arXiv subjects

Lunjia Hu

Publications and source records attributed to Lunjia Hu.

At least 19 recordsLinked to original sources

arXiv2026

Truthful Calibration Measures for Sequential Prediction

Calibration requires probabilistic reports to be conditionally unbiased and reliably interpretable as probabilities. A calibration measure assigns numerical error to miscalibrated reports. Haghtalab et al. (2024) proposed an approximately truthful calibration measure for online prediction, leaving open whether exact truthfulness is compatible with completeness and soundness. We resolve this question negatively for sequential binary prediction: exact truthfulness is incompatible with completeness and soundness, even for independent outcomes. We then show that this impossibility is specific to exact truthfulness. We give two general reductions from a base calibration measure, producing additively and multiplicatively approximately truthful calibration measures, respectively. Applying the multiplicative reduction, for every $0 < \varepsilon < 1$ we construct a sound and complete calibration measure that is $(1+\exp(-T^{(1-\varepsilon)/2}/2))$-multiplicatively truthful. This improves the approximate-truthfulness guarantee of Haghtalab et al. (2024).

cs.DS↗

arXiv2026

Optimal Recalibration of an Online Predictor

We study the problem of recalibrating an online predictor [KE17, OKS24]: given an arbitrary "hint" sequence of forecasts, the learner must output new predictions that are calibrated while incurring small excess error relative to the original forecasts, under a proper loss. We give an online algorithm that achieves $(\varepsilon, \varepsilon^2)$-recalibration for Lipschitz proper losses in $T \approx \varepsilon^{-3}$ rounds, using an imbalanced extension of the recent simultaneous Blackwell approachability reduction framework of [HTY26]. We show that this tradeoff is optimal by proving a matching lower bound for recalibrating against the squared loss. We also prove a companion $\mathcal{K}_2$-recalibration theorem that obtains the same tradeoffs up to a logarithmic factor. As our main application, we show how our recalibration algorithms can be combined with the online refinement method of [FH23] to obtain simultaneous $\varepsilon$-calibration and $\varepsilon^2$-calibeating for smooth proper losses at the same asymptotic rate, improving upon prior works that achieved these properties separately or with a worse $\varepsilon$ dependence. In particular, the $\mathcal{K}_2$ variant answers a question of [CHJL26] on simultaneously achieving near-optimal calibeating and calibration rates. We also derive extensions to settings with multiple hint sequences. Finally, we empirically evaluate our algorithms on a classification dataset undergoing distribution shift.

stat.ML↗

arXiv2026

How Many Domains Suffice for Domain Generalization? A Tight Characterization via the Domain Shattering Dimension

We study a fundamental question of domain generalization: given a family of domains (i.e., data distributions), how many randomly sampled domains do we need to collect data from in order to learn a model that performs reasonably well on every seen and unseen domain in the family? We model this problem in the PAC framework and introduce a new combinatorial measure, which we call the domain shattering dimension. We show that this dimension characterizes the domain sample complexity. Furthermore, we establish a tight quantitative relationship between the domain shattering dimension and the classic VC dimension, demonstrating that every hypothesis class that is learnable in the standard PAC setting is also learnable in our setting.

cs.LG↗

arXiv2026

Testable and Actionable Calibration for Full Swap Regret

AI generated predictions increasingly inform decision making in critical tasks, and therefore must be trustworthy. One widely used measure of trustworthiness is calibration, which requires that the predictions match the true frequencies and can be treated like real probabilities of a given outcome. However, defining calibration is subtle, and designing good measures of calibration error has been an active topic of recent research. The first goal is to find calibration measures that are actionable, meaning they can inform decision makers about their utility loss when predictions are treated as true probabilities, which is known as swap regret. The second goal is to find calibration measures that are testable, meaning that calibration error can be measured from a small sample of predictions and outcomes. Although these are very basic requirements, there is no existing calibration measure that fully satisfies both properties, and all existing measures relax actionability by bounding a weaker notion of swap regret, or relax testability by having suboptimal estimation error. We introduce a new calibration measure, Soft-Binned Calibration Decision Loss (SCDL), which we prove is fully actionable without weakening either requirement, and testable with nearly optimal error rate. In addition, SCDL satisfies other desired properties such as continuity and consistency. We also provide a set of experiments confirming that the theoretical advantages of SCDL compared to other measures lead to better performance in practice.

