Large language models (LLMs) can generate structured artifacts, but using them as dependable optimizers for scientific design requires a mechanism for iterative improvement under black-box evaluation. Here, we cast quantum circuit synthesis as a closed-loop, test-time optimization problem: an LLM proposes edits to a fixed-length gate list, and an external simulator evaluates the resulting state with the Meyer-Wallach (MW) global entanglement measure. We introduce a lightweight test-time learning recipe that can reuse prior high-performing candidates as an explicit memory trace, augments prompts with a score-difference feedback, and applies restart-from-the-best sampling to escape potential plateaus. Across fixed 20-qubit settings, the loop without feedback and restart-from-the-best improves random initial circuits over a range of gate budgets. To lift up this performance and success rate, we use the full learning strategy. For the 25-qubit, it mitigates a pronounced performance plateau when naive querying is used. Beyond raw scores, we analyze the structure of synthesized states and find that high MW solutions can correspond to stabilizer or graph-state-like constructions, but full connectivity is not guaranteed due to the metric property and prompt design. These results illustrate both the promise and the pitfalls of memory evaluator-guided LLM optimization for circuit synthesis, highlighting the critical role of prior human-made theoretical theorems to optimally design a custom tool in support of research.

Despite the fact that experimental neural scaling laws have substantially guided empirical progress in large-scale machine learning, no existing theory can quantitatively predict the exponents of these important laws for any modern LLM trained on any natural language dataset. We provide the first such theory in the case of data-limited scaling laws. We isolate two key statistical properties of language that alone can predict neural scaling exponents: (i) the decay of pairwise token correlations with time separation between token pairs, and (ii) the decay of the next-token conditional entropy with the length of the conditioning context. We further derive a simple formula in terms of these statistics that predicts data-limited neural scaling exponents from first principles without any free parameters or synthetic data models. Our theory exhibits a remarkable match with experimentally measured neural scaling laws obtained from training GPT-2 and LLaMA style models from scratch on two qualitatively different benchmarks, TinyStories and WikiText.

Applied work with interference typically models outcomes as functions of own treatment and a low-dimensional exposure mapping of others' treatments, even when that mapping may be misspecified. This raises a basic question: what policy object are exposure-based estimands implicitly targeting, and how should we interpret their direct and spillover components relative to the underlying policy question? We take as primitive the marginal policy effect, defined as the effect of a small change in the treatment probability under the actual experimental design, and show that any researcher-chosen exposure mapping induces a unique pseudo-true outcome model. This model is the best approximation to the underlying potential outcomes that depends only on the user-chosen exposure. Utilizing that representation, the marginal policy effect admits a canonical decomposition into exposure-based direct and spillover effects, and each component provides its optimal approximation to the corresponding oracle objects that would be available if interference were fully known. We then focus on a setting that nests important empirical and theoretical applications in which both local network spillovers and global spillovers, such as market equilibrium, operate. There, the marginal policy effect further decomposes asymptotically into direct, local, and global channels. An important implication is that many existing methods are more robust than previously understood once we reinterpret their targets as channel-specific components of this pseudo-true policy estimand. Simulations and a semi-synthetic experiment calibrated to a large cash-transfer experiment show that these components can be recovered in realistic experimental designs.

Classical PAC generalization bounds on the prediction risk of a classifier are insufficient to provide theoretical guarantees on fairness when the goal is to learn models balancing predictive risk and fairness constraints. We propose a PAC-Bayesian framework for deriving generalization bounds for fairness, covering both stochastic and deterministic classifiers. For stochastic classifiers, we derive a fairness bound using standard PAC-Bayes techniques. Whereas for deterministic classifiers, as usual PAC-Bayes arguments do not apply directly, we leverage a recent advance in PAC-Bayes to extend the fairness bound beyond the stochastic setting. Our framework has two advantages: (i) It applies to a broad class of fairness measures that can be expressed as a risk discrepancy, and (ii) it leads to a self-bounding algorithm in which the learning procedure directly optimizes a trade-off between generalization bounds on the prediction risk and on the fairness. We empirically evaluate our framework with three classical fairness measures, demonstrating not only its usefulness but also the tightness of our bounds.

We present a new mixed-integer programming (MIP) approach for offline multiple change-point detection by casting the problem as a globally optimal piecewise linear (PWL) fitting problem. Our main contribution is a family of strengthened MIP formulations whose linear programming (LP) relaxations admit integral projections onto the segment assignment variables, which encode the segment membership of each data point. This property yields provably tighter relaxations than existing formulations for offline multiple change-point detection. We further extend the framework to two settings of active research interest: (i) multidimensional PWL models with shared change-points, and (ii) sparse change-point detection, where only a subset of dimensions undergo structural change. Extensive computational experiments on benchmark real-world datasets demonstrate that the proposed formulations achieve reductions in solution times under both $\ell_1$ and $\ell_2$ loss functions in comparison to the state-of-the-art.

