arXiv subjects

Vahab Mirrokni

Publications and source records attributed to Vahab Mirrokni.

At least 19 recordsLinked to original sources

arXiv2026

Near-Optimal Space and Time Cut Sparsification in Turnstile Streams

A seminal result of Ahn, Guha, and McGregor (PODS'12) showed that a cut sparsifier of an unweighted graph can be computed from a near-linear number of linear measurements. Later work extended this to spectral sparsifiers (Kapralov et al., FOCS'14) and to near-linear recovery time (Kapralov et al., SODA'20). For weighted graphs, these sketches rely on grouping edges by weight, which general turnstile streams do not allow. Chen, Khanna, and Li (FOCS'22) removed this limitation, using $\tilde{O}(n/\varepsilon^3)$ bits of space. We improve the space to $\tilde{O}(n/\varepsilon^2)$ bits. Our algorithm makes one pass over a general turnstile stream of a weighted graph, processes each update in $\tilde{O}(1)$ time, and recovers, in $\tilde{O}(n/\varepsilon^2)$ time, a reweighted subgraph with $O(n\varepsilon^{-2}\log^2 n)$ edges that preserves every cut within a factor of $1\pm\varepsilon$. Its space, update time, and recovery time are all optimal up to polylogarithmic factors.

cs.DS↗

arXiv2026

PrivacyFlow: Privacy in AI Agents through Information Flow Analysis and Consequence-Aware Risk Management

Autonomous LLM agents executing multi-step workflows over personal data often misalign with user privacy expectations and leak sensitive, out-of-context information. Although Contextual Integrity (CI) formalizes this problem, existing benchmarks and defenses rarely operationalize CI for realistic multi-step task sequences in evolving environments. In this work, we first introduce agentic PrivacyBench, a benchmark of 5873 tasks in 986 task sequences with dynamic environment updates and CI-grounded allow-sets for sensitive information. We also introduce PrivacyFlow, a defense that separates deliberative privacy planning (System 2) from task execution (System 1), treating each sharing decision as a structured risk-management problem. Before execution, PrivacyFlow traces the provenance and purpose of each planned data flow, models recipient background knowledge, and simulates what external actors could infer. This allows us to model the downstream consequences of disclosure, including inference attacks and context-dependent harms. It then weighs the remaining disclosure risk against task utility and executes a minimized plan, blocks it, or consults the user. Across six models, PrivacyFlow significantly reduces step-level privacy failures and raises the share of completed task sequences without a detected privacy violation by 22--55 percentage points. We further highlight a new privacy--utility--autonomy tradeoff: allowing PrivacyFlow to proactively prompt the user in case of uncertainty can increase authorized task completion at the cost of shifting decision effort back to the user. Our results suggest that proactive CI reasoning can substantially improve agentic privacy and motivate further study of this tradeoff.

cs.CR↗

arXiv2026

On the Accuracy of Approximate Leave-One-Out with Blockwise Nonsmooth Regularization in High-dimensions

Approximate leave-one-out (ALO) refers to methods that provide computationally efficient and statistically accurate approximations to leave-one-out risk estimation. These methods have recently attracted increased attention because of their connections to other important problems, including machine unlearning and data valuation. For empirical risk minimization, ALO methods often rely on a few Newton steps and therefore require smooth loss functions and regularizers. In high-dimensional problems, however, regularizers are frequently non-smooth. This has motivated the development of heuristic ALO methods for non-smooth regularization, whose empirical accuracy is often strong but whose theoretical justification remains limited. In this paper, we study the accuracy of a heuristic ALO method for a broad class of non-differentiable penalties that promote block sparsity in the proportional high-dimensional regime. Assuming uniformly bounded block sizes, we prove that the expected average absolute discrepancy between the exact leave-one-out loss and its ALO approximation is $O(\log^{3/2}(en)/\sqrt{n})$. Our proof technique differs substantially from existing approaches and relies on obtaining sharp bounds on the number of active groups whose status changes between the full-data and leave-one-out estimators. In addition to yielding stronger guarantees under weaker assumptions than prior work, our analysis provides new insight into the stability of sparse estimators under perturbations of the training data.

