Compressed linear algebra for large-scale machine learning

Abstract

Large-scale machine learning algorithms are often iterative, using repeated read-only data access and I/O-bound matrix-vector multiplications to converge to an optimal model. It is crucial for performance to fit the data into single-node or distributed main memory and enable fast matrix-vector operations on in-memory data. General-purpose, heavy- and lightweight compression techniques struggle to achieve both good compression ratios and fast decompression speed to enable block-wise uncompressed operations. Therefore, we initiate work—inspired by database compression and sparse matrix formats—on value-based compressed linear algebra (CLA), in which heterogeneous, lightweight database compression techniques are applied to matrices, and then linear algebra operations such as matrix-vector multiplication are executed directly on the compressed representation. We contribute effective column compression schemes, cache-conscious operations, and an efficient sampling-based compression algorithm. Our experiments show that CLA achieves in-memory operations performance close to the uncompressed case and good compression ratios, which enables fitting substantially larger datasets into available memory. We thereby obtain significant end-to-end performance improvements up to \(9.2\mathrm{x}\).

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Notes

  1. 1.

    Dummy coding transforms a categorical feature having d possible values into d Boolean features, each indicating the rows in which a given value occurs. The larger the value of d, the greater the sparsity (from adding \(d-1\) zeros per row).

  2. 2.

    The results with native BLAS libraries would be similar because memory bandwidth and I/O are the bottlenecks.

  3. 3.

    For consistency with previously published results [32], we use Snappy, which was the default codec in Spark 1.x. However, we also include LZ4, which is the default in Spark 2.x.

  4. 4.

    For Mnist with its original 10 classes, we created the labels with \(\mathbf {y} \leftarrow (\mathbf {y}==7)\) (i.e., class 7 against the rest), whereas for ImageNet with its 1000 classes, we created the labels with \(\mathbf {y}\leftarrow (\mathbf {y}_0 > (\max (\mathbf {y}_0) - (\max (\mathbf {y}_0)-\min (\mathbf {y}_0))/2))\), where we derived \(\mathbf {y}_0 = \mathbf {X}\mathbf {w}\) from the data \(\mathbf {X}\) and a random model \(\mathbf {w}\).

  5. 5.

    We enabled code generation for cell-wise operations only because SystemML 0.14 does not yet support operator fusion, i.e., code generation, for compressed matrices.

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Acknowledgements

We thank Alexandre Evfimievski and Prithviraj Sen for thoughtful discussions on compressed linear algebra and code generation, Srinivasan Parthasarathy for pointing us to the related work on graph compression, as well as our reviewers for their valuable comments and suggestions.

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Elgohary, A., Boehm, M., Haas, P.J. et al. Compressed linear algebra for large-scale machine learning. The VLDB Journal 27, 719–744 (2018). https://doi.org/10.1007/s00778-017-0478-1

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Keywords

  • Machine learning
  • Large-scale
  • Declarative
  • Linear algebra
  • Lossless compression