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We present differentially private algorithms for high-dimensional mean estimation. Previous private estimators on distributions over $\mathbb{R}^d$ suffer from a curse of dimensionality, as they require $\Omega(d^{1/2})$ samples to achieve non-trivial error, even in cases where $O(1)$ samples suffice without privacy. This rate is unavoidable when the distribution is isotropic, namely, when the covariance is a multiple of the identity matrix, or when accuracy is measured with respect to the affine-invariant Mahalanobis distance. Yet, real-world data is often highly anisotropic, with signals concentrated on a small number of principal components. We develop estimators that are appropriate for such signals$\unicode{x2013}$our estimators are $(\varepsilon,\delta)$-differentially private and have sample complexity that is dimension-independent for anisotropic subgaussian distributions. Given $n$ samples from a distribution with known covariance-proxy $\Sigma$ and unknown mean $\mu$, we present an estimator $\hat{\mu}$ that achieves error $\|\hat{\mu}-\mu\|_2\leq \alpha$, as long as $n\gtrsim\mathrm{tr}(\Sigma)/\alpha^2+ \mathrm{tr}(\Sigma^{1/2})/(\alpha\varepsilon)$. In particular, when $\pmb{\sigma}^2=(\sigma_1^2, \ldots, \sigma_d^2)$ are the singular values of $\Sigma$, we have $\mathrm{tr}(\Sigma)=\|\pmb{\sigma}\|_2^2$ and $\mathrm{tr}(\Sigma^{1/2})=\|\pmb{\sigma}\|_1$, and hence our bound avoids dimension-dependence when the signal is concentrated in a few principal components. We show that this is the optimal sample complexity for this task up to logarithmic factors. Moreover, for the case of unknown covariance, we present an algorithm whose sample complexity has improved dependence on the dimension, from $d^{1/2}$ to $d^{1/4}$.

Variational inequalities (VIs) are a broad class of optimization problems encompassing machine learning problems ranging from standard convex minimization to more complex scenarios like min-max optimization and computing the equilibria of multi-player games. In convex optimization, strong convexity allows for fast statistical learning rates requiring only $\Theta(1/\epsilon)$ stochastic first-order oracle calls to find an $\epsilon$-optimal solution, rather than the standard $\Theta(1/\epsilon^2)$ calls. In this paper, we explain how one can similarly obtain fast $\Theta(1/\epsilon)$ rates for learning VIs that satisfy strong monotonicity, a generalization of strong convexity. Specifically, we demonstrate that standard stability-based generalization arguments for convex minimization extend directly to VIs when the domain admits a small covering, or when the operator is integrable and suboptimality is measured by potential functions; such as when finding equilibria in multi-player games.

Local Differential Privacy (LDP) offers strong privacy guarantees without requiring users to trust external parties. However, LDP applies uniform protection to all data features, including less sensitive ones, which degrades performance of downstream tasks. To overcome this limitation, we propose a Bayesian framework, Bayesian Coordinate Differential Privacy (BCDP), that enables feature-specific privacy quantification. This more nuanced approach complements LDP by adjusting privacy protection according to the sensitivity of each feature, enabling improved performance of downstream tasks without compromising privacy. We characterize the properties of BCDP and articulate its connections with standard non-Bayesian privacy frameworks. We further apply our BCDP framework to the problems of private mean estimation and ordinary least-squares regression. The BCDP-based approach obtains improved accuracy compared to a purely LDP-based approach, without compromising on privacy.

Reinforcement Learning from Human Feedback (RLHF) has become a popular approach to align language models (LMs) with human preferences. This method involves collecting a large dataset of human pairwise preferences across various text generations and using it to infer (implicitly or explicitly) a reward model. Numerous methods have been proposed to learn the reward model and align a LM with it. However, the costly process of collecting human preferences has received little attention and could benefit from theoretical insights. This paper addresses this issue and aims to formalize the reward training model in RLHF. We frame the selection of an effective dataset as a simple regret minimization task, using a linear contextual dueling bandit method. Given the potentially large number of arms, this approach is more coherent than the best-arm identification setting. We then propose an offline framework for solving this problem. Under appropriate assumptions - linearity of the reward model in the embedding space, and boundedness of the reward parameter - we derive bounds on the simple regret. Finally, we provide a lower bound that matches our upper bound up to constant and logarithmic terms. To our knowledge, this is the first theoretical contribution in this area to provide an offline approach as well as worst-case guarantees.