math.ST↗

arXiv2026

DuoDiscover: Scaling Autonomous Discovery Beyond the Evaluation Bottleneck

Evaluation has been central to advances in AI, and it is equally critical to autonomous discovery. Yet scaling autonomous discovery to real-world engineering challenges faces a fundamental bottleneck: the cost of evaluating progress. In real settings, a single true evaluation, such as a full benchmark or serving trace, can take hours, so a search can afford only a handful of true evaluations. We present DuoDiscover, a search system that jointly discovers what to optimize and how to evaluate progress efficiently, through three mechanisms. First, each worker in a fleet of coding agents co-discovers an optimization target and a fast, tailored proxy evaluator, which we call a yardstick. The agent then hill-climbs the yardstick rather than the true evaluation. Second, since search can exploit gaps between a yardstick and the true objective, fidelity tests continually recheck each yardstick's ranking and invalidate any that drift. Third, the fleet is bottom-up, with no central decomposition of the problem. A lightweight orchestrator spawns and terminates agents, briefs running ones, and seeds new agents with others' confirmed work. DuoDiscover enables autonomous optimization from software to chip architecture. We demonstrate DuoDiscover on two repository-level benchmarks and four GPU and TPU kernel suites covering hundreds of kernels. On SWE-fficiency, it doubles the average speedup of human experts. On SOL-Exec, it ranks first on 121 of 235 kernels and in the top three on 212. On the heavily optimized vLLM and SQLite engines, DuoDiscover achieves 1.92$\times$ and 1.38$\times$ gains over the base code. On a CPU microarchitecture optimization task with evaluations of up to 28 hours, DuoDiscover surpasses both AlphaEvolve-based ArchAgent v2 and the best human design, with 1.65$\times$ ArchAgent v2's gain in 16.7 days versus more than 40 days, advancing the frontier of autonomous chip design.

cs.CL↗

arXiv2026

Optimal and Efficient Online Inverse Optimization

In online inverse linear optimization, a learner recommends an action and then observes the choice of an expert who maximizes a fixed, unknown linear objective on $\mathbb{R}^{d}$; the goal is to learn to optimize this objective without observing it. Sakaue recently obtained the optimal regret $O(\sqrt d)$ with a randomized algorithm making $(dT)^{O(d)}$ linear optimizations per round, and asked whether it can be attained in polynomial time. We answer positively: our deterministic algorithm has regret $O(\sqrt d)$ for every horizon $T$ and runs in time polynomial in $d$ and $T$. It is a variant of the variable-metric algorithms of Sakaue et al.\ and Cai et al., in which a metric update is revoked once the query point moves far enough from where the update was made.

cs.LG↗

arXiv2026

BRANCH-MoE: Balance-Aware Tree Routing for Large Embedding Models

Mixture-of-experts (MoE) layers increase model capacity without a proportional increase in per-example computation. However, conventional flat routers can yield imbalanced expert utilization and treat experts as an unstructured collection, whose indices carry no topological meaning. We introduce {\bf BRANCH-MoE}, a routing architecture that places $E$ experts at the leaves of a binary decision tree of depth $\log_2 E$. At each internal node the branching probability is centered on the arrival-weighted mean score of the traffic reaching that node. This mean is estimated using an exponential moving average, which promotes utilization of both child subtrees without an auxiliary load-balancing loss. We show that this moving-average estimate admits an explicit noise-lag trade-off. We prove that for linear node maps and log-concave arrival distributions, this mechanism prevents routing-mass collapse. We further establish that, under a frozen router, an expert's execution frequency controls its stochastic-gradient convergence rate, and that confident decisions near the root bound cross-device communication when experts are assigned to devices by tree prefix. We evaluate BRANCH-MoE against Switch softmax, DeepSeek-V3 dynamic-bias, Skywork logit-normalized, and deterministic hash routing on Criteo click-through-rate prediction, Forest Covertype, HIGGS, and YearPredictionMSD, using $E=16$, top-$4$ routing, and five random seeds. Our results show that hierarchical routing can preserve task quality and balanced utilization while inducing a topology that supports localized expert co-activation and reduced communication.

