Gödel's seminal work directly contradicted one of the great minds of mathematics and limited the field forever Kurt Gödel, the man who ruined mathematics, was one of the most important thinkers of the 20th century. He was born in 1906, smack-bang in the middle of the greatest crisis that maths has ever known. Just a few decades later, he would help resolve this turmoil, but in doing so doom mathematicians to a smaller world than the one that came before. Mathematics, as an intellectual framework, is incredibly powerful. The entire point is taking one set of logical ideas and using them to build another, making maths the closest thing we have to a cognitive perpetual-motion machine - there is always a new mathematical idea lurking across the horizon, and we just need to assemble the steps to get there.

We present a novel statistical framework for analyzing stereotypes in large language models (LLMs) by systematically estimating the bias and variation in their generation. Current evaluation metrics in the alignment literature often overlook the randomness of stereotypes caused by the inconsistent generative behavior of LLMs. For example, this inconsistency can result in LLMs displaying contradictory stereotypes, including those related to gender or race, for identical professions across varied contexts. Neglecting such inconsistency could lead to misleading conclusions in alignment evaluations and hinder the accurate assessment of the risk of LLM applications perpetuating or amplifying social stereotypes and unfairness.This work proposes a Bias-Volatility Framework (BVF) that estimates the probability distribution function of LLM stereotypes. Specifically, since the stereotype distribution fully captures an LLM's generation variation, BVF enables the assessment of both the likelihood and extent to which its outputs are against vulnerable groups, thereby allowing for the quantification of the LLM's aggregated discrimination risk. Furthermore, we introduce a mathematical framework to decompose an LLM's aggregated discrimination risk into two components: bias risk and volatility risk, originating from the mean and variation of LLM's stereotype distribution, respectively. We apply BVF to assess 12 commonly adopted LLMs and compare their risk levels. Our findings reveal that: i) Bias risk is the primary cause of discrimination risk in LLMs; ii) Most LLMs exhibit significant pro-male stereotypes for nearly all careers; iii) Alignment with reinforcement learning from human feedback lowers discrimination by reducing bias, but increases volatility; iv) Discrimination risk in LLMs correlates with key sociol-economic factors like professional salaries. Finally, we emphasize that BVF can also be used to assess other dimensions of generation inconsistency's impact on LLM behavior beyond stereotypes, such as knowledge mastery.

The development of generative language models that can create long and coherent textual outputs via autoregression has lead to a proliferation of uses and a corresponding sweep of analyses as researches work to determine the limitations of this new paradigm. Unlike humans, these ' ' (LLMs) are highly sensitive to small changes in their inputs, leading to unwanted inconsistency in their behavior. One problematic inconsistency when LLMs are used to answer multiple-choice questions or analyze multiple inputs is: the output of an LLM can (and often does) change significantly when sub-sequences are swapped, despite both orderings being semantically identical. In this paper we present, a technique that the output of an LLM will not have order dependence on a specified set of sub-sequences. We show that this method eliminates order dependency, and that it can be applied to transformer-based LLM to enable text generation that is unaffected by re-orderings. Delving into the implications of our method, we show that, despite our inputs being out of distribution, the impact on expected accuracy is small, where the expectation is over the order of uniformly chosen shuffling of the candidate responses, and usually significantly less in practice. Thus, can be used as a ' ' method on fully trained models.

Neural representations for 3D scenes have made substantial advancements recently, yet object removal remains a challenging yet practical issue, due to the absence of multi-view supervision over occluded areas. Diffusion Models (DMs), trained on extensive 2D images, show diverse and high-fidelity generative capabilities in the 2D domain. However, due to not being specifically trained on 3D data, their application to multi-view data often exacerbates inconsistency, hence impacting the overall quality of the 3D output. To address these issues, we introduce In-N-Out, a novel approach that begins by inpainting a prior, i.e., the occluded area from a single view using DMs, followed by outstretching it to create multi-view inpaintings via latents alignments. Our analysis identifies that the variability in DMs' outputs mainly arises from initially sampled latents and intermediate latents predicted in the denoising process. We explicitly align of initial latents using a Neural Radiance Field (NeRF) to establish a consistent foundational structure in the inpainted area, complemented by an implicit alignment of intermediate latents through cross-view attention during the denoising phases, enhancing appearance consistency across views. To further enhance rendering results, we apply a patch-based hybrid loss to optimize NeRF. We demonstrate that our techniques effectively mitigate the challenges posed by inconsistencies in DMs and substantially improve the fidelity and coherence of inpainted 3D representations.

This paper proposes ProEdit - a simple yet effective framework for high-quality 3D scene editing guided by diffusion distillation in a novel progressive manner. Inspired by the crucial observation that multi-view inconsistency in scene editing is rooted in the diffusion model's large feasible output space (FOS), our framework controls the size of FOS and reduces inconsistency by decomposing the overall editing task into several subtasks, which are then executed progressively on the scene.

We have the following lemma. Using the notation of Lemma A.1, we have E The third inequality uses the Lipschitz assumption of the loss function. Figure 10 supplements'Relation to disagreement ' at the end of Section 2. It shows an example where the behavior of inconsistency is different from disagreement. All the experiments were done using GPUs (A100 or older). The goal of the experiments reported in Section 3.1 was to find whether/how the predictiveness of The arrows indicate the direction of training becoming longer.

Huaibo Huang is the corresponding author.