Practitioners have consistently observed three puzzling phenomena in transformer-based large language models (LLMs): attention sinks, value-state drains, and residual-state peaks, collectively referred to as extreme-token phenomena. These phenomena are characterized by certain so-called "sink tokens" receiving disproportionately high attention weights, exhibiting significantly smaller value states, and having much larger residual-state norms than those of other tokens. These extreme tokens give rise to various challenges in LLM inference, quantization, and interpretability. We elucidate the mechanisms behind extreme-token phenomena. First, we show that these phenomena arise in very simple architectures -- transformers with one to three layers -- trained on a toy model, the Bigram-Backcopy (BB) task. In this setting, we identify an active-dormant mechanism, where attention heads become sinks for specific input domains while remaining non-sinks for others. Our theoretical analysis of the training dynamics reveals that these phenomena are driven by a mutual reinforcement mechanism. Building on these insights, we propose strategies to mitigate extreme-token phenomena during pretraining, including replacing softmax with ReLU and Adam with SGD. Next, we extend our analysis to pretrained LLMs, including Llama and OLMo, showing that many attention heads exhibit a similar active-dormant mechanism as in the BB task, and that the mutual reinforcement mechanism also governs the emergence of extreme-token phenomena during LLM pretraining. Our results reveal that many of the static and dynamic properties of extreme-token phenomena predicted by the BB task align with observations in pretrained LLMs.

Emerging marketplaces for large language models and other large-scale machine learning (ML) models appear to exhibit market concentration, which has raised concerns about whether there are insurmountable barriers to entry in such markets. In this work, we study this issue from both an economic and an algorithmic point of view, focusing on a phenomenon that reduces barriers to entry. Specifically, an incumbent company risks reputational damage unless its model is sufficiently aligned with safety objectives, whereas a new company can more easily avoid reputational damage. To study this issue formally, we define a multi-objective high-dimensional regression framework that captures reputational damage, and we characterize the number of data points that a new company needs to enter the market. Our results demonstrate how multi-objective considerations can fundamentally reduce barriers to entry -- the required number of data points can be significantly smaller than the incumbent company's dataset size. En route to proving these results, we develop scaling laws for high-dimensional linear regression in multi-objective environments, showing that the scaling rate becomes slower when the dataset size is large, which could be of independent interest.

We provide a unified analysis of two-timescale gradient descent ascent (TTGDA) for solving structured nonconvex minimax optimization problems in the form of $\min_\textbf{x} \max_{\textbf{y} \in Y} f(\textbf{x}, \textbf{y})$, where the objective function $f(\textbf{x}, \textbf{y})$ is nonconvex in $\textbf{x}$ and concave in $\textbf{y}$, and the constraint set $Y \subseteq \mathbb{R}^n$ is convex and bounded. In the convex-concave setting, the single-timescale gradient descent ascent (GDA) algorithm is widely used in applications and has been shown to have strong convergence guarantees. In more general settings, however, it can fail to converge. Our contribution is to design TTGDA algorithms that are effective beyond the convex-concave setting, efficiently finding a stationary point of the function $\Phi(\cdot) := \max_{\textbf{y} \in Y} f(\cdot, \textbf{y})$. We also establish theoretical bounds on the complexity of solving both smooth and nonsmooth nonconvex-concave minimax optimization problems. To the best of our knowledge, this is the first systematic analysis of TTGDA for nonconvex minimax optimization, shedding light on its superior performance in training generative adversarial networks (GANs) and in other real-world application problems., Comment: A preliminary version [arXiv:1906.00331] of this paper, with a subset of the results that are presented here, was presented at ICML 2020; 44 Pages, 10 Figures

Collaborative learning offers a promising avenue for leveraging decentralized data. However, collaboration in groups of strategic learners is not a given. In this work, we consider strategic agents who wish to train a model together but have sampling distributions of different quality. The collaboration is organized by a benevolent aggregator who gathers samples so as to maximize total welfare, but is unaware of data quality. This setting allows us to shed light on the deleterious effect of adverse selection in collaborative learning. More precisely, we demonstrate that when data quality indices are private, the coalition may undergo a phenomenon known as unravelling, wherein it shrinks up to the point that it becomes empty or solely comprised of the worst agent. We show how this issue can be addressed without making use of external transfers, by proposing a novel method inspired by probabilistic verification. This approach makes the grand coalition a Nash equilibrium with high probability despite information asymmetry, thereby breaking unravelling.

In economic theory, the concept of externality refers to any indirect effect resulting from an interaction between players that affects the social welfare. Most of the models within which externality has been studied assume that agents have perfect knowledge of their environment and preferences. This is a major hindrance to the practical implementation of many proposed solutions. To address this issue, we consider a two-player bandit setting where the actions of one of the players affect the other player and we extend the Coase theorem [Coase, 1960]. This result shows that the optimal approach for maximizing the social welfare in the presence of externality is to establish property rights, i.e., enable transfers and bargaining between the players. Our work removes the classical assumption that bargainers possess perfect knowledge of the underlying game. We first demonstrate that in the absence of property rights, the social welfare breaks down. We then design a policy for the players which allows them to learn a bargaining strategy which maximizes the total welfare, recovering the Coase theorem under uncertainty.

