We consider stochastic optimization problems where the objective depends on some parameter, as commonly found in hyperparameter optimization for instance. We investigate the behavior of the derivatives of the iterates of Stochastic Gradient Descent (SGD) with respect to that parameter and show that they are driven by an inexact SGD recursion on a different objective function, perturbed by the convergence of the original SGD. This enables us to establish that the derivatives of SGD converge to the derivative of the solution mapping in terms of mean squared error whenever the objective is strongly convex. Specifically, we demonstrate that with constant step-sizes, these derivatives stabilize within a noise ball centered at the solution derivative, and that with vanishing step-sizes they exhibit O(\log(k) 2 / k) convergence rates. Additionally, we prove exponential convergence in the interpolation regime.

Emotions are known to mediate the relationship between users' content consumption and their online engagement, with heightened emotional intensity leading to increased engagement. Building on this insight, we propose three regressor-guided image editing approaches aimed at diminishing the emotional impact of images. These include (i) a parameter optimization approach based on global image transformations known to influence emotions, (ii) an optimization approach targeting the style latent space of a generative adversarial network, and (iii) a diffusion-based approach employing classifier guidance and classifier-free guidance. Our findings demonstrate that approaches can effectively alter the emotional properties of images while maintaining high visual quality. Optimization-based methods primarily adjust low-level properties like color hues and brightness, whereas the diffusion-based approach introduces semantic changes, such as altering appearance or facial expressions. Notably, results from a behavioral study reveal that only the diffusion-based approach successfully elicits changes in viewers' emotional responses while preserving high perceived image quality. In future work, we will investigate the impact of these image adaptations on internet user behavior.