Milan Vojnovic

Fridolin Linder, Thomas Leeper, Daniel Haimovich et al.

Estimating the prevalence of a category in a population using imperfect measurement devices (diagnostic tests, classifiers, or large language models) is fundamental to science, public health, and online trust and safety. Standard approaches correct for known device error rates but assume these rates remain stable across populations. We show this assumption fails under covariate shift and that multicalibration, which enforces calibration conditional on the input features rather than just on average, is sufficient for unbiased prevalence estimation under such shift. Standard calibration and quantification methods fail to provide this guarantee. Our work connects recent theoretical work on fairness to a longstanding measurement problem spanning nearly all academic disciplines. A simulation confirms that standard methods exhibit bias growing with shift magnitude, while a multicalibrated estimator maintains near-zero bias. While we focus the discussion mostly on LLMs, our theoretical results apply to any classification model. Two empirical applications -- estimating employment prevalence across U.S. states using the American Community Survey, and classifying political texts across four countries using an LLM -- demonstrate that multicalibration substantially reduces bias in practice, while highlighting that calibration data should cover the key feature dimensions along which target populations may differ.

Daniel Haimovich, Fridolin Linder, Lorenzo Perini et al.

Multicalibration gradient boosting has recently emerged as a scalable method that empirically produces approximately multicalibrated predictors and has been deployed at web scale. Despite this empirical success, its convergence properties are not well understood. In this paper, we bridge the gap by providing convergence guarantees for multicalibration gradient boosting in regression with squared-error loss. We show that the magnitude of successive prediction updates decays at $O(1/\sqrt{T})$, which implies the same convergence rate bound for the multicalibration error over rounds. Under additional smoothness assumptions on the weak learners, this rate improves to linear convergence. We further analyze adaptive variants, showing local quadratic convergence of the training loss, and we study rescaling schemes that preserve convergence. Experiments on real-world datasets support our theory and clarify the regimes in which the method achieves fast convergence and strong multicalibration.

Milan Vojnovic, Se-Young Yun

In this note, we examine the aggregation of preferences achieved by the Group Policy Optimisation (GRPO) algorithm, a reinforcement learning method used to train advanced artificial intelligence models such as DeepSeek-R1-Zero and DeepSeekMath. The GRPO algorithm trains a policy using a reward preference model, which is computed by sampling a set of outputs for a given context, observing the corresponding rewards, and applying shift-and-scale normalisation to these reward values. Additionally, it incorporates a penalty function to discourage deviations from a reference policy. We present a framework that enables us to characterise the stationary policies of the GRPO algorithm. This analysis reveals that the aggregation of preferences differs fundamentally from standard logarithmic pooling, which is implemented by other approaches such as RLHF. The precise form of preference aggregation arises from the way the reward preference model is defined and from the penalty function, which we show to essentially correspond to the reverse Kullback-Leibler (KL) divergence between the aggregation policy and the reference policy. Interestingly, we demonstrate that for groups of size two, the reward preference model corresponds to pairwise comparison preferences, similar to those in other alignment methods based on pairwise comparison feedback. We provide explicit characterisations of the aggregate preference for binary questions, for groups of size two, and in the limit of large group size. This provides insights into the dependence of the aggregate preference on parameters such as the regularisation constant and the confidence margin of question answers. Finally, we discuss the aggregation of preferences obtained by modifying the GRPO algorithm to use direct KL divergence as the penalty or to use rewards without scale normalisation.

Jung-hun Kim, Milan Vojnovic, Se-Young Yun

In this study, we consider the infinitely many-armed bandit problems in a rested rotting setting, where the mean reward of an arm may decrease with each pull, while otherwise, it remains unchanged. We explore two scenarios regarding the rotting of rewards: one in which the cumulative amount of rotting is bounded by $V_T$, referred to as the slow-rotting case, and the other in which the cumulative number of rotting instances is bounded by $S_T$, referred to as the abrupt-rotting case. To address the challenge posed by rotting rewards, we introduce an algorithm that utilizes UCB with an adaptive sliding window, designed to manage the bias and variance trade-off arising due to rotting rewards. Our proposed algorithm achieves tight regret bounds for both slow and abrupt rotting scenarios. Lastly, we demonstrate the performance of our algorithm using numerical experiments.