Science and technology have a growing need for effective mechanisms that ensure reliable, controlled performance from black-box machine learning algorithms. These performance guarantees should ideally hold conditionally on the input-that is the performance guarantees should hold, at least approximately, no matter what the input. However, beyond stylized discrete groupings such as ethnicity and gender, the right notion of conditioning can be difficult to define. For example, in problems such as image segmentation, we want the uncertainty to reflect the intrinsic difficulty of the test sample, but this may be difficult to capture via a conditioning event. Building on the recent work of Gibbs et al. [2023], we propose a methodology for achieving approximate conditional control of statistical risks-the expected value of loss functions-by adapting to the difficulty of test samples. Our framework goes beyond traditional conditional risk control based on user-provided conditioning events to the algorithmic, data-driven determination of appropriate function classes for conditioning. We apply this framework to various regression and segmentation tasks, enabling finer-grained control over model performance and demonstrating that by continuously monitoring and adjusting these parameters, we can achieve superior precision compared to conventional risk-control methods.

We study collaborative learning systems in which the participants are competitors who will defect from the system if they lose revenue by collaborating. As such, we frame the system as a duopoly of competitive firms who are each engaged in training machine-learning models and selling their predictions to a market of consumers. We first examine a fully collaborative scheme in which both firms share their models with each other and show that this leads to a market collapse with the revenues of both firms going to zero. We next show that one-sided collaboration in which only the firm with the lower-quality model shares improves the revenue of both firms. Finally, we propose a more equitable, *defection-free* scheme in which both firms share with each other while losing no revenue, and we show that our algorithm converges to the Nash bargaining solution.

To address issues of group-level fairness in machine learning, it is natural to adjust model parameters based on specific fairness objectives over a sensitive-attributed validation set. Such an adjustment procedure can be cast within a meta-learning framework. However, naive integration of fairness goals via meta-learning can cause hypergradient conflicts for subgroups, resulting in unstable convergence and compromising model performance and fairness. To navigate this issue, we frame the resolution of hypergradient conflicts as a multi-player cooperative bargaining game. We introduce a two-stage meta-learning framework in which the first stage involves the use of a Nash Bargaining Solution (NBS) to resolve hypergradient conflicts and steer the model toward the Pareto front, and the second stage optimizes with respect to specific fairness goals. Our method is supported by theoretical results, notably a proof of the NBS for gradient aggregation free from linear independence assumptions, a proof of Pareto improvement, and a proof of monotonic improvement in validation loss. We also show empirical effects across various fairness objectives in six key fairness datasets and two image classification tasks.

We consider a dynamic mechanism design problem where an auctioneer sells an indivisible good to groups of buyers in every round, for a total of $T$ rounds. The auctioneer aims to maximize their discounted overall revenue while adhering to a fairness constraint that guarantees a minimum average allocation for each group. We begin by studying the static case ($T=1$) and establish that the optimal mechanism involves two types of subsidization: one that increases the overall probability of allocation to all buyers, and another that favors the groups which otherwise have a lower probability of winning the item. We then extend our results to the dynamic case by characterizing a set of recursive functions that determine the optimal allocation and payments in each round. Notably, our results establish that in the dynamic case, the seller, on the one hand, commits to a participation bonus to incentivize truth-telling, and on the other hand, charges an entry fee for every round. Moreover, the optimal allocation once more involves subsidization, which its extent depends on the difference in future utilities for both the seller and buyers when allocating the item to one group versus the others. Finally, we present an approximation scheme to solve the recursive equations and determine an approximately optimal and fair allocation efficiently., Comment: A shorter conference version has been accepted at the Advances in Neural Information Processing Systems (NeurIPS) 2024

Auto-regressive large language models (LLMs) show impressive capacities to solve many complex reasoning tasks while struggling with some simple logical reasoning tasks such as inverse search: when trained on '$A \to B$' (e.g., 'Tom is the parent of John'), LLM fails to directly conclude '$B \gets A$' (e.g., 'John is the child of Tom') during inference even if the two sentences are semantically identical, which is known as the 'reversal curse'. In this paper, we theoretically analyze the reversal curse via the training dynamics of (stochastic) gradient descent for two auto-regressive models: (1) a bilinear model that can be viewed as a simplification of a one-layer transformer; (2) one-layer transformers under certain assumptions. Our analysis reveals that for both models, the reversal curse is a consequence of the (effective) model weights 'asymmetry', i.e., the increase of weights from a token $A$ to token $B$ during training does not necessarily cause the increase of the weights from $B$ to $A$, which is caused by the training dynamics under certain choice of loss function and the optimization space of model parameters. Moreover, our analysis can be naturally applied to other logical reasoning tasks such as chain-of-thought (COT), which provides a new perspective different from previous work that focuses on expressivity. Finally, we conduct experiments to validate our theory on multi-layer transformers under different settings. Our code is available at https://github.com/marlo-z/reversal_curse_analysis/., Comment: 41 pages, 18 figures, NeurIPS 2024

