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We investigate the role of pseudo-labels in the test-time adaptation (TTA) problem. When working with unlabeled samples in TTA, pseudo-labels have become a natural approach to updating the target model. However, pseudo-label learning also presents some challenges: it suffers from a memorization effect (the model learns from clean labels first, then memorizes the noisy ones) and confirmation bias (errors from noisy labels increase over time and disrupt model performance when they become significant). Our work first identifies two underlying mechanisms leading to these obstacles. On the one hand, existing methods follow a "slow" adaptation to the target domain, allowing sufficient time for the model to memorize noisy labels (memorization effect) and accumulate errors (confirmation bias). Furthermore, training with noisy labels blurs the decision boundary with nearby classes. To address the first issue, we propose a novel loss function, namely sparse cross logit (sparse-CL), that operates in the logit space and allows the model to take larger learning steps in a stable training manner. This helps the target model reach a better solution faster under the same number of updating steps. To address the second issue, we introduce a regularization that penalizes negative pseudo-labels while encouraging positive ones, which can increase the boundary between nearby classes. We demonstrate that our methods outperform state-of-the-art methods in a diverse set of TTA experiments.