Learning models over factorized joins avoids redundant computations by identifying and pre-computing shared cofactors. Previous work has investigated the performance gain when computing cofactors on traditional disk-based database systems. Due to the absence of published code, the experiments could not be reproduced on in-memory database systems. This work describes the implementation when using cofactors for in-database factorized learning. We benchmark our open-source implementation for learning linear regression on factorized joins with PostgreSQL -- as a disk-based database system -- and HyPer -- as an in-memory engine. The evaluation shows a performance gain of factorized learning on in-memory database systems by 70\% to non-factorized learning and by a factor of 100 compared to disk-based database systems. Thus, modern database engines can contribute to the machine learning pipeline by pre-computing aggregates prior to data extraction to accelerate training.

The machine learning (ML) training over disparate data sources traditionally involves materialization, which can impose substantial time and space overhead due to data movement and replication. Factorized learning, which leverages direct computation on disparate sources through linear algebra (LA) rewriting, has emerged as a viable alternative to improve computational efficiency. However, the adaptation of factorized learning to leverage the full capabilities of modern LA-friendly hardware like GPUs has been limited, often requiring manual intervention for algorithm compatibility. This paper introduces Ilargi, a novel factorized learning framework that utilizes matrix-represented data integration (DI) metadata to facilitate automatic factorization across CPU and GPU environments without the need for costly relational joins. Ilargi incorporates an ML-based cost estimator to intelligently selects between factorization and materialization based on data properties, algorithm complexity, hardware environments, and their interactions. This strategy ensures up to 8.9x speedups on GPUs and achieves over 20% acceleration in batch ML training workloads, thereby enhancing the practicability of ML training across diverse data integration scenarios and hardware platforms. To our knowledge, this work is the very first effort in GPU-compatible factorized learning.