Evaluating the causal impacts of possible interventions is crucial for informing decision-making, especially towards improving access to opportunity. However, if causal effects are heterogeneous and predictable from covariates, personalized treatment decisions can improve individual outcomes and contribute to both efficiency and equity. In practice, however, causal researchers do not have a single outcome in mind a priori and often collect multiple outcomes of interest that are noisy estimates of the true target of interest. For example, in government-assisted social benefit programs, policymakers collect many outcomes to understand the multidimensional nature of poverty. The ultimate goal is to learn an optimal treatment policy that in some sense maximizes multiple outcomes simultaneously. To address such issues, we present a data-driven dimensionality-reduction methodology for multiple outcomes in the context of optimal policy learning with multiple objectives. We learn a low-dimensional representation of the true outcome from the observed outcomes using reduced rank regression. We develop a suite of estimates that use the model to denoise observed outcomes, including commonly-used index weightings. These methods improve estimation error in policy evaluation and optimization, including on a case study of real-world cash transfer and social intervention data. Reducing the variance of noisy social outcomes can improve the performance of algorithmic allocations.

Collaboration between different data centers is often challenged by heterogeneity across sites. To account for the heterogeneity, the state-of-the-art method is to re-weight the covariate distributions in each site to match the distribution of the target population. Nevertheless, this method could easily fail when a certain site couldn't cover the entire population. Moreover, it still relies on the concept of traditional meta-analysis after adjusting for the distribution shift. In this work, we propose a collaborative inverse propensity score weighting estimator for causal inference with heterogeneous data. Instead of adjusting the distribution shift separately, we use weighted propensity score models to collaboratively adjust for the distribution shift. Our method shows significant improvements over the methods based on meta-analysis when heterogeneity increases. To account for the vulnerable density estimation, we further discuss the double machine method and show the possibility of using nonparametric density estimation with d<8 and a flexible machine learning method to guarantee asymptotic normality. We propose a federated learning algorithm to collaboratively train the outcome model while preserving privacy. Using synthetic and real datasets, we demonstrate the advantages of our method., Comment: submitted to ICML

We study price-discrimination games between buyers and a seller where privacy arises endogenously--that is, utility maximization yields equilibrium strategies where privacy occurs naturally. In this game, buyers with a high valuation for a good have an incentive to keep their valuation private, lest the seller charge them a higher price. This yields an equilibrium where some buyers will send a signal that misrepresents their type with some probability; we refer to this as buyer-induced privacy. When the seller is able to publicly commit to providing a certain privacy level, we find that their equilibrium response is to commit to ignore buyers' signals with some positive probability; we refer to this as seller-induced privacy. We then turn our attention to a repeated interaction setting where the game parameters are unknown and the seller cannot credibly commit to a level of seller-induced privacy. In this setting, players must learn strategies based on information revealed in past rounds. We find that, even without commitment ability, seller-induced privacy arises as a result of reputation building. We characterize the resulting seller-induced privacy and seller's utility under no-regret and no-policy-regret learning algorithms and verify these results through simulations., Comment: To appear in Symposium on Foundations of Responsible Computing (FORC 2024)

Decision-making pipelines are generally characterized by tradeoffs among various risk functions. It is often desirable to manage such tradeoffs in a data-adaptive manner. As we demonstrate, if this is done naively, state-of-the art uncertainty quantification methods can lead to significant violations of putative risk guarantees. To address this issue, we develop methods that permit valid control of risk when threshold and tradeoff parameters are chosen adaptively. Our methodology supports monotone and nearly-monotone risks, but otherwise makes no distributional assumptions. To illustrate the benefits of our approach, we carry out numerical experiments on synthetic data and the large-scale vision dataset MS-COCO., Comment: 27 pages, 10 figures

The acquisition of training data is crucial for machine learning applications. Data markets can increase the supply of data, particularly in data-scarce domains such as healthcare, by incentivizing potential data providers to join the market. A major challenge for a data buyer in such a market is choosing the most valuable data points from a data seller. Unlike prior work in data valuation, which assumes centralized data access, we propose a federated approach to the data acquisition problem that is inspired by linear experimental design. Our proposed data acquisition method achieves lower prediction error without requiring labeled validation data and can be optimized in a fast and federated procedure. The key insight of our work is that a method that directly estimates the benefit of acquiring data for test set prediction is particularly compatible with a decentralized market setting., Comment: 31 pages, 16 figures, To appear in NeurIPS 2024

The evaluation of machine learning models using human-labeled validation data can be expensive and time-consuming. AI-labeled synthetic data can be used to decrease the number of human annotations required for this purpose in a process called autoevaluation. We suggest efficient and statistically principled algorithms for this purpose that improve sample efficiency while remaining unbiased. These algorithms increase the effective human-labeled sample size by up to 50% on experiments with GPT-4., Comment: New experiments, fix fig 1

Large Language Models (LLMs) have unlocked new capabilities and applications; however, evaluating the alignment with human preferences still poses significant challenges. To address this issue, we introduce Chatbot Arena, an open platform for evaluating LLMs based on human preferences. Our methodology employs a pairwise comparison approach and leverages input from a diverse user base through crowdsourcing. The platform has been operational for several months, amassing over 240K votes. This paper describes the platform, analyzes the data we have collected so far, and explains the tried-and-true statistical methods we are using for efficient and accurate evaluation and ranking of models. We confirm that the crowdsourced questions are sufficiently diverse and discriminating and that the crowdsourced human votes are in good agreement with those of expert raters. These analyses collectively establish a robust foundation for the credibility of Chatbot Arena. Because of its unique value and openness, Chatbot Arena has emerged as one of the most referenced LLM leaderboards, widely cited by leading LLM developers and companies. Our demo is publicly available at \url{https://chat.lmsys.org}.

