How can we design a market of human-generated content for use in training AI models that both enables technological progress and preserves individual incentives for high-quality content creation? Existing approaches take polar positions: a "free-for-all" model based on fair use and a "strong intellectual property rights" model. We show that both fail: Free-for-all does not compensate creators, and -- by modeling as a static Stackelberg game -- strong intellectual property rights also underpower creative incentives. We find this especially true for more innovative creators, a phenomenon we term the "originality penalty." Extending this insight to a dynamic model, we find another market failure undermining AI model performance, even for an initially good model: Such a model induces greater reliance by humans on AI-assisted creation, resulting in homogenized content feeding back into training, which degrades the model performance -- a "curse of precision." We further propose a market design with a data intermediary internalizing cross-creator externalities and subsidizing innovative contributions, thereby restoring efficiency.

Persistence diagrams are common representations in topological data analysis, but they do not naturally live in a vector space, and the statistical tools developed for comparing them have largely evolved separately from those used for downstream prediction. We introduce STRAND (Survival Topological Representation ANalysis of Diagrams), which treats (collections of) PDs as survival data: each topological feature with persistence value $p = d - b$ is a fully observed time-to-event, and the persistence survival function $S(t) = \mathbb{P}(p > t)$ is the central object for comparing diagrams. From this single representation we derive (i) a non-parametric two-sample test with calibrated Type I error and high power from a small number of diagrams; (ii) interpretable effect sizes; and (iii) a 1-Wasserstein-stable feature vector for downstream machine learning. We validate calibration and power on synthetic manifolds with controlled topology, demonstrate competitive vectorisation across 14 graph and 3D point cloud benchmarks, and apply the method to study functional brain connectivity in fMRI/neuroscience data. To our knowledge, STRAND is the first method to provide hypothesis testing and vectorisation for persistence diagrams from a single coherent and interpretable representation.

Given this law, the evolution of a continuoustime autonomous1 dynamical system is determined by its Uncertainty has become a central topic in machine learning initial state. Or, stated differently, the evolution is given by (ML), with increasing interest in the distinction between aleatoric and epistemic uncertainty. Aleatoric uncertainty the solution to the initial value problem reflects randomness inherent in a process, whereas epistemicx (t) = f(x(t)), x(0) = x0, (1) uncertainty originates from a lack of knowledge about that process. The former is therefore irreducible, while the latterwhere x(t) X is the state at time t and x0 X the can, in principle, be reduced by gathering more information (Hullermeier and Waegeman, 2021).

Compression progress is a long-standing proposal for intrinsic motivation: reward an agent when its world model becomes better at predicting or compressing experience. The folk claim is that this reward is "credible" because it is paid only for learning. We make this precise and prove it. If intrinsic reward is the signed decrease of a fixed sealed-audit loss, r_t = E(theta_{t-1}) - E(theta_t), then cumulative reward telescopes exactly to endpoint audit improvement, so no policy can push reward up indefinitely while true audit performance stagnates or degrades. For finite audit panels the same result holds with a sharp false-positive budget: cumulative empirical reward is at most true audit improvement plus 2 Delta_n(F, delta), the uniform audit deviation of the model class. This is horizon-free: adaptivity over time costs nothing once the sealed panel uniformly controls the class. The theorem also identifies the failure modes: the guarantee disappears if progress is clipped, scored on the agent's own stream, exposed to a high-capacity model on a reusable panel, or applied to a neural class that makes Delta_n vacuous. We give a Lean 4 mechanization of the structural core (telescoping, the finite-audit bound, finite Gibbs, and the entropy floor) and an experiment suite on ARC-TGI grid-transformation generators with adaptive holdout attacks. Experiments confirm the theory: finite-audit deviation scales as n^{-0.527}; signed progress resists clip-farming, stream leakage, and noisy-TV curiosity; naive reusable audits are exploitable by black-box scalar feedback, while standard release defenses keep the attack below the 2 Delta_n threshold. Signed compression progress on a sealed audit is an accounting signal of genuine improvement.

Online learning with delayed feedback typically assumes that the learner can track all pending rounds until their feedback arrives. In practice, tracking resources are finite, and feedback from untracked rounds is permanently lost. In this paper, we study delayed online convex optimization (OCO) under a hard capacity constraint, where at most $C$ pending rounds can be tracked at any time. To model delay information, we introduce a semi-clairvoyant model that refines the clairvoyant assumption from prior work: rather than requiring delays to be known at prediction time, the learner observes delay expirations online, consistent with the classical unconstrained delayed setting. Our approach proceeds via a reduction to a novel ``delayed and weighted'' OCO problem, using a scheduler that randomizes tracking decisions and importance-weights the resulting observations. For this base problem, we propose and analyze Delayed-Weighted FTRL and its bandit analogue, establishing regret bounds that explicitly characterize the interaction between time-varying weights and delayed feedback. Combining these base learners with our schedulers yields the first regret guarantees for capacity-constrained OCO under convex and strongly convex losses, for both first-order and bandit feedback. For first-order feedback, capacity $C = Ω(\log T)$ suffices to recover standard delayed OCO rates up to logarithmic factors. For bandit feedback, the regret rates are modulated by powers of $(1 + σ_{\text{max}}/C)$, where $σ_{\text{max}}$ is the maximum number of pending observations at any time. This allows the regret bound to degrade gracefully when $C < σ_{\text{max}}$, while remaining sublinear.