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arXiv2026

Truthful Calibration Errors for Multi-Class Prediction

Calibrated predictions are useful because their numerical values can be interpreted as probabilities. Calibration errors are therefore widely used to evaluate, compare, and tune probabilistic predictors. Recently, Haghtalab et al. (2024) introduced an additional requirement for such measures: truthfulness. A calibration measure is truthful if a predictor minimizes its expected measured error by reporting the true conditional label distribution. Many standard empirical calibration errors are non-truthful: a predictor may appear better calibrated by distorting its probabilities rather than reporting them truthfully. We study the practical role of truthfulness for calibration measurement in multiclass prediction. First, we introduce perfectly truthful calibration errors for multidimensional linear properties of the label distribution, generalizing the truthful calibration error for binary predictions in Hartline et al. (2025). This framework includes full multiclass calibration and classwise calibration. We also identify a truthful correction for confidence calibration. Second, we characterize the decision-theoretic implications of these truthful errors. For calibrated predictors, truthful calibration errors preserve the Blackwell dominance: a more informative calibrated predictor receives no larger expected error. Third, we show that this decision-theoretic interpretation explains and mitigates the well-observed ranking robustness problem of binned calibration errors. Empirically, non-truthful confidence-based errors can reverse model rankings when the number of bins changes, while our truthful errors give more stable rankings across binning choices.

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arXiv2026

A Perfectly Truthful Calibration Measure

Calibration requires that predictions are conditionally unbiased and, therefore, reliably interpretable as probabilities. A calibration measure quantifies how far a predictor is from perfect calibration. As introduced by Haghtalab et al. (2024), a calibration measure is truthful if it is minimized in expectation when a predictor outputs the ground-truth probabilities. Predicting the true probabilities guarantees perfect calibration, but in reality, when calibration is evaluated on a random sample, all known calibration measures incentivize predictors to lie in order to appear more calibrated. Such lack of truthfulness motivated Haghtalab et al. (2024) and Qiao and Zhao (2025) to construct approximately truthful calibration measures in the sequential prediction setting, but no perfectly truthful calibration measure was known to exist even in the more basic batch setting. We design a simple, perfectly and strictly truthful, sound and complete calibration measure in the batch setting: averaged two-bin calibration error (ATB). ATB is quadratically related to two existing calibration measures: the smooth calibration error smCal and the lower distance to calibration distCal. The simplicity in our definition of ATB makes it efficient and straightforward to compute, allowing us to give the first linear-time calibration testing algorithm, improving a result of Hu et al. (2024). We also introduce a general recipe for constructing truthful measures based on the variance additivity of independent random variables, which proves the truthfulness of ATB as a special case and allows us to construct other truthful calibration measures such as quantile-binned l_2-ECE.

cs.LG↗

arXiv2026

Equitable Evaluation via Elicitation

Individuals with similar qualifications and skills may vary in their demeanor, or outward manner: some tend toward self-promotion while others are modest to the point of omitting crucial information. Comparing the self-descriptions of equally qualified job-seekers with different self-presentation styles is therefore problematic. We build an interactive AI for skill elicitation that provides accurate determination of skills while simultaneously allowing individuals to speak in their own voice. Such a system can be deployed, for example, when a new user joins a professional networking platform, or when matching employees to needs during a company reorganization. To obtain sufficient training data, we train an LLM to act as synthetic humans. Elicitation mitigates endogenous bias arising from individuals' own self-reports. To address systematic model bias we enforce a mathematically rigorous notion of equitability ensuring that the covariance between self-presentation manner and skill evaluation error is small.

cs.LG↗

arXiv2026

Simultaneous Blackwell Approachability and Applications to Multiclass Omniprediction

Omniprediction is a learning problem that requires suboptimality bounds for each of a family of losses $\mathcal{L}$ against a family of comparator predictors $\mathcal{C}$. We initiate the study of omniprediction in a multiclass setting, where the comparator family $\mathcal{C}$ may be infinite. Our main result is an extension of the recent binary omniprediction algorithm of [OKK25] to the multiclass setting, with sample complexity (in statistical settings) or regret horizon (in online settings) $\approx \varepsilon^{-(k+1)}$, for $\varepsilon$-omniprediction in a $k$-class prediction problem. En route to proving this result, we design a framework of potential broader interest for solving Blackwell approachability problems where multiple sets must simultaneously be approached via coupled actions.