This work considers a repeated principal-agent bandit game, where the principal can only interact with her environment through the agent. The principal and the agent have misaligned objectives and the choice of action is only left to the agent. However, the principal can influence the agent's decisions by offering incentives which add up to his rewards. The principal aims to iteratively learn an incentive policy to maximize her own total utility. This framework extends usual bandit problems and is motivated by several practical applications, such as healthcare or ecological taxation, where traditionally used mechanism design theories often overlook the learning aspect of the problem. We present nearly optimal (with respect to a horizon $T$) learning algorithms for the principal's regret in both multi-armed and linear contextual settings. Finally, we support our theoretical guarantees through numerical experiments.

Online platforms and regulators face a continuing problem of designing effective evaluation metrics. While tools for collecting and processing data continue to progress, this has not addressed the problem of "unknown unknowns", or fundamental informational limitations on part of the evaluator. To guide the choice of metrics in the face of this informational problem, we turn to the evaluated agents themselves, who may have more information about how to measure their own outcomes. We model this interaction as an agency game, where we ask: "When does an agent have an incentive to reveal the observability of a metric to their evaluator?" We show that an agent will prefer to reveal metrics that differentiate the most difficult tasks from the rest, and conceal metrics that differentiate the easiest. We further show that the agent can prefer to reveal a metric "garbled" with noise over both fully concealing and fully revealing. This indicates an economic value to privacy that yields Pareto improvement for both the agent and evaluator. We demonstrate these findings on data from online rideshare platforms.

We study a three-layer data market comprising users (data owners), platforms, and a data buyer. Each user benefits from platform services in exchange for data, incurring privacy loss when their data, albeit noisily, is shared with the buyer. The user chooses platforms to share data with, while platforms decide on data noise levels and pricing before selling to the buyer. The buyer selects platforms to purchase data from. We model these interactions via a multi-stage game, focusing on the subgame Nash equilibrium. We find that when the buyer places a high value on user data (and platforms can command high prices), all platforms offer services to the user who joins and shares data with every platform. Conversely, when the buyer's valuation of user data is low, only large platforms with low service costs can afford to serve users. In this scenario, users exclusively join and share data with these low-cost platforms. Interestingly, increased competition benefits the buyer, not the user: as the number of platforms increases, the user utility does not necessarily improve while the buyer utility improves. However, increasing the competition improves the overall utilitarian welfare. Building on our analysis, we then study regulations to improve the user utility. We discover that banning data sharing maximizes user utility only when all platforms are low-cost. In mixed markets of high- and low-cost platforms, users prefer a minimum noise mandate over a sharing ban. Imposing this mandate on high-cost platforms and banning data sharing for low-cost ones further enhances user utility.

We study a producer's problem of selling a product to a continuum of privacy-conscious consumers, where the producer can implement third-degree price discrimination, offering different prices to different market segments. We consider a privacy mechanism that provides a degree of protection by probabilistically masking each market segment. We establish that the resultant set of all consumer-producer utilities forms a convex polygon, characterized explicitly as a linear mapping of a certain high-dimensional convex polytope into $\mathbb{R}^2$. This characterization enables us to investigate the impact of the privacy mechanism on both producer and consumer utilities. In particular, we establish that the privacy constraint always hurts the producer by reducing both the maximum and minimum utility achievable. From the consumer's perspective, although the privacy mechanism ensures an increase in the minimum utility compared to the non-private scenario, interestingly, it may reduce the maximum utility. Finally, we demonstrate that increasing the privacy level does not necessarily intensify these effects. For instance, the maximum utility for the producer or the minimum utility for the consumer may exhibit nonmonotonic behavior in response to an increase of the privacy level.

Reinforcement Learning from Human Feedback (RLHF) is a pivotal technique that aligns language models closely with human-centric values. The initial phase of RLHF involves learning human values using a reward model from ranking data. It is observed that the performance of the reward model degrades after one epoch of training, and optimizing too much against the learned reward model eventually hinders the true objective. This paper delves into these issues, leveraging the theoretical insights to design improved reward learning algorithm termed 'Iterative Data Smoothing' (IDS). The core idea is that during each training epoch, we not only update the model with the data, but also update the date using the model, replacing hard labels with soft labels. Our empirical findings highlight the superior performance of this approach over the traditional methods.

