Two-Sample Testing

Hyperparameter tuning remains a significant challenge in the training of deep neural networks (DNNs), requiring manual search or time-intensive grid searches that increase resource costs and limit the accessibility of machine learning. The global initial learning rate is among the most consequential of these hyperparameters. Adaptive and scheduling-based methods manage the learning rate during training but still require manual selection of an initial global value; learning-rate-free alternatives remove this selection at the cost of performance or stability on non-convex problems. We present ExpTest, an autonomous learning-rate controller that treats the training loss curve as an online signal and performs sequential statistical tests on theoretically motivated windows to detect convergent behavior and trigger learning-rate reductions. The framework combines a covariance-based initial learning-rate estimate, curvature-motivated window sizing, and two-phase test-driven decay, drawing on the approximately exponential decay behavior predicted under linearized network dynamics. We provide a mathematical motivation for ExpTest and evaluate it on regression, classification, forecasting, and natural-language tasks across fully-connected, convolutional, transformer-based, and pretrained architectures. Across these tasks, ExpTest achieves competitive performance relative to hand-tuned SGD-based baselines and recent learning-rate-free methods, without manual initial learning-rate selection or predefined scheduling.

Two-sample tests have been extensively employed in various scientific fields and machine learning to discriminate whether two sets of samples come from the same distribution or not. Kernel-based procedures for hypothetical testing have been proposed to efficiently disentangle high-dimensional complex structures in data to obtain accurate results in a model-free way by embedding the data into the reproducing kernel Hilbert space (RKHS). While the choice of kernels plays a crucial role for their performance, little is understood about how to choose kernel especially for small datasets. Here we construct a hypothetical test which can be effective even for small datasets, based on the theoretical foundation of kernel-based tests using maximum mean discrepancy, which is called MMD-FUSE. We enhance the MMD-FUSE framework by incorporating quantum kernels and propose a novel hybrid testing strategy that fuses classical and quantum kernels. This approach creates a powerful and adaptive test by combining the domain-specific inductive biases of classical kernels with the unique expressive power of quantum kernels. We evaluate our method on various synthetic and real-world clinical datasets, and our experiments reveal two key findings: 1) With appropriate hyperparameter tuning, MMD-FUSE with quantum kernels consistently improves test power over classical counterparts, especially for small and high-dimensional data. 2) The proposed hybrid framework demonstrates remarkable robustness, adapting to different data characteristics and achieving high test power across diverse scenarios. These results highlight the potential of quantum-inspired and hybrid kernel strategies to build more effective statistical tests, offering versatile tools for data analysis where sample sizes are limited.