Covariate shift, a widely used assumption in tackling {\it distributional shift} (when training and test distributions differ), focuses on scenarios where the distribution of the labels conditioned on the feature vector is the same, but the distribution of features in the training and test data are different. Despite the significance and extensive work on covariate shift, theoretical guarantees for algorithms in this domain remain sparse. In this paper, we distill the essence of the covariate shift problem and focus on estimating the average $\mathbb{E}_{\tilde{\mathbf{x}}\sim p_{\mathrm{test}}}\mathbf{f}(\tilde{\mathbf{x}})$, of any unknown and bounded function $\mathbf{f}$, given labeled training samples $(\mathbf{x}_i, \mathbf{f}(\mathbf{x}_i))$, and unlabeled test samples $\tilde{\mathbf{x}}_i$; this is a core subroutine for several widely studied learning problems. We give several efficient algorithms, with provable sample complexity and computational guarantees. Moreover, we provide the first rigorous analysis of algorithms in this space when $\mathbf{f}$ is unrestricted, laying the groundwork for developing a solid theoretical foundation for covariate shift problems.

How should one judge whether a given large language model (LLM) can reliably perform economic reasoning? Most existing LLM benchmarks focus on specific applications and fail to present the model with a rich variety of economic tasks. A notable exception is Raman et al. [2024], who offer an approach for comprehensively benchmarking strategic decision-making; however, this approach fails to address the non-strategic settings prevalent in microeconomics, such as supply-and-demand analysis. We address this gap by taxonomizing microeconomic reasoning into $58$ distinct elements, focusing on the logic of supply and demand, each grounded in up to $10$ distinct domains, $5$ perspectives, and $3$ types. The generation of benchmark data across this combinatorial space is powered by a novel LLM-assisted data generation protocol that we dub auto-STEER, which generates a set of questions by adapting handwritten templates to target new domains and perspectives. Because it offers an automated way of generating fresh questions, auto-STEER mitigates the risk that LLMs will be trained to over-fit evaluation benchmarks; we thus hope that it will serve as a useful tool both for evaluating and fine-tuning models for years to come. We demonstrate the usefulness of our benchmark via a case study on $27$ LLMs, ranging from small open-source models to the current state of the art. We examined each model's ability to solve microeconomic problems across our whole taxonomy and present the results across a range of prompting strategies and scoring metrics.

We study an online learning problem where, over $T$ rounds, a learner observes both time-varying sets of feasible actions and an agent's optimal actions, selected by solving linear optimization over the feasible actions. The learner sequentially makes predictions of the agent's underlying linear objective function, and their quality is measured by the regret, the cumulative gap between optimal objective values and those achieved by following the learner's predictions. A seminal work by B\"armann et al. (ICML 2017) showed that online learning methods can be applied to this problem to achieve regret bounds of $O(\sqrt{T})$. Recently, Besbes et al. (COLT 2021, Oper. Res. 2023) significantly improved the result by achieving an $O(n^4\ln T)$ regret bound, where $n$ is the dimension of the ambient space of objective vectors. Their method, based on the ellipsoid method, runs in polynomial time but is inefficient for large $n$ and $T$. In this paper, we obtain an $O(n\ln T)$ regret bound, improving upon the previous bound of $O(n^4\ln T)$ by a factor of $n^3$. Our method is simple and efficient: we apply the online Newton step (ONS) to appropriate exp-concave loss functions. Moreover, for the case where the agent's actions are possibly suboptimal, we establish an $O(n\ln T+\sqrt{\Delta_Tn\ln T})$ regret bound, where $\Delta_T$ is the cumulative suboptimality of the agent's actions. This bound is achieved by using MetaGrad, which runs ONS with $\Theta(\ln T)$ different learning rates in parallel. We also provide a simple instance that implies an $\Omega(n)$ lower bound, showing that our $O(n\ln T)$ bound is tight up to an $O(\ln T)$ factor. This gives rise to a natural question: can the $O(\ln T)$ factor in the upper bound be removed? For the special case of $n=2$, we show that an $O(1)$ regret bound is possible, while we delineate challenges in extending this result to higher dimensions.

The emergence of Large Language Models has fundamentally transformed the capabilities of AI agents, enabling a new class of autonomous agents capable of interacting with their environment through dynamic code generation and execution. These agents possess the theoretical capacity to operate as independent economic actors within digital markets, offering unprecedented potential for value creation through their distinct advantages in operational continuity, perfect replication, and distributed learning capabilities. However, contemporary digital infrastructure, architected primarily for human interaction, presents significant barriers to their participation. This work presents a systematic analysis of the infrastructure requirements necessary for AI agents to function as autonomous participants in digital markets. We examine four key areas - identity and authorization, service discovery, interfaces, and payment systems - to show how existing infrastructure actively impedes agent participation. We argue that addressing these infrastructure challenges represents more than a technical imperative; it constitutes a fundamental step toward enabling new forms of economic organization. Much as traditional markets enable human intelligence to coordinate complex activities beyond individual capability, markets incorporating AI agents could dramatically enhance economic efficiency through continuous operation, perfect information sharing, and rapid adaptation to changing conditions. The infrastructure challenges identified in this work represent key barriers to realizing this potential.