cs.DS↗

arXiv2026

Near-optimal Swap Regret Minimization for Convex Losses

We give a randomized online algorithm that guarantees near-optimal $\widetilde O(\sqrt T)$ expected swap regret against any sequence of $T$ adaptively chosen Lipschitz convex losses on the unit interval. This improves the previous best bound of $\widetilde O(T^{2/3})$ and answers an open question of Fishelson et al. [2025b]. In addition, our algorithm is efficient: it runs in $\mathsf{poly}(T)$ time. A key technical idea we develop to obtain this result is to discretize the unit interval into bins at multiple scales of granularity and simultaneously use all scales to make randomized predictions, which we call multi-scale binning and may be of independent interest. A direct corollary of our result is an efficient online algorithm for minimizing the calibration error for general elicitable properties. This result does not require the Lipschitzness assumption of the identification function needed in prior work, making it applicable to median calibration, for which we achieve the first $\widetilde O(\sqrt T)$ calibration error guarantee.

cs.LG↗

arXiv2025

Efficient Swap Multicalibration of Elicitable Properties

Multicalibration [HJKRR18] is an algorithmic fairness perspective that demands that the predictions of a predictor are correct conditional on themselves and membership in a collection of potentially overlapping subgroups of a population. The work of [NR23] established a surprising connection between multicalibration for an arbitrary property $Γ$ (e.g., mean or median) and property elicitation: a property $Γ$ can be multicalibrated if and only if it is elicitable, where elicitability is the notion that the true property value of a distribution can be obtained by solving a regression problem over the distribution. In the online setting, [NR23] proposed an inefficient algorithm that achieves $\sqrt T$ $\ell_2$-multicalibration error for a hypothesis class of group membership functions and an elicitable property $Γ$, after $T$ rounds of interaction between a forecaster and adversary. In this paper, we generalize multicalibration for an elicitable property $Γ$ from group membership functions to arbitrary bounded hypothesis classes and introduce a stronger notion -- swap multicalibration, following [GKR23]. Subsequently, we propose an oracle-efficient algorithm which, when given access to an online agnostic learner, achieves $T^{1/(r+1)}$ $\ell_r$-swap multicalibration error with high probability (for $r\ge2$) for a hypothesis class with bounded sequential Rademacher complexity and an elicitable property $Γ$. For the special case of $r=2$, this implies an oracle-efficient algorithm that achieves $T^{1/3}$ $\ell_2$-swap multicalibration error, which significantly improves on the previously established bounds for the problem [NR23, GMS25, LSS25a], and completely resolves an open question raised in [GJRR24] on the possibility of an oracle-efficient algorithm that achieves $\sqrt{T}$ $\ell_2$-mean multicalibration error by answering it in a strongly affirmative sense.

cs.LG↗

arXiv2025

Calibration through the Lens of Indistinguishability

Calibration is a classical notion from the forecasting literature which aims to address the question: how should predicted probabilities be interpreted? In a world where we only get to observe (discrete) outcomes, how should we evaluate a predictor that hypothesizes (continuous) probabilities over possible outcomes? The study of calibration has seen a surge of recent interest, given the ubiquity of probabilistic predictions in machine learning. This survey describes recent work on the foundational questions of how to define and measure calibration error, and what these measures mean for downstream decision makers who wish to use the predictions to make decisions. A unifying viewpoint that emerges is that of calibration as a form of indistinguishability, between the world hypothesized by the predictor and the real world (governed by nature or the Bayes optimal predictor). In this view, various calibration measures quantify the extent to which the two worlds can be told apart by certain classes of distinguishers or statistical measures.

