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: 750 samples with 1 evaluation per sample.
 Range (min … max):  4.643 ms … 20.766 ms  ┊ GC (min … max):  0.00% … 57.78%
 Time  (median):     5.703 ms              ┊ GC (median):     0.00%
 Time  (mean ± σ):   6.648 ms ±  1.815 ms  ┊ GC (mean ± σ):  17.05% ± 18.20%

       ▂▆█▄▂▁▂                                                
  ▃▂▂▃████████▇▆▃▃▁▂▁▂▂▁▁▂▁▁▁▁▁▁▂▁▂▃▂▄▅▅▇▇▆▆▆▅▅▄▄▃▃▂▂▂▂▁▁▁▁▃ ▃
  4.64 ms        Histogram: frequency by time        10.4 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: 258 samples with 1 evaluation per sample.
 Range (min … max):  17.048 ms … 45.700 ms  ┊ GC (min … max):  0.00% … 47.52%
 Time  (median):     19.474 ms              ┊ GC (median):    11.25%
 Time  (mean ± σ):   19.420 ms ±  3.307 ms  ┊ GC (mean ± σ):   8.45% ±  7.88%

  █▅▃       ▂▅▅▄▄▃▂                                            
  ███▆▄▄▄▁▄▇███████▇▇▇▁▁▁▁▄▁▁▄▁▁▁▁▁▁▁▁▁▁▁▁▄▄▆▇▆▄▁▁▁▁▁▁▁▁▁▁▁▁▄ ▆
  17 ms        Histogram: log(frequency) by time      29.6 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.