Trajectory optimization is an essential tool for generating efficient and dynamically consistent gaits in legged locomotion. This paper explores the indirect method of trajectory optimization, emphasizing its application in creating optimal periodic gaits for legged systems and contrasting it with the more commonly used direct method. While the direct method provides considerable flexibility in its implementation, it is limited by its input space parameterization. In contrast, the indirect method improves accuracy by defining control inputs as functions of the system's states and costates. We tackle the convergence challenges associated with indirect shooting methods, particularly through the systematic development of gait libraries by utilizing numerical continuation methods. Our contributions include: (1) the formalization of a general periodic trajectory optimization problem that extends existing first-order necessary conditions for a broader range of cost functions and operating conditions; (2) a methodology for efficiently generating libraries of optimal trajectories (gaits) utilizing a single shooting approach combined with numerical continuation methods, including a novel approach for reconstructing Lagrange multipliers and costates from passive gaits; and (3) a comparative analysis of the indirect and direct shooting methods using a compass-gait walker as a case study, demonstrating the former's superior accuracy in generating optimal gaits. The findings underscore the potential of the indirect method for generating families of optimal gaits, thereby advancing the field of trajectory optimization in legged robotics.

Large Language Models have shown tremendous performance on a large variety of natural language processing tasks, ranging from text comprehension to common sense reasoning. However, the mechanisms responsible for this success remain opaque, and it is unclear whether LLMs can achieve human-like cognitive capabilities or whether these models are still fundamentally circumscribed. Abstract reasoning is a fundamental task for cognition, consisting of finding and applying a general pattern from few data. Evaluating deep neural architectures on this task could give insight into their potential limitations regarding reasoning and their broad generalisation abilities, yet this is currently an under-explored area. In this paper, we introduce a new benchmark for evaluating language models beyond memorization on abstract reasoning tasks. We perform extensive evaluations of state-of-the-art LLMs, showing that they currently achieve very limited performance in contrast with other natural language tasks, even when applying techniques that have been shown to improve performance on other NLP tasks. We argue that guiding LLM generation to follow causal paths could help improve the generalisation and reasoning abilities of LLMs.

Motivated by the successes of deep learning, we propose a class of neural network-based discrete choice models, called RUMnets, inspired by the random utility maximization (RUM) framework. This model formulates the agents' random utility function using a sample average approximation. We show that RUMnets sharply approximate the class of RUM discrete choice models: any model derived from random utility maximization has choice probabilities that can be approximated arbitrarily closely by a RUMnet. Reciprocally, any RUMnet is consistent with the RUM principle. We derive an upper bound on the generalization error of RUMnets fitted on choice data, and gain theoretical insights on their ability to predict choices on new, unseen data depending on critical parameters of the dataset and architecture. By leveraging open-source libraries for neural networks, we find that RUMnets are competitive against several choice modeling and machine learning methods in terms of predictive accuracy on two real-world datasets.

In recent years, Reinforcement Learning (RL) has been applied to real-world problems with increasing success. Such applications often require to put constraints on the agent's behavior. Existing algorithms for constrained RL (CRL) rely on gradient descent-ascent, but this approach comes with a caveat. While these algorithms are guaranteed to converge on average, they do not guarantee last-iterate convergence, i.e., the current policy of the agent may never converge to the optimal solution. In practice, it is often observed that the policy alternates between satisfying the constraints and maximizing the reward, rarely accomplishing both objectives simultaneously. Here, we address this problem by introducing Reinforcement Learning with Optimistic Ascent-Descent (ReLOAD), a principled CRL method with guaranteed last-iterate convergence. We demonstrate its empirical effectiveness on a wide variety of CRL problems including discrete MDPs and continuous control. In the process we establish a benchmark of challenging CRL problems.

ChatGPT has taken the Internet by storm. People all over the world use ChatGPT to generate ideas for content, essays, emails, and codes and solve mathematical questions. The AI chatbot has surpassed 100 million users in less than two months making it the fastest growing consumer internet app ever. It's normal for any app or website to struggle with such a large user base. We, too, often struggled with problems when using ChatGPT.

I am writing in response to Douglas Rooks' op-ed piece about the limits of computers and artificial intelligence ("A prescription for the new year," Dec. 29). While I agree with Rooks more often than not, on this topic, he is way out of his league. He claims to have read up on Artificial Intelligence and dismisses the use of the term intelligence to apply to computers -- with very little supporting evidence. So, let me provide some. Robert Frost, a man of wit Nature's beauty in his pen Words that softly sit These were composed by ChatGPT, an "artificial intelligence" that was developed by OpenAI, and while the AI does not reach the complexity and depth of either Frost or Eliot's poetry, it does capture something of the meaning, tone and themes of both poet's work -- and we are early on in the development of these chats.