Daniel Haimovich, Dima Karamshuk, Fridolin Linder et al.

We investigate the convergence rates and data sample sizes required for training a machine learning model using a stochastic gradient descent (SGD) algorithm, where data points are sampled based on either their loss value or uncertainty value. These training methods are particularly relevant for active learning and data subset selection problems. For SGD with a constant step size update, we present convergence results for linear classifiers and linearly separable datasets using squared hinge loss and similar training loss functions. Additionally, we extend our analysis to more general classifiers and datasets, considering a wide range of loss-based sampling strategies and smooth convex training loss functions. We propose a novel algorithm called Adaptive-Weight Sampling (AWS) that utilizes SGD with an adaptive step size that achieves stochastic Polyak's step size in expectation. We establish convergence rate results for AWS for smooth convex training loss functions. Our numerical experiments demonstrate the efficiency of AWS on various datasets by using either exact or estimated loss values.

Lorenzo Perini, Daniel Haimovich, Fridolin Linder et al.

We propose MCGrad, a novel and scalable multicalibration algorithm. Multicalibration - calibration in sub-groups of the data - is an important property for the performance of machine learning-based systems. Existing multicalibration methods have thus far received limited traction in industry. We argue that this is because existing methods (1) require such subgroups to be manually specified, which ML practitioners often struggle with, (2) are not scalable, or (3) may harm other notions of model performance such as log loss and Area Under the Precision-Recall Curve (PRAUC). MCGrad does not require explicit specification of protected groups, is scalable, and often improves other ML evaluation metrics instead of harming them. MCGrad has been in production at Meta, and is now part of hundreds of production models. We present results from these deployments as well as results on public datasets.

Jung-hun Kim, Milan Vojnovic, Se-Young Yun

We consider the infinitely many-armed bandit problem with rotting rewards, where the mean reward of an arm decreases at each pull of the arm according to an arbitrary trend with maximum rotting rate $\varrho=o(1)$. We show that this learning problem has an $Ω(\max\{\varrho^{1/3}T,\sqrt{T}\})$ worst-case regret lower bound where $T$ is the horizon time. We show that a matching upper bound $\tilde{O}(\max\{\varrho^{1/3}T,\sqrt{T}\})$, up to a poly-logarithmic factor, can be achieved by an algorithm that uses a UCB index for each arm and a threshold value to decide whether to continue pulling an arm or remove the arm from further consideration, when the algorithm knows the value of the maximum rotting rate $\varrho$. We also show that an $\tilde{O}(\max\{\varrho^{1/3}T,T^{3/4}\})$ regret upper bound can be achieved by an algorithm that does not know the value of $\varrho$, by using an adaptive UCB index along with an adaptive threshold value.

Jung-hun Kim, Milan Vojnovic

We consider the problem of scheduling in multi-class, parallel-server queuing systems with uncertain rewards from job-server assignments. In this scenario, jobs incur holding costs while awaiting completion, and job-server assignments yield observable stochastic rewards with unknown mean values. The mean rewards for job-server assignments are assumed to follow a bilinear model with respect to features that characterize jobs and servers. Our objective is to minimize regret by maximizing the cumulative reward of job-server assignments over a time horizon, while keeping the total job holding cost bounded to ensure the stability of the queueing system. This problem is motivated by applications requiring resource allocation in network systems. In this problem, it is essential to control the tradeoff between reward maximization and fair allocation for the stability of the underlying queuing system (i.e., maximizing network throughput). To address this problem, we propose a scheduling algorithm based on a weighted proportional fair criteria augmented with marginal costs for reward maximization, incorporating a bandit algorithm tailored for bilinear rewards. Our algorithm achieves a sub-linear regret bound and a sub-linear mean holding cost (and queue length bound) of $\tilde{O}(\sqrt{T})$, respectively, with respect to the time horizon $T$, thus guaranteeing queuing system stability. Additionally, we establish stability conditions for distributed iterative algorithms for computing allocations, which are relevant to large-scale system applications. Finally, we demonstrate the efficiency of our algorithm through numerical experiments.

