Next, we apply our proposed fitting method (see Section 2) on this data set.

Large language models (LLMs) are increasingly used in economics as predictive tools--both to generate synthetic responses in place of human subjects (Horton, 2023; Anthis et al., 2025), and to forecast economic outcomes directly (Hewitt et al., 2024a; Faria-e Castro and Leibovici, 2024; Chan-Lau et al., 2025). Their appeal in these roles is obvious: A pretrained LLM embeds a vast amount of information and can be deployed at negligible cost, often in settings where collecting new, domain-specific human data would be expensive or infeasible. What remains unclear is how to assess the quality of these predictions. This paper proposes a measure that quantifies the domain-specific value of LLMs in an interpretable unit: the amount of human data they substitute for. Specifically, we ask how much human data would be required for a conventional model trained on that data to match the predictive performance of the pretrained LLM in that domain.

We initiate the study of active learning algorithms for classifying strategic agents. Active learning is a well-established framework in machine learning in which the learner selectively queries labels, often achieving substantially higher accuracy and efficiency than classical supervised methods-especially in settings where labeling is costly or time-consuming, such as hiring, admissions, and loan decisions. Strategic classification, however, addresses scenarios where agents modify their features to obtain more favorable outcomes, resulting in observed data that is not truthful. Such manipulation introduces challenges beyond those in learning from clean data. Our goal is to design active and noise-tolerant algorithms that remain effective in strategic environments-algorithms that classify strategic agents accurately while issuing as few label requests as possible. The central difficulty is to simultaneously account for strategic manipulation and preserve the efficiency gains of active learning. Our main result is an algorithm for actively learning linear separators in the strategic setting that preserves the exponential improvement in label complexity over passive learning previously obtained only in the non-strategic case. Specifically, for data drawn uniformly from the unit sphere, we show that a modified version of the Active Perceptron algorithm [DKM05,YZ17] achieves excess error $ε$ using only $\tilde{O}(d \ln \frac{1}ε)$ label queries and incurs at most $\tilde{O}(d \ln \frac{1}ε)$ additional mistakes relative to the optimal classifier, even in the nonrealizable case, when a $\tildeΩ(ε)$ fraction of inputs have inconsistent labels with the optimal classifier. The algorithm is computationally efficient and, under these distributional assumptions, requires substantially fewer label queries than prior work on strategic Perceptron [ABBN21].

Neural Information Processing Systems http://nips.cc/

Large language models (LLMs) can already identify patterns and reason effectively, yet their variable accuracy hampers adoption in high-stakes decision-making applications. In this paper, we study this issue from a venture capital perspective by predicting idea-stage startup success based on founder traits. (i) To build a reliable prediction model, we introduce LLM-AR, a pipeline inspired by neural-symbolic systems that distils LLM-generated heuristics into probabilistic rules executed by the ProbLog automated-reasoning engine. (ii) An iterative policy-evolution loop incorporates association-rule mining to progressively refine the prediction rules. On unseen folds, LLM-AR achieves 59.5% precision and 8.7% recall, 5.9x the random baseline precision, while exposing every decision path for human inspection. The framework is interpretable and tunable via hyperparameters, showing promise to extend into other domains.

Next, we apply our proposed fitting method (see Section 2) on this data set.

Learning, whether natural or artificial, is a process of selection. It starts with a set of candidate options and selects the more successful ones. In the case of machine learning the selection is done based on empirical estimates of prediction accuracy of candidate prediction rules on some data. Due to randomness of data sampling the empirical estimates are inherently noisy, leading to selection under uncertainty. The book provides statistical tools to obtain theoretical guarantees on the outcome of selection under uncertainty. We start with concentration of measure inequalities, which are the main statistical instrument for controlling how much an empirical estimate of expectation of a function deviates from the true expectation. The book covers a broad range of inequalities, including Markov's, Chebyshev's, Hoeffding's, Bernstein's, Empirical Bernstein's, Unexpected Bernstein's, kl, and split-kl. We then study the classical (offline) supervised learning and provide a range of tools for deriving generalization bounds, including Occam's razor, Vapnik-Chervonenkis analysis, and PAC-Bayesian analysis. The latter is further applied to derive generalization guarantees for weighted majority votes. After covering the offline setting, we turn our attention to online learning. We present the space of online learning problems characterized by environmental feedback, environmental resistance, and structural complexity. A common performance measure in online learning is regret, which compares performance of an algorithm to performance of the best prediction rule in hindsight, out of a restricted set of prediction rules. We present tools for deriving regret bounds in stochastic and adversarial environments, and under full information and bandit feedback.