cs.LG↗

arXiv2025

Generalized and Unified Equivalences between Hardness and Pseudoentropy

Pseudoentropy characterizations give quantitatively precise formulations of the relationship between computational hardness and computational randomness. We prove a unified pseudoentropy characterization that generalizes and strengthens previous results in both uniform and nonuniform models of computation. Our characterization applies to a general family of entropy notions, including Shannon entropy and min-entropy as special cases. Moreover, the characterizations for these different entropy notions can be witnessed simultaneously by a single universal function, which captures both computational hardness and computational randomness. A key technical insight is that weight-restricted calibration, from the recent literature on algorithmic fairness, together with standard computational indistinguishability (known as multiaccuracy in the fairness literature), suffices for proving pseudoentropy characterizations for general entropy notions. To obtain this combination of properties, we prove an enhanced version of the Leakage Simulation Lemma (Jetchev and Pietrzak, 2014), which in turn extends the Complexity Theoretic-Regularity Lemma (Trevisan, Tulsiani, and Vadhan, 2009) from boolean functions to ones over a larger alphabet. Our Enhanced Regularity/Leakage-Simulation Lemma enables us to obtain an exponential improvement in the dependence on the alphabet size compared with the pseudoentropy characterizations of Casacuberta, Dwork, and Vadhan (2024), which are based on the stronger notion of multicalibration. We also show that this exponential dependence on the alphabet size is inevitable for multicalibration and even for the weaker notion of calibrated multiaccuracy.

cs.CC↗

arXiv2025

Inducing Efficient and Equitable Professional Networks through Link Recommendations

Professional networks are a key determinant of individuals' labor market outcomes. They may also play a role in either exacerbating or ameliorating inequality of opportunity across demographic groups. In a theoretical model of professional network formation, we show that inequality can increase even without exogenous in-group preferences, confirming and complementing existing theoretical literature. Increased inequality emerges from the differential leverage privileged and unprivileged individuals have in forming connections due to their asymmetric ex ante prospects. This is a formalization of a source of inequality in the labor market which has not been previously explored. We next show how inequality-aware platforms may reduce inequality by subsidizing connections, through link recommendations that reduce costs, between privileged and unprivileged individuals. Indeed, mixed-privilege connections turn out to be welfare improving, over all possible equilibria, compared to not recommending links or recommending some smaller fraction of cross-group links. Taken together, these two findings reveal a stark reality: professional networking platforms that fail to foster integration in the link formation process risk reducing the platform's utility to its users and exacerbating existing labor market inequality.

cs.GT↗

arXiv2025

Omnipredicting Single-Index Models with Multi-Index Models

Recent work on supervised learning [GKR+22] defined the notion of omnipredictors, i.e., predictor functions $p$ over features that are simultaneously competitive for minimizing a family of loss functions $\mathcal{L}$ against a comparator class $\mathcal{C}$. Omniprediction requires approximating the Bayes-optimal predictor beyond the loss minimization paradigm, and has generated significant interest in the learning theory community. However, even for basic settings such as agnostically learning single-index models (SIMs), existing omnipredictor constructions require impractically-large sample complexities and runtimes, and output complex, highly-improper hypotheses. Our main contribution is a new, simple construction of omnipredictors for SIMs. We give a learner outputting an omnipredictor that is $\varepsilon$-competitive on any matching loss induced by a monotone, Lipschitz link function, when the comparator class is bounded linear predictors. Our algorithm requires $\approx \varepsilon^{-4}$ samples and runs in nearly-linear time, and its sample complexity improves to $\approx \varepsilon^{-2}$ if link functions are bi-Lipschitz. This significantly improves upon the only prior known construction, due to [HJKRR18, GHK+23], which used $\gtrsim \varepsilon^{-10}$ samples. We achieve our construction via a new, sharp analysis of the classical Isotron algorithm [KS09, KKKS11] in the challenging agnostic learning setting, of potential independent interest. Previously, Isotron was known to properly learn SIMs in the realizable setting, as well as constant-factor competitive hypotheses under the squared loss [ZWDD24]. As they are based on Isotron, our omnipredictors are multi-index models with $\approx \varepsilon^{-2}$ prediction heads, bringing us closer to the tantalizing goal of proper omniprediction for general loss families and comparators.

