound for k-color discrepancy: For any k ≥ 2, there exists a hypergraph with n vertices such that its k-color discrepancy is at least Ω(√n). This improves on the previously known lower bound of Ω(√n/ log k) due to Caragiannis et al. [CLS25]. As an application, we show that our result implies improved lower bounds for group fair division.
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We study the d-dimensional knapsack problem. We are given a set of items, each with a d-dimensional cost vector and a profit, along with a d-dimensional budget vector. The goal is to select a set of items that do not exceed the budget in all dimensions and maximize the total profit. A polynomial-time approximation scheme (PTAS) with running time n^{Θ(d/{ε})} has long been known for this problem, where {ε} is the error parameter and n is the encoding size. Despite decades of active research, the best running time of a PTAS has remained O(n^{⌈ d/{ε} ⌉ - d}). Unfortunately, existing lower bounds only cover the special case with two dimensions d = 2, and do not answer whether there is a n^{o(d/({ε)})}-time PTAS for larger values of d.
In this work, we show that the running times of the best-known PTAS cannot be improved up to a polylogarithmic factor assuming the Exponential Time Hypothesis (ETH). Our techniques are based on a robust reduction from 2-CSP, which embeds 2-CSP constraints into a desired number of dimensions. Then, using a recent result of [Bafna Karthik and Minzer, STOC'25], we succeed in exhibiting tight trade-off between d and {ε} for all regimes of the parameters assuming d is sufficiently large. Informally, our result also shows that under ETH, for any function f there is no f(d/({ε)}) ⋅ n^{õ(d/({ε)})}-time (1-{ε})-approximation for d-dimensional knapsack, where n is the number of items and õ hides polylogarithmic factors in d/({ε)}.
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We study the fair allocation of indivisible goods with variable groups. In this model, the goal is to partition the agents into groups of given sizes and allocate the goods to the groups in a fair manner. We show that for any number of groups and corresponding sizes, there always exists an envy-free up to one good (EF1) outcome, thereby generalizing an important result from the individual setting. Our result holds for arbitrary monotonic utilities and comes with an efficient algorithm. We also prove that the EF1 existence can be guaranteed even when the goods lie on a path and each group must receive a connected bundle. In addition, we consider a probabilistic model where the utilities are additive and drawn randomly from a distribution. We show that if there are n agents and the number of goods m is divisible by the number of groups k, then an envy-free outcome exists with high probability if m = ω(log n), and this bound is tight. On the other hand, if m is not divisible by k, then an envy-free outcome is unlikely to exist as long as m = o(√n).
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We consider a setting where we have a ground set ℳ together with real-valued set functions f₁, … , f_n, and the goal is to partition ℳ into two sets S₁,S₂ such that |f_i(S₁) - f_i(S₂)| is small for every i. Many results in discrepancy theory can be stated in this form with the functions f_i being additive. In this work, we initiate the study of the unstructured case where f_i is not assumed to be additive. We show that even without the additivity assumption, the upper bound remains at most O(√{n log n}).
Our result has implications on the fair allocation of indivisible goods. In particular, we show that a consensus halving up to O(√{n log n}) goods always exists for n agents with monotone utilities. Previously, only an O(n) bound was known for this setting.
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Rank aggregation is a task of combining the rankings of items from multiple users into a single ranking that best represents the users' rankings. Alabi et al. (AAAI'22) presents differentially-private (DP) polynomial-time approximation schemes (PTASes) and 5-approximation algorithms with certain additive errors for the Kemeny rank aggregation problem in both central and local models. In this paper, we present improved DP PTASes with smaller additive error in the central model. Furthermore, we are first to study the footrule rank aggregation problem under DP. We give a near-optimal algorithm for this problem; as a corollary, this leads to 2-approximation algorithms with the same additive error as the 5-approximation algorithms of Alabi et al. for the Kemeny rank aggregation problem in both central and local models.
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Balls-and-Bins Sampling for DP-SGD
Lynn Chua
Charlie Harrison
Pritish Kamath
Ethan Leeman
Amer Sinha
Chiyuan Zhang
AISTATS (2025)
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We introduce the Balls-and-Bins sampling for differentially private (DP) optimization methods such as DP-SGD. While it has been common practice to use some form of shuffling in DP-SGD implementations, privacy accounting algorithms have typically assumed that Poisson subsampling is used instead. Recent work by Chua et al. (2024) however pointed out that shuffling based DP-SGD can have a much larger privacy cost in practical regime of parameters. We show that the Balls-and-Bins sampling achieves the “best-of-both” samplers, namely, the implementation of Balls-and-Bins sampling is similar to that of Shuffling and models trained with Balls-and-Bins based DP-SGD achieve utility comparable to those trained with Shuffle based DP-SGD at the same noise multiplier, and yet, Balls-and-Bins sampling enjoys similar-or-better privacy amplification as compared to Poisson subsampling.