cs.LG↗

arXiv2026

Robust Budget Pacing with a Single Sample

Major Internet advertising platforms offer budget pacing tools as a standard service for advertisers to manage their ad campaigns. Given the inherent non-stationarity in an advertiser's value and also competing advertisers' values over time, a commonly used approach is to learn a target expenditure plan that specifies a target spend as a function of time, and then run a controller that tracks this plan. This raises the question: how many historical samples are required to learn a good expenditure plan? We study this question by considering an advertiser repeatedly participating in $T$ second-price auctions, where the tuple of her value and the highest competing bid is drawn from an unknown time-varying distribution. The advertiser seeks to maximize her total utility subject to her budget constraint. Prior work has shown the sufficiency of $T\log T$ samples per distribution to achieve the optimal $O(\sqrt{T})$-regret. We dramatically improve this state-of-the-art and show that just one sample per distribution is enough to achieve the near-optimal $\tilde O(\sqrt{T})$-regret, while still being robust to noise in the sampling distributions.

cs.LG↗

arXiv2026

SSTQ:Privacy-Preserving Vector Quantization via Subsampled Stochastic TurboQuant

Achieving local differential privacy in distributed optimization while maintaining low communication cost remains challenging. Existing vector quantization methods, such as vqSGD, rely on high-dimensional geometric constructions but incur unfavorable dimension-dependent variance. In this work, we propose Subsampled Stochastic TurboQuant (SSTQ), a framework that combines a bounded Kashin representation, data-independent coordinate subsampling, and privacy-aware one-dimensional quantization. SSTQ includes two variants: (1) a Flat Randomized Response variant that is unbiased and, for a fixed codebook bit-width, frame redundancy, and dimension-independent Kashin level, achieves reconstruction MSE that scales linearly with the ambient dimension $d$, while using only $\lceil \log_2 N \rceil + b$ bits per message. Here, $N = Θ(d)$ denotes the frame size in the Kashin transform and $b$ is the codebook bit-width; and (2) a metric-aware truncated-Laplace variant that removes the exponential dependence on bit-width at the cost of a non-vanishing bias. We also derive a convex uniform-surrogate codebook objective whose worst-case codebook-dependent upper bound improves from $O(4^b)$ to $O(2^b)$. Experiments on synthetic regression, Fashion-MNIST, and CIFAR-10 compare the per-message privacy-utility and uplink-communication trade-offs of SSTQ with those of established baselines, demonstrating favorable utility and communication efficiency.

cs.LG↗

arXiv2026

A Proof of the Most Informative Boolean Function Conjecture

Let $X$ be uniform on $\{-1,1\}^n$, let $Y$ be obtained by passing its coordinates independently through a binary symmetric channel with crossover probability $p$, and let $g:\{-1,1\}^n\to\{0,1\}$ be a Boolean function. We give a computer-assisted proof of the Courtade--Kumar conjecture $I(g(X);Y)\le1-H_2(p)$, where $H_2$ is binary entropy, with equality attained by dictator functions. The present work builds on the differential-equation method, itself a limiting form of the auxiliary-receiver approach in network information theory using a continuum of degraded receivers. The proof proceeds from a local inequality to a dimension-independent bound on entropy production. Differentiation along the Boolean noise semigroup expresses entropy production as an average of edge costs. The key estimate is therefore an unrestricted Bellman inequality with two mean constraints and two entropy constraints, allowing arbitrary couplings of the edge variables. This paper and its supplement provide the proofs and computational verification records. The document is lengthy because it is designed to be entirely self-contained, deriving all proofs from first principles and reproducing the proofs of cited results. We also give a self-contained expository note explaining the reduction to a low-dimensional inequality and the ideas behind the key lower bounds. The entire proof, including all numerical certificates, has been formally verified in Lean end-to-end, and is available online.

