We consider the $[0,1]$-valued regression problem in the stochastic setting. In a related problem called cost-sensitive classification, Foster and Krishnamurthy (2021) have shown that the log loss minimizer achieves an improved generalization bound compared to that of the squared loss minimizer in the sense that the bound scales with the cost of the best classifier, which can be arbitrarily small depending on the problem instance. Such a result is often called a first-order bound. For $[0,1]$-valued regression, we first show that the log loss minimizer leads to a similar first-order bound. We then ask if there exists a loss function that achieves a variance-dependent bound, also known as a second-order bound, which is a strict improvement upon first-order bounds. We answer this question in the affirmative by proposing a novel loss function called betting loss. Our result is variance-adaptive in the sense that the bound is attained by an algorithm without any knowledge about the variance, which is in contrast to the existing works such as weighted least squares with known variances or those that model label variance or its distribution such as distributional reinforcement learning.

We introduce Grade School Math with Distracting Context (GSM-DC 1 ), a synthetic benchmark to evaluate Large Language Models' (LLMs) reasoning robustness against systematically controlled irrelevant context (IC).GSM-DC constructs symbolic reasoning graphs with precise distractor injections, enabling rigorous, reproducible evaluation.Our experiments demonstrate that LLMs are significantly sensitive to IC, affecting both reasoning path selection and arithmetic accuracy.Additionally, training models with strong distractors improves performance in both in-distribution and out-of-distribution scenarios.We further propose a stepwise tree search guided by a process reward model, which notably enhances robustness in out-of-distribution conditions.