Self-Consistency (SC) is a widely used method to mitigate hallucinations in Large Language Models (LLMs) by sampling the LLM multiple times and outputting the most frequent solution. Despite its benefits, SC results in significant computational costs proportional to the number of samples generated. Previous early-stopping approaches, such as Early Stopping Self Consistency and Adaptive Consistency, have aimed to reduce these costs by considering output consistency, but they do not analyze the quality of the reasoning paths (RPs) themselves. To address this issue, we propose Reasoning-Aware Self-Consistency (RASC), an innovative early-stopping framework that dynamically adjusts the number of sample generations by considering both the output answer and the RPs from Chain of Thought (CoT) prompting. RASC assigns confidence scores sequentially to the generated samples, stops when certain criteria are met, and then employs weighted majority voting to optimize sample usage and enhance answer reliability. We comprehensively test RASC with multiple LLMs across varied QA datasets. RASC outperformed existing methods and significantly reduces sample usage by an average of 80% while maintaining or improving accuracy up to 5% compared to the original SC

Structured statistical estimation problems are often solved by Conditional Gradient (CG) type methods to avoid the computationally expensive projection operation. However, the existing CG type methods are not robust to data corruption. To address this, we propose to robustify CG type methods against Huber's corruption model and heavy-tailed data. First, we show that the two Pairwise CG methods are stable, i.e., do not accumulate error. Combined with robust mean gradient estimation techniques, we can therefore guarantee robustness to a wide class of problems, but now in a projection-free algorithmic framework. Next, we consider high dimensional problems. Robust mean estimation based approaches may have an unacceptably high sample complexity. When the constraint set is a $\ell_0$ norm ball, Iterative-Hard-Thresholding-based methods have been developed recently. Yet extension is non-trivial even for general sets with $O(d)$ extreme points. For setting where the feasible set has $O(\text{poly}(d))$ extreme points, we develop a novel robustness method, based on a new condition we call the Robust Atom Selection Condition (RASC). When RASC is satisfied, our method converges linearly with a corresponding statistical error, with sample complexity that scales correctly in the sparsity of the problem, rather than the ambient dimension as would be required by any approach based on robust mean estimation.