We study contextual dynamic pricing with linear valuations and bounded-support agnostic noise, whose induced demand curve may be non-Lipschitz with arbitrary jumps and atoms. Such discontinuities break the cross-context interpolation arguments used by smooth-demand pricing algorithms, while the best previous method achieved only $\tilde O(T^{3/4})$ regret. We propose Conservative-Markdown Redirect-UCB Pricing, a polynomial-time algorithm that combines randomized parameter estimation, conservative residual-grid probing, and confidence-based one-step redirection. Our algorithm achieves $\tilde O(T^{2/3})$ optimal regret, matching the known lower bounds of Kleinberg and Leighton (2003) up to logarithmic factors and improving over the previous upper bound of Xu and Wang (2022). Under stochastic well-conditioned contexts, this closes the long-existing open regret gap in linear-valuation contextual pricing under agnostic non-Lipschitz noise distribution.

Self-supervised image denoising methods have traditionally relied on either architectural constraints or specialized loss functions that require prior knowledge of the noise distribution to avoid the trivial identity mapping. Among these, approaches such as Noisier2Noise or Recorrupted2Recorrupted, create training pairs by adding synthetic noise to the noisy images. While effective, these recorruption-based approaches require precise knowledge of the noise distribution, which is often not available. We present Learning to Recorrupt (L2R), a noise distribution-agnostic denoising technique that eliminates the need for knowledge of the noise distribution. Our method introduces a learnable monotonic neural network that learns the recorruption process through a min-max saddle-point objective. The proposed method achieves state-of-the-art performance across unconventional and heavy-tailed noise distributions, such as log-gamma, Laplace, and spatially correlated noise, as well as signal-dependent noise models such as Poisson-Gaussian noise.

Stochastic regularisation is an important weapon in the arsenal of a deep learning practitioner. However, despite recent theoretical advances, our understanding of how noise influences signal propagation in deep neural networks remains limited. By extending recent work based on mean field theory, we develop a new framework for signal propagation in stochastic regularised neural networks. Our \textit{noisy signal propagation} theory can incorporate several common noise distributions, including additive and multiplicative Gaussian noise as well as dropout. We use this framework to investigate initialisation strategies for noisy ReLU networks. We show that no critical initialisation strategy exists using additive noise, with signal propagation exploding regardless of the selected noise distribution. For multiplicative noise (e.g.\ dropout), we identify alternative critical initialisation strategies that depend on the second moment of the noise distribution. Simulations and experiments on real-world data confirm that our proposed initialisation is able to stably propagate signals in deep networks, while using an initialisation disregarding noise fails to do so.

Generative models are being increasingly used in science and industry applications.