Flore Sentenac, Jialin Yi, Clément Calauzènes et al.

Finding an optimal matching in a weighted graph is a standard combinatorial problem. We consider its semi-bandit version where either a pair or a full matching is sampled sequentially. We prove that it is possible to leverage a rank-1 assumption on the adjacency matrix to reduce the sample complexity and the regret of off-the-shelf algorithms up to reaching a linear dependency in the number of vertices (up to poly log terms).

Dabeen Lee, Milan Vojnovic

We study a single-server scheduling problem for the objective of minimizing the expected cumulative holding cost incurred by jobs, where parameters defining stochastic job holding costs are unknown to the scheduler. We consider a general setting allowing for different job classes, where jobs of the same class have statistically identical holding costs and service times, with an arbitrary number of jobs across classes. In each time step, the server can process a job and observes random holding costs of the jobs that are yet to be completed. We consider a learning-based $cμ$ rule scheduling which starts with a preemption period of fixed duration, serving as a learning phase, and having gathered data about jobs, it switches to nonpreemptive scheduling. Our algorithms are designed to handle instances with large and small gaps in mean job holding costs and achieve near-optimal performance guarantees. The performance of algorithms is evaluated by regret, where the benchmark is the minimum possible total holding cost attained by the $cμ$ rule scheduling policy when the parameters of jobs are known. We show regret lower bounds and algorithms that achieve nearly matching regret upper bounds. Our numerical results demonstrate the efficacy of our algorithms and show that our regret analysis is nearly tight.

Dabeen Lee, Milan Vojnovic, Se-Young Yun

Motivated by recent developments in designing algorithms based on individual item scores for solving utility maximization problems, we study the framework of using test scores, defined as a statistic of observed individual item performance data, for solving the budgeted stochastic utility maximization problem. We extend an existing scoring mechanism, namely the replication test scores, to incorporate heterogeneous item costs as well as item values. We show that a natural greedy algorithm that selects items solely based on their replication test scores outputs solutions within a constant factor of the optimum for a broad class of utility functions. Our algorithms and approximation guarantees assume that test scores are noisy estimates of certain expected values with respect to marginal distributions of individual item values, thus making our algorithms practical and extending previous work that assumes noiseless estimates. Moreover, we show how our algorithm can be adapted to the setting where items arrive in a streaming fashion while maintaining the same approximation guarantee. We present numerical results, using synthetic data and data sets from the Academia.StackExchange Q&A forum, which show that our test score algorithm can achieve competitiveness, and in some cases better performance than a benchmark algorithm that requires access to a value oracle to evaluate function values.

Milan Vojnovic, Seyoung Yun, Kaifang Zhou

In this paper, we study a popular method for inference of the Bradley-Terry model parameters, namely the MM algorithm, for maximum likelihood estimation and maximum a posteriori probability estimation. This class of models includes the Bradley-Terry model of paired comparisons, the Rao-Kupper model of paired comparisons allowing for tie outcomes, the Luce choice model, and the Plackett-Luce ranking model. We establish tight characterizations of the convergence rate for the MM algorithm, and show that it is essentially equivalent to that of a gradient descent algorithm. For the maximum likelihood estimation, the convergence is shown to be linear with the rate crucially determined by the algebraic connectivity of the matrix of item pair co-occurrences in observed comparison data. For the Bayesian inference, the convergence rate is also shown to be linear, with the rate determined by a parameter of the prior distribution in a way that can make the convergence arbitrarily slow for small values of this parameter. We propose a simple modification of the classical MM algorithm that avoids the observed slow convergence issue and accelerates the convergence. The key component of the accelerated MM algorithm is a parameter rescaling performed at each iteration step that is carefully chosen based on theoretical analysis and characterisation of the convergence rate. Our experimental results, performed on both synthetic and real-world data, demonstrate the identified slow convergence issue of the classic MM algorithm, and show that significant efficiency gains can be obtained by our new proposed method.

Moez Draief, Konstantin Kutzkov, Kevin Scaman et al.