cs.LG↗

arXiv2024

Calibration Error for Decision Making

Calibration allows predictions to be reliably interpreted as probabilities by decision makers. We propose a decision-theoretic calibration error, the Calibration Decision Loss (CDL), defined as the maximum improvement in decision payoff obtained by calibrating the predictions, where the maximum is over all payoff-bounded decision tasks. Vanishing CDL guarantees the payoff loss from miscalibration vanishes simultaneously for all downstream decision tasks. We show separations between CDL and existing calibration error metrics, including the most well-studied metric Expected Calibration Error (ECE). Our main technical contribution is a new efficient algorithm for online calibration that achieves near-optimal $O(\frac{\log T}{\sqrt{T}})$ expected CDL, bypassing the $Ω(T^{-0.472})$ lower bound for ECE by Qiao and Valiant (2021).

cs.LG↗

arXiv2024

Testing Calibration in Nearly-Linear Time

In the recent literature on machine learning and decision making, calibration has emerged as a desirable and widely-studied statistical property of the outputs of binary prediction models. However, the algorithmic aspects of measuring model calibration have remained relatively less well-explored. Motivated by [BGHN23], which proposed a rigorous framework for measuring distances to calibration, we initiate the algorithmic study of calibration through the lens of property testing. We define the problem of calibration testing from samples where given $n$ draws from a distribution $\mathcal{D}$ on $(predictions, binary outcomes)$, our goal is to distinguish between the case where $\mathcal{D}$ is perfectly calibrated, and the case where $\mathcal{D}$ is $\varepsilon$-far from calibration. We make the simple observation that the empirical smooth calibration linear program can be reformulated as an instance of minimum-cost flow on a highly-structured graph, and design an exact dynamic programming-based solver for it which runs in time $O(n\log^2(n))$, and solves the calibration testing problem information-theoretically optimally in the same time. This improves upon state-of-the-art black-box linear program solvers requiring $Ω(n^ω)$ time, where $ω> 2$ is the exponent of matrix multiplication. We also develop algorithms for tolerant variants of our testing problem improving upon black-box linear program solvers, and give sample complexity lower bounds for alternative calibration measures to the one considered in this work. Finally, we present experiments showing the testing problem we define faithfully captures standard notions of calibration, and that our algorithms scale efficiently to accommodate large sample sizes.

cs.LG↗

arXiv2024

On Computationally Efficient Multi-Class Calibration

Consider a multi-class labelling problem, where the labels can take values in $[k]$, and a predictor predicts a distribution over the labels. In this work, we study the following foundational question: Are there notions of multi-class calibration that give strong guarantees of meaningful predictions and can be achieved in time and sample complexities polynomial in $k$? Prior notions of calibration exhibit a tradeoff between computational efficiency and expressivity: they either suffer from having sample complexity exponential in $k$, or needing to solve computationally intractable problems, or give rather weak guarantees. Our main contribution is a notion of calibration that achieves all these desiderata: we formulate a robust notion of projected smooth calibration for multi-class predictions, and give new recalibration algorithms for efficiently calibrating predictors under this definition with complexity polynomial in $k$. Projected smooth calibration gives strong guarantees for all downstream decision makers who want to use the predictor for binary classification problems of the form: does the label belong to a subset $T \subseteq [k]$: e.g. is this an image of an animal? It ensures that the probabilities predicted by summing the probabilities assigned to labels in $T$ are close to some perfectly calibrated binary predictor for that task. We also show that natural strengthenings of our definition are computationally hard to achieve: they run into information theoretic barriers or computational intractability. Underlying both our upper and lower bounds is a tight connection that we prove between multi-class calibration and the well-studied problem of agnostic learning in the (standard) binary prediction setting.

cs.LG↗

arXiv2024

Multigroup Robustness

To address the shortcomings of real-world datasets, robust learning algorithms have been designed to overcome arbitrary and indiscriminate data corruption. However, practical processes of gathering data may lead to patterns of data corruption that are localized to specific partitions of the training dataset. Motivated by critical applications where the learned model is deployed to make predictions about people from a rich collection of overlapping subpopulations, we initiate the study of multigroup robust algorithms whose robustness guarantees for each subpopulation only degrade with the amount of data corruption inside that subpopulation. When the data corruption is not distributed uniformly over subpopulations, our algorithms provide more meaningful robustness guarantees than standard guarantees that are oblivious to how the data corruption and the affected subpopulations are related. Our techniques establish a new connection between multigroup fairness and robustness.

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