cs.DS↗

arXiv2026

DLB: Distributed Load Balancing at Scale for Generative AI Inference

The reliance on scarce and expensive accelerators such as GPUs and TPUs in modern datacenters places unprecedented demands on backend infrastructure. For workloads characterized by heterogeneous service times and complex multi-stage processing, such as Generative AI, conventional load balancing techniques are often inadequate, relying heavily on costly overprovisioning to maintain service level objectives. This paper introduces DLB, the Distributed Load Balancer, a novel system designed to minimize end-to-end user latency for large-scale, heterogeneous workloads. DLB employs a scalable, distributed design with peer-to-peer probing to maintain real-time visibility into server capacity across large-scale, geographically distributed infrastructure. The system continuously learns latency models to estimate the latency impact of routing decisions, allowing it to effectively manage heterogeneous hardware and diverse model architectures. We provide a novel theoretical analysis of our routing algorithms that establishes their stability and global performance guarantees over time. We also evaluate DLB through extensive simulations, which show substantial gains compared to state-of-the-art load balancing algorithms. Finally, following a 22-month deployment of DLB at Google, where it facilitates large-scale Generative AI inference for thousands of different machine learning models and millions of requests per second, we detail the design choices and practical experiences gained from the system in production. Analysis of production migrations demonstrates that DLB yields statistically significant latency reductions compared to the legacy baseline, including a 17\% decrease in median latency and a 13\% decrease at the p95 tail.

cs.DC↗

arXiv2026

Stellar Colosseum: A Many-Agent Harness for Long-Horizon Research in Mathematics and Theoretical Computer Science

Language models can produce plausible short proofs, but may still be unreliable on long-horizon research problems, where progress depends on a sequence of uncertain and interdependent decisions. We introduce Stellar Colosseum, a model-agnostic harness for allocating inference across research in mathematics and theoretical computer science. Colosseum explores alternative strategies before proof construction, uses a readiness gate to decide when a route is mature enough to decompose, represents the proof plan as interdependent section-level subproblems, and routes verifier findings back to the affected part of the argument. Across these stages, it generates candidates in parallel, attacks them with targeted falsification, and combines candidates and their critiques into a single research artifact through overlapping random-sample tree aggregation. The Colosseum workflow has been integrated into Google Antigravity's Teamwork framework as the Long Proof pattern. We demonstrate the capabilities of Colosseum through open-ended research and evaluations on theorem-proving and competitive programming benchmarks. Using Colosseum with Gemini 3.1 Pro, we obtain several new results that address open problems arising from papers published at top venues such as FOCS and JMLR. On TCS-Bench, a benchmark of research-level theorem-proving tasks drawn from papers published at FOCS, STOC, and SODA, Colosseum achieves 71.0% accuracy using Gemini 3.1 Pro and Gemini 3.7 Flash. In a separate Codeforces evaluation using Gemini 3.1 Pro, the proof-oriented pipeline with execution feedback solves 218 of 222 problems.

cs.AI↗

arXiv2026

Causal inference with bipartite designs: A generalized propensity score approach

Bipartite experiments, in which one set of units receives a treatment while outcomes are measured on another set, have recently garnered attention for their ability to capture interference across two distinct populations, such as buyers and sellers in online marketplaces. However, analyzing these experiments can be challenging, given that exposure is neither purely binary nor independent across units. In this paper, we propose a unified framework for causal inference in bipartite designs that leverages generalized propensity scores (GPS) to estimate exposure-response functions. Under standard unconfoundedness assumptions, we show that our GPS-based estimators are unbiased and derive theoretical bounds on their variance. We further introduce practical modeling and weighting strategies, along with double deconfounding methods, that integrate the GPS into both the outcome model and the assignment mechanism. Through extensive simulations, we demonstrate that these approaches achieve substantial bias reduction compared to naive methods. We also illustrate their effectiveness in a real-world application using an Amazon Pet Supplies dataset, where controlling for network structure proves critical to drawing valid causal conclusions. Our results underscore the importance of bipartite designs in contexts with significant interference and highlight how GPS-based methods can bolster the reliability of causal effect estimates in such settings.