We present novel graph kernels for graphs with node and edge labels that have ordered neighborhoods, i.e. when neighbor nodes follow an order. Graphs with ordered neighborhoods are a natural data representation for evolving graphs where edges are created over time, which induces an order. Combining convolutional subgraph kernels and string kernels, we design new scalable algorithms for generation of explicit graph feature maps using sketching techniques. We obtain precise bounds for the approximation accuracy and computational complexity of the proposed approaches and demonstrate their applicability on real datasets. In particular, our experiments demonstrate that neighborhood ordering results in more informative features. For the special case of general graphs, i.e. graphs without ordered neighborhoods, the new graph kernels yield efficient and simple algorithms for the comparison of label distributions between graphs.

Virag Shah, Lennart Gulikers, Laurent Massoulie et al.

A matching in a two-sided market often incurs an externality: a matched resource may become unavailable to the other side of the market, at least for a while. This is especially an issue in online platforms involving human experts as the expert resources are often scarce. The efficient utilization of experts in these platforms is made challenging by the fact that the information available about the parties involved is usually limited. To address this challenge, we develop a model of a task-expert matching system where a task is matched to an expert using not only the prior information about the task but also the feedback obtained from the past matches. In our model the tasks arrive online while the experts are fixed and constrained by a finite service capacity. For this model, we characterize the maximum task resolution throughput a platform can achieve. We show that the natural greedy approaches where each expert is assigned a task most suitable to her skill is suboptimal, as it does not internalize the above externality. We develop a throughput optimal backpressure algorithm which does so by accounting for the `congestion' among different task types. Finally, we validate our model and confirm our theoretical findings with data-driven simulations via logs of Math.StackExchange, a StackOverflow forum dedicated to mathematics.

Dan Alistarh, Demjan Grubic, Jerry Li et al.

Parallel implementations of stochastic gradient descent (SGD) have received significant research attention, thanks to excellent scalability properties of this algorithm, and to its efficiency in the context of training deep neural networks. A fundamental barrier for parallelizing large-scale SGD is the fact that the cost of communicating the gradient updates between nodes can be very large. Consequently, lossy compression heuristics have been proposed, by which nodes only communicate quantized gradients. Although effective in practice, these heuristics do not always provably converge, and it is not clear whether they are optimal. In this paper, we propose Quantized SGD (QSGD), a family of compression schemes which allow the compression of gradient updates at each node, while guaranteeing convergence under standard assumptions. QSGD allows the user to trade off compression and convergence time: it can communicate a sublinear number of bits per iteration in the model dimension, and can achieve asymptotically optimal communication cost. We complement our theoretical results with empirical data, showing that QSGD can significantly reduce communication cost, while being competitive with standard uncompressed techniques on a variety of real tasks. In particular, experiments show that gradient quantization applied to training of deep neural networks for image classification and automated speech recognition can lead to significant reductions in communication cost, and end-to-end training time. For instance, on 16 GPUs, we are able to train a ResNet-152 network on ImageNet 1.8x faster to full accuracy. Of note, we show that there exist generic parameter settings under which all known network architectures preserve or slightly improve their full accuracy when using quantization.

Fajwel Fogel, Alexandre d'Aspremont, Milan Vojnovic

We describe a seriation algorithm for ranking a set of items given pairwise comparisons between these items. Intuitively, the algorithm assigns similar rankings to items that compare similarly with all others. It does so by constructing a similarity matrix from pairwise comparisons, using seriation methods to reorder this matrix and construct a ranking. We first show that this spectral seriation algorithm recovers the true ranking when all pairwise comparisons are observed and consistent with a total order. We then show that ranking reconstruction is still exact when some pairwise comparisons are corrupted or missing, and that seriation based spectral ranking is more robust to noise than classical scoring methods. Finally, we bound the ranking error when only a random subset of the comparions are observed. An additional benefit of the seriation formulation is that it allows us to solve semi-supervised ranking problems. Experiments on both synthetic and real datasets demonstrate that seriation based spectral ranking achieves competitive and in some cases superior performance compared to classical ranking methods.