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We consider the question: what is the abstraction that should be implemented by the computational engine of a machine learning system? Current machine learning systems typically push whole tensors through a series of compute kernels such as matrix multiplications or activation functions, where each kernel runs on an AI accelerator (ASIC) such as a GPU. This implementation abstraction provides little built-in support for ML systems to scale past a single machine, or for handling large models with matrices or tensors that do not easily fit into the RAM of an ASIC. In this paper, we present an alternative implementation abstraction called the tensor relational algebra (TRA). The TRA is a set-based algebra based on the relational algebra. Expressions in the TRA operate over binary tensor relations, where keys are multi-dimensional arrays and values are tensors. The TRA is easily executed with high efficiency in a parallel or distributed environment, and amenable to automatic optimization. Our empirical study shows that the optimized TRA-based back-end can significantly outperform alternatives for running ML workflows in distributed clusters. 

When numerical and machine learning (ML) computations are expressed relationally, classical query execution strategies (hash-based joins and aggregations) can do a poor job distributing the computation. In this paper, we propose a two-phase execution strategy for numerical computations that are expressed relationally, as aggregated join trees (that is, expressed as a series of relational joins followed by an aggregation). In a pilot run, lineage information is collected; this lineage is used to optimally plan the computation at the level of individual records. Then, the computation is actually executed. We show experimentally that a relational system making use of this two-phase strategy can be an excellent platform for distributed ML computations.