We study statistical watermarking by formulating it as a hypothesis testing problem, a general framework which subsumes all previous statistical watermarking methods. Key to our formulation is a coupling of the output tokens and the rejection region, realized by pseudo-random generators in practice, that allows non-trivial trade-offs between the Type I error and Type II error. We characterize the Uniformly Most Powerful (UMP) watermark in the general hypothesis testing setting and the minimax Type II error in the model-agnostic setting. In the common scenario where the output is a sequence of $n$ tokens, we establish nearly matching upper and lower bounds on the number of i.i.d. tokens required to guarantee small Type I and Type II errors. Our rate of $\Theta(h^{-1} \log (1/h))$ with respect to the average entropy per token $h$ highlights potentials for improvement from the rate of $h^{-2}$ in the previous works. Moreover, we formulate the robust watermarking problem where the user is allowed to perform a class of perturbations on the generated texts, and characterize the optimal Type II error of robust UMP tests via a linear programming problem. To the best of our knowledge, this is the first systematic statistical treatment on the watermarking problem with near-optimal rates in the i.i.d. setting, which might be of interest for future works.

Calibration of machine learning classifiers is necessary to obtain reliable and interpretable predictions, bridging the gap between model confidence and actual probabilities. One prominent technique, isotonic regression (IR), aims at calibrating binary classifiers by minimizing the cross entropy on a calibration set via monotone transformations. IR acts as an adaptive binning procedure, which allows achieving a calibration error of zero, but leaves open the issue of the effect on performance. In this paper, we first prove that IR preserves the convex hull of the ROC curve -- an essential performance metric for binary classifiers. This ensures that a classifier is calibrated while controlling for overfitting of the calibration set. We then present a novel generalization of isotonic regression to accommodate classifiers with K classes. Our method constructs a multidimensional adaptive binning scheme on the probability simplex, again achieving a multi-class calibration error equal to zero. We regularize this algorithm by imposing a form of monotony that preserves the K-dimensional ROC surface of the classifier. We show empirically that this general monotony criterion is effective in striking a balance between reducing cross entropy loss and avoiding overfitting of the calibration set.

Recent developments in domains such as non-local games, quantum interactive proofs, and quantum generative adversarial networks have renewed interest in quantum game theory and, specifically, quantum zero-sum games. Central to classical game theory is the efficient algorithmic computation of Nash equilibria, which represent optimal strategies for both players. In 2008, Jain and Watrous proposed the first classical algorithm for computing equilibria in quantum zero-sum games using the Matrix Multiplicative Weight Updates (MMWU) method to achieve a convergence rate of $\mathcal{O}(d/\epsilon^2)$ iterations to $\epsilon$-Nash equilibria in the $4^d$-dimensional spectraplex. In this work, we propose a hierarchy of quantum optimization algorithms that generalize MMWU via an extra-gradient mechanism. Notably, within this proposed hierarchy, we introduce the Optimistic Matrix Multiplicative Weights Update (OMMWU) algorithm and establish its average-iterate convergence complexity as $\mathcal{O}(d/\epsilon)$ iterations to $\epsilon$-Nash equilibria. This quadratic speed-up relative to Jain and Watrous' original algorithm sets a new benchmark for computing $\epsilon$-Nash equilibria in quantum zero-sum games., Comment: 53 pages, 7 figures, QTML 2023 (Accepted (Long Talk))

We study the role of regulatory inspections in a contract design problem in which a principal interacts separately with multiple agents. Each agent's hidden action includes a dimension that determines whether they undertake an extra costly step to adhere to safety protocols. The principal's objective is to use payments combined with a limited budget for random inspections to incentivize agents towards safety-compliant actions that maximize the principal's utility. We first focus on the single-agent setting with linear contracts and present an efficient algorithm that characterizes the optimal linear contract, which includes both payment and random inspection. We further investigate how the optimal contract changes as the inspection cost or the cost of adhering to safety protocols vary. Notably, we demonstrate that the agent's compensation increases if either of these costs escalates. However, while the probability of inspection decreases with rising inspection costs, it demonstrates nonmonotonic behavior as a function of the safety action costs. Lastly, we explore the multi-agent setting, where the principal's challenge is to determine the best distribution of inspection budgets among all agents. We propose an efficient approach based on dynamic programming to find an approximately optimal allocation of inspection budget across contracts. We also design a random sequential scheme to determine the inspector's assignments, ensuring each agent is inspected at most once and at the desired probability. Finally, we present a case study illustrating that a mere difference in the cost of inspection across various agents can drive the principal's decision to forego inspecting a significant fraction of them, concentrating its entire budget on those that are less costly to inspect.

