Performance

Data structures for sparsity pattern representations

The most efficient internal data structure for sparsity pattern representations depends on the number of inputs and the computational graph / sparsity of a given function.

Let's use a convolutional layer from Flux.jl as an example. By default, SCT uses BitSet for Jacobian sparsity detection, which is well suited for small to medium sized functions.

using SparseConnectivityTracer, Flux, BenchmarkTools

x = rand(28, 28, 3, 1)
layer = Conv((3, 3), 3 => 2)

detector_bitset = TracerSparsityDetector()
jacobian_sparsity(layer, x, detector_bitset)

1352×2352 SparseArrays.SparseMatrixCSC{Bool, Int64} with 36504 stored entries:
⎡⠙⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎤
⎢⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⡀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⎥
⎢⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠦⎥
⎢⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⡀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⎥
⎣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⎦

@benchmark jacobian_sparsity(layer, x, detector_bitset);

BenchmarkTools.Trial: 764 samples with 1 evaluation per sample.
 Range (min … max):  5.028 ms … 20.653 ms  ┊ GC (min … max):  0.00% … 48.41%
 Time  (median):     5.471 ms              ┊ GC (median):     0.00%
 Time  (mean ± σ):   6.529 ms ±  1.813 ms  ┊ GC (mean ± σ):  17.10% ± 17.83%

   ▃█▆▂                                                       
  ▄████▅▄▃▂▁▁▁▂▁▁▂▁▁▂▁▁▂▁▂▂▂▃▄▅▅▅▅▅▄▃▅▄▃▃▂▃▁▂▁▁▁▁▁▁▁▁▁▁▂▁▂▁▂ ▃
  5.03 ms        Histogram: frequency by time          11 ms <

 Memory estimate: 20.35 MiB, allocs estimate: 255954.

Instead of BitSet, we can use any concrete subtype of AbstractSet{<:Integer}, for example Set{UInt}. To set the sparsity pattern type for Jacobian sparsity detection, we use the keyword argument gradient_pattern_type:

detector_set = TracerSparsityDetector(; gradient_pattern_type=Set{UInt})
@benchmark jacobian_sparsity(layer, x, detector_set);

BenchmarkTools.Trial: 257 samples with 1 evaluation per sample.
 Range (min … max):  17.087 ms … 38.319 ms  ┊ GC (min … max):  0.00% … 36.89%
 Time  (median):     19.589 ms              ┊ GC (median):    10.81%
 Time  (mean ± σ):   19.490 ms ±  2.669 ms  ┊ GC (mean ± σ):   8.21% ±  7.65%

   █                                                           
  ▇█▆▄▄▃▁▂▃▅▇▇█▇▅▅▄▃▃▂▁▂▁▁▁▁▂▁▂▁▁▁▁▁▁▁▁▁▁▁▂▃▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂ ▃
  17.1 ms         Histogram: frequency by time        31.1 ms <

 Memory estimate: 27.45 MiB, allocs estimate: 256928.

While this is slower for the given input size, the performance is highly dependant on the problem. For larger inputs (e.g. of size $224 \times 224 \times 3 \times 1$), detector_set will outperform detector_bitset. Note that memory requirement will vary as well.

For Hessians sparsity detection, the internal sparsity pattern representation uses either concrete subtypes of AbstractDict{I, AbstractSet{I}} or AbstractSet{Tuple{I, I}}, where I <: Integer. By default, Dict{Int, BitSet) is used. To set the sparsity pattern type, use the keyword argument hessian_pattern_type:

detector = TracerSparsityDetector(; hessian_pattern_type=Dict{UInt, Set{UInt}})

TracerSparsityDetector{SparseConnectivityTracer.GradientTracer{Int64, BitSet},SparseConnectivityTracer.HessianTracer{UInt64, Set{UInt64}, Dict{UInt64, Set{UInt64}}, SparseConnectivityTracer.NotShared}}()

Data structures can also be set analogously for TracerLocalSparsityDetector. If both Jacobian and Hessian sparsity patterns are needed, gradient_pattern_type and hessian_pattern_type can be set separately.