stat.ME↗

arXiv2026

TCS-BENCH: Benchmarking State-of-the-Art Generative AI Theoretical Computer Science Research Ability

We introduce TCS-Bench, a benchmark for evaluating Large Language Models (LLMs) on research-level Theoretical Computer Science (TCS) proof generation. TCS-Bench consists of theorem-proving tasks from papers published at top theoretical computer science venues (STOC, FOCS, and SODA). Each task provides the necessary context to derive a self-contained proof for a target result. We evaluate state-of-the-art models on this benchmark. We verify the correctness of generated proofs via a verification agent, and further benchmark the verifier against human-expert proof judgements on a set of target statements and generated proofs pairs. Our reference verifier achieves over 90% accuracy on the expert labeled set.

cs.CL↗

arXiv2026

Nearly Optimal Strong Coresets for $\ell_p$ Subspace Approximation

We study strong coresets for $\ell_p$ subspace approximation. Given a matrix $A\in\mathbb{R}^{n\times d}$, the goal is to sample and rescale a small number of its rows to obtain $SA$ such that $\left\|SA(I-P_F)\right\|_{p,2}^p=(1\pm\varepsilon)\left\|A(I-P_F)\right\|_{p,2}^p$ simultaneously for every subspace $F\subseteq\mathbb{R}^d$ of dimension at most $k$, where $P_F$ is the orthogonal projector onto $F$. Woodruff and Yasuda [WY25] (FOCS 2025) obtained coreset sizes $\widetilde{O}_p(k\varepsilon^{-4/p})$ for $1\leq p<2$ and $\widetilde{O}_p(k^{p/2}\varepsilon^{-p})$ for $p>2$. We improve these bounds to $\widetilde{O}_p(k\varepsilon^{-2})$ and $\widetilde{O}_p(k^{p/2}\varepsilon^{-2})$, respectively. For $1\leq p<2$, our algorithm runs in $\widetilde{O}_p(\mathrm{nnz}(A)+d^ω+k\varepsilon^{-2})$ time. The resulting coreset size matches the sampling lower bound [LWW21] up to logarithmic factors when $k+1\geq C\log(1/\varepsilon)$ for an absolute constant $C$. For $p>2$, our algorithm runs in $\widetilde{O}_p(\mathrm{nnz}(A)+d^ω)$ time, matching the running time of the framework of Woodruff and Yasuda. We use different techniques in the two regimes. For $1\leq p<2$, we combine a bicriteria low-rank split with Lewis-weight sampling and empirical-process bounds independent of the output dimension. For $p>2$, we give a sharper analysis of the Woodruff-Yasuda construction. By retaining the truncation in its sampling probabilities throughout the row-count recurrence, we show that it achieves the improved $\varepsilon^{-2}$ dependence.

cs.DS↗

arXiv2026

Proteus: Incremental Memory Activation for Long-Context Sequence Modeling

The quadratic cost of attention-based sequence models for long contexts has motivated a growing line of research on memory-based models that can compress context into a compact state. However, most existing memory models expose a static memory throughout the entire sequence. Because early tokens face no compression pressure, they occupy too many degrees of freedom and "pollute" the memory state, leaving little capacity for later context and increasing interference between what is stored and what arrives next. We study a new paradigm of incremental memory activation, where the effective capacity of memory is progressively expanded as the context grows. Imposing an early bottleneck forces the model to compress history more effectively, while unlocking fresh capacity over time reduces interference and improves retention of later context. We instantiate this paradigm in Proteus, a straightforward mechanism that can be incorporated into a broad class of neural memory architectures at no additional cost. We apply Proteus to state-of-the-art models, including SWLA, Comba, Titans, and Hope-Attention, and observe consistent improvements on standard language modeling and reasoning, as well as on long-context retrieval and understanding, with gains that grow at longer context lengths. Overall, our results show that static memory is suboptimal and that scheduling effective capacity is a simple and broadly applicable tool for sequence modeling.