Kernel-based optimal transport (OT) estimators offer an alternative, functional estimation procedure to address OT problems from samples. Recent works suggest that these estimators are more statistically efficient than plug-in (linear programming-based) OT estimators when comparing probability measures in high-dimensions~\citep{Vacher-2021-Dimension}. Unfortunately, that statistical benefit comes at a very steep computational price: because their computation relies on the short-step interior-point method (SSIPM), which comes with a large iteration count in practice, these estimators quickly become intractable w.r.t. sample size $n$. To scale these estimators to larger $n$, we propose a nonsmooth fixed-point model for the kernel-based OT problem, and show that it can be efficiently solved via a specialized semismooth Newton (SSN) method: We show, exploring the problem's structure, that the per-iteration cost of performing one SSN step can be significantly reduced in practice. We prove that our SSN method achieves a global convergence rate of $O(1/\sqrt{k})$, and a local quadratic convergence rate under standard regularity conditions. We show substantial speedups over SSIPM on both synthetic and real datasets., Comment: Accepted by AISTATS 2024; Fix some inaccuracy in the definition and proof; 24 pages, 36 figures

Online gradient descent (OGD) is well known to be doubly optimal under strong convexity or monotonicity assumptions: (1) in the single-agent setting, it achieves an optimal regret of $\Theta(\log T)$ for strongly convex cost functions; and (2) in the multi-agent setting of strongly monotone games, with each agent employing OGD, we obtain last-iterate convergence of the joint action to a unique Nash equilibrium at an optimal rate of $\Theta(\frac{1}{T})$. While these finite-time guarantees highlight its merits, OGD has the drawback that it requires knowing the strong convexity/monotonicity parameters. In this paper, we design a fully adaptive OGD algorithm, \textsf{AdaOGD}, that does not require a priori knowledge of these parameters. In the single-agent setting, our algorithm achieves $O(\log^2(T))$ regret under strong convexity, which is optimal up to a log factor. Further, if each agent employs \textsf{AdaOGD} in strongly monotone games, the joint action converges in a last-iterate sense to a unique Nash equilibrium at a rate of $O(\frac{\log^3 T}{T})$, again optimal up to log factors. We illustrate our algorithms in a learning version of the classical newsvendor problem, where due to lost sales, only (noisy) gradient feedback can be observed. Our results immediately yield the first feasible and near-optimal algorithm for both the single-retailer and multi-retailer settings. We also extend our results to the more general setting of exp-concave cost functions and games, using the online Newton step (ONS) algorithm., Comment: Accepted by Operations Research; 47 pages

We introduce Conformal Decision Theory, a framework for producing safe autonomous decisions despite imperfect machine learning predictions. Examples of such decisions are ubiquitous, from robot planning algorithms that rely on pedestrian predictions, to calibrating autonomous manufacturing to exhibit high throughput and low error, to the choice of trusting a nominal policy versus switching to a safe backup policy at run-time. The decisions produced by our algorithms are safe in the sense that they come with provable statistical guarantees of having low risk without any assumptions on the world model whatsoever; the observations need not be I.I.D. and can even be adversarial. The theory extends results from conformal prediction to calibrate decisions directly, without requiring the construction of prediction sets. Experiments demonstrate the utility of our approach in robot motion planning around humans, automated stock trading, and robot manufacturing., Comment: 8 pages, 5 figures

The rapid progress in machine learning in recent years has been based on a highly productive connection to gradient-based optimization. Further progress hinges in part on a shift in focus from pattern recognition to decision-making and multi-agent problems. In these broader settings, new mathematical challenges emerge that involve equilibria and game theory instead of optima. Gradient-based methods remain essential -- given the high dimensionality and large scale of machine-learning problems -- but simple gradient descent is no longer the point of departure for algorithm design. We provide a gentle introduction to a broader framework for gradient-based algorithms in machine learning, beginning with saddle points and monotone games, and proceeding to general variational inequalities. While we provide convergence proofs for several of the algorithms that we present, our main focus is that of providing motivation and intuition., Comment: 36 pages, 7 figures; minor corrections

Motivated by the emergence of decentralized machine learning (ML) ecosystems, we study the delegation of data collection. Taking the field of contract theory as our starting point, we design optimal and near-optimal contracts that deal with two fundamental information asymmetries that arise in decentralized ML: uncertainty in the assessment of model quality and uncertainty regarding the optimal performance of any model. We show that a principal can cope with such asymmetry via simple linear contracts that achieve 1-1/e fraction of the optimal utility. To address the lack of a priori knowledge regarding the optimal performance, we give a convex program that can adaptively and efficiently compute the optimal contract. We also study linear contracts and derive the optimal utility in the more complex setting of multiple interactions.

We present Scaff-PD, a fast and communication-efficient algorithm for distributionally robust federated learning. Our approach improves fairness by optimizing a family of distributionally robust objectives tailored to heterogeneous clients. We leverage the special structure of these objectives, and design an accelerated primal dual (APD) algorithm which uses bias corrected local steps (as in Scaffold) to achieve significant gains in communication efficiency and convergence speed. We evaluate Scaff-PD on several benchmark datasets and demonstrate its effectiveness in improving fairness and robustness while maintaining competitive accuracy. Our results suggest that Scaff-PD is a promising approach for federated learning in resource-constrained and heterogeneous settings.