We study distributed optimization with stochastic gradients and finite-bit communication modeled by random (unbiased) quantization. We propose q-PDGD, a quantized stochastic primal-dual method, and analyze it under relaxed global geometry. Under restricted secant inequality (RSI), a constant step-size yields linear contraction to an explicit neighborhood determined by gradient noise, quantization distortion, and network connectivity, while a diminishing step-size achieves O(1/k) convergence without shared-minimizer assumptions. Under Polyak-Lojasiewicz (PL) inequality, we obtain linear-to-neighborhood convergence in the same stochastic quantized setting. Our results match the best-known centralized stochastic rates in oracle complexity, and are supported by experiments demonstrating the predicted tradeoffs between quantization level, step-size choice, and graph structure.

This paper studies efficient online algorithms for multinomial logistic bandits (MLogB), where the feedback distribution over $K+1$ outcomes follows a multinomial logistic model of $d$-dimensional action vectors. A representative UCB-type algorithm, OFUL-MLogB, achieves a regret bound of $\tilde{\mathcal{O}}(Kd\sqrt{T})$, but still requires $\mathcal{O}(K^3d^3)$ time and $\mathcal{O}(K^2d^2)$ space per round due to parameter estimation and optimistic reward construction, which is prohibitive in high-dimensional settings. To address this limitation, we propose EOFD-MLogB, which integrates frequent directions matrix sketching into OFUL-MLogB. By maintaining a low-rank SVD sketch of the accumulated Hessian, constrained online Newton updates in parameter estimation and $Kd \times K$ spectral-norm computations in the reward bonus are reduced to one-dimensional root-finding tasks and $K \times K$ eigenvalue computations, respectively. This yields dominant per-round time complexity $\mathcal{O}(Kd(m+K)^2)$ and space complexity $\mathcal{O}(Kd(m+K))$, where $m \ll d$ is the sketch size. We further prove a regret bound of $\tilde{\mathcal{O}}(Δ_T(Kd\lnΔ_T+m)\sqrt{T})$, where the sketching error factor $Δ_T$ is controlled by the $m$-truncated spectral tail of the Hessian. Thus, when the Hessian is approximately low-rank, the regret is close to that of OFUL-MLogB. Experiments validate the computational efficiency and competitive performance.

We propose a spectral-based, unsupervised representation learning framework to derive low-dimensional embeddings for clinical concepts and patients in rare disease cohorts from electronic health records, where data are high-dimensional but sample sizes are limited. To overcome this challenge, we incorporate a knowledge matrix extracted from a broader population that shares a partially overlapping subspace with the rare-disease cohort. Our method departs from existing approaches by relaxing restrictive one-to-one signal-alignment assumptions between the latent data matrix and knowledge matrix, allowing more flexible and realistic forms of structured sharing. We introduce a novel two-step spectral embedding procedure: first, we identify and remove irrelevant components from the knowledge matrix; then, we apply a projection-based method to separately recover shared and heterogeneous components. Simulations and an analysis of a real-world multiple sclerosis cohort show that the proposed method outperforms competing approaches, particularly in challenging scenarios where shared signals are weak and only partially aligned, as is common in rare-disease data.

We study a dynamic assortment problem on a two-sided service platform with incomplete information and heterogeneous customers in a discrete-time setting. In each period, a customer arrives seeking service, and the platform chooses an assortment of sellers to display. The customer then proposes a transaction to at most one seller in the assortment according to a multinomial logit choice model. After a fixed number of periods, sellers review the proposals they have received and each chooses at most one customer according to another multinomial logit choice model, after which the cycle repeats. A key challenge is that the platform does not know the choice-model parameters of either customers or sellers in advance. To our knowledge, this is the first study of a dynamic assortment problem in which both sides' choice parameters are unknown. We develop a data-driven algorithm that learns these parameters while optimizing the platform's objective over time. We evaluate performance using regret, which measures revenue loss relative to a clairvoyant benchmark that knows all parameters and customer arrivals in advance. We show that the algorithm's worst-case regret grows polylogarithmically over time, and we derive a matching lower bound, establishing its rate optimality.

Discrete state space diffusion models have shown significant advantages in applications involving discrete data, such as text and image generation. It has also been observed that their performance is highly sensitive to the choice of rate matrices, particularly between uniform and absorbing rate matrices. While empirical results suggest that absorbing rate matrices often yield better generation quality compared to uniform rate matrices, existing theoretical works have largely focused on the uniform rate matrices case. Notably, convergence guarantees and error analyses for absorbing diffusion models are still missing. In this work, we provide the first finite-time error bounds and convergence rate analysis for discrete diffusion models using absorbing rate matrices.