cs.LG↗

arXiv2026

A Near-Optimal Lower Bound for Prefix-Matrix Factorizations

For the $n\times n$ lower-triangular all-ones matrix $Q$, we prove a near-optimal lower bound \[ γ_{2,1}(Q) := \inf_{Q=AB} \|A\|_{2\to\infty}\|B\|_{1\to1} = Ω\!\left( \frac{\log^{3/2}n}{(\log\log n)^{3/2}} \right), \] where the infimum ranges over real factorizations of arbitrary finite inner dimension. This cost is a central parameter in space bounds for factorization-based rank and quantile estimation in turnstile streams and in error bounds for matrix mechanisms for continual counting under pure differential privacy. The proof combines right-sided Haar projections with a scale-dependent numerical-sparsity decomposition of the rows of $B$. At each scale, a rank--Frobenius argument shows that the numerically sparse rows cannot account for all of the required Schatten $2/3$ mass, while a Haar projection estimate bounds the contribution of the remaining rows. Summing these bounds over the dyadic scales yields the result. The proof was obtained using a fully automated Gemini-based agentic system developed internally at Google. The authors verified the proof and made minor revisions.

cs.DS↗

arXiv2026

Pairwise-Independent Dithering for Single-Stage Hadamard Quantization

Quantizing high-dimensional vectors is fundamental to similarity search, distributed learning, and model compression. Feng, Indyk, Kapralov, Krachun, and Prokhorov established sharp guarantees for an unbiased dithered quantizer based on a randomized Hadamard transform [FIK+26]. Their $1/d$-scale inner-product estimator, however, uses a second randomized transform and residual quantization, increasing both communication and the leading constant in the proved bound. We show that this extra stage is unnecessary: pairwise-independent dithers across Hadamard coordinates suffice. The resulting unbiased single-stage estimator uses $b$ bits per coordinate and achieves \[ \mathbb{E}\!\left[ \left|\left\langle y,\widehat{x}-x\right\rangle\right|^2 \right] \leq \left(\frac{3π\sqrt{3}}{2}+o(1)\right) \frac{\lVert y\rVert_2^2}{d\,4^b}, \] as $b\to\infty$, with a dimension-free $o(1)$ term uniform over unit inputs and fixed queries. Compared with the two-stage construction of Feng et al., it eliminates the residual-stage $O(d)$-bit payload and reduces the leading upper-bound constant by a factor of approximately $5.93$. The proof was first obtained using a fully automated Gemini-based agentic system developed internally at Google. The authors have verified the proof and edited it for clarity of presentation.

cs.DS↗

arXiv2026

The Condition-Number Barrier in Sparse Least Squares

In [AS21], Axiotis and Sviridenko conjectured that the linear dependence on the restricted condition number in sparse convex optimization cannot be improved by a polynomial-time algorithm. We establish their conjectured lower bound for least-squares objectives, conditional on the randomized exact-volume Small-Set Expansion Hypothesis in the weighted regular-graph formulation of Raghavendra, Steurer, and Tulsiani [RST12]. Concretely, for every fixed $γ\in(0,1]$, there is no randomized polynomial-time algorithm that, with probability at least $2/3$, returns a vector $x$ such that, writing $s=\lVert x\rVert_0$, \[ \lVert Ax-b\rVert_2^2 \leq \min_{\lVert z\rVert_0\leq k}\lVert Az-b\rVert_2^2+\varepsilon \quad\text{and}\quad s=O\!\left(k\,κ_{s+k}^{\,1-γ}\right), \] where $κ_r$ is the restricted condition number at sparsity level $r$. The result holds even on rational instances with $A$ of full column rank. The proof was first obtained using a fully automated Gemini-based agentic system developed internally at Google. The authors have verified the proof and edited it for clarity of presentation.

cs.DS↗