Contemporary scientific research is a distributed, collaborative endeavor, carried out by teams of researchers, regulatory institutions, funding agencies, commercial partners, and scientific bodies, all interacting with each other and facing different incentives. To maintain scientific rigor, statistical methods should acknowledge this state of affairs. To this end, we study hypothesis testing when there is an agent (e.g., a researcher or a pharmaceutical company) with a private prior about an unknown parameter and a principal (e.g., a policymaker or regulator) who wishes to make decisions based on the parameter value. The agent chooses whether to run a statistical trial based on their private prior and then the result of the trial is used by the principal to reach a decision. We show how the principal can conduct statistical inference that leverages the information that is revealed by an agent's strategic behavior -- their choice to run a trial or not. In particular, we show how the principal can design a policy to elucidate partial information about the agent's private prior beliefs and use this to control the posterior probability of the null. One implication is a simple guideline for the choice of significance threshold in clinical trials: the type-I error level should be set to be strictly less than the cost of the trial divided by the firm's profit if the trial is successful.

We present a method for solving general nonconvex-strongly-convex bilevel optimization problems. Our method -- the \emph{Restarted Accelerated HyperGradient Descent} (\texttt{RAHGD}) method -- finds an $\epsilon$-first-order stationary point of the objective with $\tilde{\mathcal{O}}(\kappa^{3.25}\epsilon^{-1.75})$ oracle complexity, where $\kappa$ is the condition number of the lower-level objective and $\epsilon$ is the desired accuracy. We also propose a perturbed variant of \texttt{RAHGD} for finding an $\big(\epsilon,\mathcal{O}(\kappa^{2.5}\sqrt{\epsilon}\,)\big)$-second-order stationary point within the same order of oracle complexity. Our results achieve the best-known theoretical guarantees for finding stationary points in bilevel optimization and also improve upon the existing upper complexity bound for finding second-order stationary points in nonconvex-strongly-concave minimax optimization problems, setting a new state-of-the-art benchmark. Empirical studies are conducted to validate the theoretical results in this paper.

Numerous applications in machine learning and data analytics can be formulated as equilibrium computation over Riemannian manifolds. Despite the extensive investigation of their Euclidean counterparts, the performance of Riemannian gradient-based algorithms remain opaque and poorly understood. We revisit the original scheme of Riemannian gradient descent (RGD) and analyze it under a geodesic monotonicity assumption, which includes the well-studied geodesically convex-concave min-max optimization problem as a special case. Our main contribution is to show that, despite the phenomenon of distance distortion, the RGD scheme, with a step size that is agnostic to the manifold's curvature, achieves a curvature-independent and linear last-iterate convergence rate in the geodesically strongly monotone setting. To the best of our knowledge, the possibility of curvature-independent rates and/or last-iterate convergence in the Riemannian setting has not been considered before.

As the scale of machine learning models increases, trends such as scaling laws anticipate consistent downstream improvements in predictive accuracy. However, these trends take the perspective of a single model-provider in isolation, while in reality providers often compete with each other for users. In this work, we demonstrate that competition can fundamentally alter the behavior of these scaling trends, even causing overall predictive accuracy across users to be non-monotonic or decreasing with scale. We define a model of competition for classification tasks, and use data representations as a lens for studying the impact of increases in scale. We find many settings where improving data representation quality (as measured by Bayes risk) decreases the overall predictive accuracy across users (i.e., social welfare) for a marketplace of competing model-providers. Our examples range from closed-form formulas in simple settings to simulations with pretrained representations on CIFAR-10. At a conceptual level, our work suggests that favorable scaling trends for individual model-providers need not translate to downstream improvements in social welfare in marketplaces with multiple model providers., Comment: Appeared at NeurIPS 2023; this is the full version

Standard conformal prediction methods provide a marginal coverage guarantee, which means that for a random test point, the conformal prediction set contains the true label with a user-specified probability. In many classification problems, we would like to obtain a stronger guarantee--that for test points of a specific class, the prediction set contains the true label with the same user-chosen probability. For the latter goal, existing conformal prediction methods do not work well when there is a limited amount of labeled data per class, as is often the case in real applications where the number of classes is large. We propose a method called clustered conformal prediction that clusters together classes having "similar" conformal scores and performs conformal prediction at the cluster level. Based on empirical evaluation across four image data sets with many (up to 1000) classes, we find that clustered conformal typically outperforms existing methods in terms of class-conditional coverage and set size metrics.

Identifying clients with similar objectives and learning a model-per-cluster is an intuitive and interpretable approach to personalization in federated learning. However, doing so with provable and optimal guarantees has remained an open challenge. We formalize this problem as a stochastic optimization problem, achieving optimal convergence rates for a large class of loss functions. We propose simple iterative algorithms which identify clusters of similar clients and train a personalized model-per-cluster, using local client gradients and flexible constraints on the clusters. The convergence rates of our algorithms asymptotically match those obtained if we knew the true underlying clustering of the clients and are provably robust in the Byzantine setting where some fraction of the clients are malicious.