Commonly used caching policies, such as LRU (Least Recently Used) or LFU (Least Frequently Used), exhibit optimal performance only under specific traffic patterns. Even advanced machine learning-based methods, which detect patterns in historical request data, struggle when future requests deviate from past trends. Recently, a new class of policies has emerged that are robust to varying traffic patterns. These algorithms address an online optimization problem, enabling continuous adaptation to the context. They offer theoretical guarantees on the regret metric, which measures the performance gap between the online policy and the optimal static cache allocation in hindsight. However, the high computational complexity of these solutions hinders their practical adoption. In this study, we introduce a new variant of the gradient-based online caching policy that achieves groundbreaking logarithmic computational complexity relative to catalog size, while also providing regret guarantees. This advancement allows us to test the policy on large-scale, real-world traces featuring millions of requests and items - a significant achievement, as such scales have been beyond the reach of existing policies with regret guarantees. To the best of our knowledge, our experimental results demonstrate for the first time that the regret guarantees of gradient-based caching policies offer substantial benefits in practical scenarios.