Existing methods typically rely on either interpretable but inflexible models or highly flexible black-box approaches that sacrifice interpretability; moreover, most impose a single global decision rule across patients. We introduce the Locally Interpretable Individualized Treatment Rule (LI-ITR) method, which combines flexible machine learning models to accurately learn complex treatment outcomes with locally interpretable approximations to construct subject-specific treatment rules. LI-ITR employs variational autoencoders to generate realistic local synthetic samples and learns individualized decision rules through a mixture of interpretable experts. Simulation studies show that LI-ITR accurately recovers true subject-specific local coefficients and optimal treatment strategies. An application to precision side-effect management in breast cancer illustrates the necessity of flexible predictive modeling and highlights the practical utility of LI-ITR in estimating optimal treatment rules while providing transparent, clinically interpretable explanations.

Scale-invariance in games has recently emerged as a widely valued desirable property. Yet, almost all fast convergence guarantees in learning in games require prior knowledge of the utility scale. To address this, we develop learning dynamics that achieve fast convergence while being both scale-free, requiring no prior information about utilities, and scale-invariant, remaining unchanged under positive rescaling of utilities. For two-player zero-sum games, we obtain scale-free and scale-invariant dynamics with external regret bounded by $\tilde{O}(A_{\mathrm{diff}})$, where $A_{\mathrm{diff}}$ is the payoff range, which implies an $\tilde{O}(A_{\mathrm{diff}} / T)$ convergence rate to Nash equilibrium after $T$ rounds. For multiplayer general-sum games with $n$ players and $m$ actions, we obtain scale-free and scale-invariant dynamics with swap regret bounded by $O(U_{\mathrm{max}} \log T)$, where $U_{\mathrm{max}}$ is the range of the utilities, ignoring the dependence on the number of players and actions. This yields an $O(U_{\mathrm{max}} \log T / T)$ convergence rate to correlated equilibrium. Our learning dynamics are based on optimistic follow-the-regularized-leader with an adaptive learning rate that incorporates the squared path length of the opponents' gradient vectors, together with a new stopping-time analysis that exploits negative terms in regret bounds without scale-dependent tuning. For general-sum games, scale-free learning is enabled also by a technique called doubling clipping, which clips observed gradients based on past observations.

We introduce the problem of learning conditional averages in the PAC framework. The learner receives a sample labeled by an unknown target concept from a known concept class, as in standard PAC learning. However, instead of learning the target concept itself, the goal is to predict, for each instance, the average label over its neighborhood -- an arbitrary subset of points that contains the instance. In the degenerate case where all neighborhoods are singletons, the problem reduces exactly to classic PAC learning. More generally, it extends PAC learning to a setting that captures learning tasks arising in several domains, including explainability, fairness, and recommendation systems. Our main contribution is a complete characterization of when conditional averages are learnable, together with sample complexity bounds that are tight up to logarithmic factors. The characterization hinges on the joint finiteness of two novel combinatorial parameters, which depend on both the concept class and the neighborhood system, and are closely related to the independence number of the associated neighborhood graph.

Gaussian processes (GPs) are powerful and widely used probabilistic regression models, but their effectiveness in practice is often limited by the choice of kernel function. This kernel function is typically handcrafted from a small set of standard functions, a process that requires expert knowledge, results in limited adaptivity to data, and imposes strong assumptions on the hypothesis space. We study Empirical GPs, a principled framework for constructing flexible, data-driven GP priors that overcome these limitations. Rather than relying on standard parametric kernels, we estimate the mean and covariance functions empirically from a corpus of historical observations, enabling the prior to reflect rich, non-trivial covariance structures present in the data. Theoretically, we show that the resulting model converges to the GP that is closest (in KL-divergence sense) to the real data generating process. Practically, we formulate the problem of learning the GP prior from independent datasets as likelihood estimation and derive an Expectation-Maximization algorithm with closed-form updates, allowing the model handle heterogeneous observation locations across datasets. We demonstrate that Empirical GPs achieve competitive performance on learning curve extrapolation and time series forecasting benchmarks.

We study nonparametric regression over Besov spaces from noisy observations under sub-exponential noise, aiming to achieve minimax-optimal guarantees on the integrated squared error that hold with high probability and adapt to the unknown noise level. To this end, we propose a wavelet-based online learning algorithm that dynamically adjusts to the observed gradient noise by adaptively clipping it at an appropriate level, eliminating the need to tune parameters such as the noise variance or gradient bounds. As a by-product of our analysis, we derive high-probability adaptive regret bounds that scale with the $\ell_1$-norm of the competitor. Finally, in the batch statistical setting, we obtain adaptive and minimax-optimal estimation rates for Besov spaces via a refined online-to-batch conversion. This approach carefully exploits the structure of the squared loss in combination with self-normalized concentration inequalities.