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Crosslingual Capabilities and Knowledge Barriers in Multilingual Large Language Models
Lynn Chua
Yangsibo Huang
Pritish Kamath
Amer Sinha
Chulin Xie
Chiyuan Zhang
COLM (2025)
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Large language models (LLMs) are typically multilingual due to pretraining on diverse multilingual corpora. But can these models relate corresponding concepts across languages, i.e., be crosslingual? This study evaluates state-of-the-art LLMs on inherently crosslingual tasks. We observe that while these models show promising surface-level crosslingual abilities on machine translation and embedding space analyses, they struggle with deeper crosslingual knowledge transfer, revealing a crosslingual knowledge barrier in both general (MMLU benchmark) and domain-specific (Harry Potter quiz and TOFU benchmark) contexts. Since simple inference-time mitigation methods offer only limited improvement, we propose fine-tuning of LLMs on mixed-language data, which effectively reduces these gaps, even when using out-of-domain datasets like WikiText. Our findings suggest the need for explicit optimization to unlock the full crosslingual potential of LLMs. Our code is available at https://github.com/google-research/crosslingual-knowledge-barriers.
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Several resource allocation settings involve agents with unequal entitlements represented by weights. We analyze weighted fair division from an asymptotic perspective: if m items are divided among n agents whose utilities are independently sampled from a probability distribution, when is it likely that a fair allocation exist? We show that if the ratio between the weights is bounded, a weighted envy-free allocation exists with high probability provided that m = Ω(n log n/ log log n), generalizing a prior unweighted result. For weighted proportionality, we establish a sharp threshold of m = n/(1 − μ) for the transition from non-existence to existence, where μ ∈ (0, 1) denotes the mean of the distribution. In addition, we prove that for two agents, a weighted envy-free (and weighted proportional) allocation is likely to exist if m = ω(√r), where r denotes the ratio between the two weights.
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User-level differentially private stochastic convex optimization (DP-SCO) has garnered significant attention due to the paramount importance in safeguarding user privacy in large-scale machine learning applications. Current methods, such as those based on Differentially Private Stochastic Gradient Descent (DP-SGD), often struggle with high noise accumulation and suboptimal utility due to the need to privatize every intermediate iterate. In this work, we introduce a novel linear-time algorithm that leverages robust statistics, specifically the geometric median and trimmed mean, to overcome these challenges. Our approach uniquely bounds the sensitivity of all intermediate iterates of SGD with gradient estimation based on robust statistics, thereby significantly reducing the gradient estimation noise and enhancing the privacy-utility trade-off. By sidestepping the repeated privatization required by previous methods, our algorithm not only achieves an improved theoretical privacy-utility balance but also maintains computational efficiency. This work sets the stage for more robust and efficient privacy-preserving techniques in machine learning, with implications for future research and application in the field.
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The conventional approach in differential privacy (DP) literature formulates the privacy-utility tradeoff with a "privacy-first" perspective: for a predetermined level of privacy, a certain utility is achievable. However, practitioners often operate under a "utility-first" paradigm, prioritizing a desired level of utility and then determining the corresponding privacy cost.
Wu et al. [2019] initiated a formal study of this ``utility-first'' perspective by introducing ex-post DP. They demonstrated that by adding correlated Laplace noise and progressively reducing it on demand, a sequence of increasingly accurate estimates of a private parameter can be generated, with the privacy cost attributed only to the least noisy iterate released. This led to a Laplace mechanism variant that achieves a specified utility with minimal privacy loss. However, their work, and similar findings by Whitehouse et al. [2023], are primarily limited to simple mechanisms based on Laplace or Gaussian noise.
In this paper, we significantly generalize these results. In particular, we extend the findings of Wu et al. [2019] and Liu and Talwar [2019] to support any sequence of private estimators, incurring at most a doubling of the original privacy budget. Furthermore, we demonstrate that hyperparameter tuning for these estimators, including the selection of an optimal privacy budget, can be performed without additional privacy cost. Finally, we extend our results to ex-post R'{e}nyi DP, further broadening the applicability of utility-first privacy mechanisms.
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