feat: reduce computations in backprop of lfilter

[yoyolicoris]

This PR update the backpropagation computation of DifferentiableIIR.
The update is based on my recent work (https://arxiv.org/abs/2406.05128), which uses just one DifferentiableIIR::apply instead of two to compute the gradients for both input and a_coeffs. The algorithm has been tested in torchlpc.

Below is the benchmark to version 2.4.1 lfilter.
The backward computation runs slightly faster especially when using just one thread.

v2.4.1

[-------------- IIR filter -------------]
                  |  forward  |  backward
1 threads: ------------------------------
      [32, 256]   |    404.0  |   1104.5
      [32, 1024]  |    694.3  |   1783.2
      [32, 4096]  |   2637.1  |   6579.2
2 threads: ------------------------------
      [32, 256]   |    381.5  |   1110.2
      [32, 1024]  |    574.7  |   1638.3
      [32, 4096]  |   1729.0  |   5724.7
4 threads: ------------------------------
      [32, 256]   |    362.9  |   1084.0
      [32, 1024]  |    502.0  |   1537.7
      [32, 4096]  |   1485.5  |   5376.4

Times are in microseconds (us).

This version

[-------------- IIR filter -------------]
                  |  forward  |  backward
1 threads: ------------------------------
      [32, 256]   |    391.9  |    935.1
      [32, 1024]  |    678.7  |   1499.8
      [32, 4096]  |   2481.2  |   5782.2
2 threads: ------------------------------
      [32, 256]   |    371.5  |    994.5
      [32, 1024]  |    550.3  |   1551.0
      [32, 4096]  |   1655.0  |   5896.0
4 threads: ------------------------------
      [32, 256]   |    349.1  |    981.6
      [32, 1024]  |    484.8  |   1518.2
      [32, 4096]  |   1380.2  |   4969.1

Times are in microseconds (us).

[pytorch-bot]

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🧪 See artifacts and rendered test results at hud.pytorch.org/pr/pytorch/audio/3831

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❌ 2 New Failures, 3 Cancelled Jobs, 6 Unrelated Failures

As of commit 28575a7 with merge base 97ed7b3 ():

NEW FAILURES - The following jobs have failed:

• Integration Test / build (3.8) (gh)
• Unit-tests on Linux GPU / tests (3.10, 11.8) / linux-job (gh)
  RuntimeError: Command docker exec -t 273ccb60ecc5704c8cc9801f73771505967f02bfa36fb7d13c0088b1622322d2 /exec failed with exit code 2

CANCELLED JOBS - The following jobs were cancelled. Please retry:

• Unit-tests on Linux CPU / tests (3.10) / linux-job (gh)
  ##[error]The operation was canceled.
• Unit-tests on Linux CPU / tests (3.8) / linux-job (gh)
  ##[error]The operation was canceled.
• Unit-tests on Linux CPU / tests (3.9) / linux-job (gh)
  ##[error]The operation was canceled.

FLAKY - The following jobs failed but were likely due to flakiness present on trunk:

• Build Linux Wheels / pytorch/audio / upload / manywheel-py3_9-cpu (gh) (detected as infra flaky with no log or failing log classifier)
• Unittests on Windows CPU / unittests-windows-cpu / windows-job (gh) (detected as infra flaky with no log or failing log classifier)

BROKEN TRUNK - The following jobs failed but were present on the merge base:

👉 Rebase onto the `viable/strict` branch to avoid these failures

• Unit-tests on Linux GPU / tests (3.8, 11.8) / linux-job (gh) (trunk failure)
  ##[error]The operation was canceled.
• Unit-tests on Linux GPU / tests (3.9, 11.8) / linux-job (gh) (trunk failure)
  torchaudio_unittest/models/hdemucs/hdemucs_gpu_test.py::HDemucsFloat32GPUTest::test_import_recreate_low_model_3
• Unit-tests on Macos CPU / tests / macos-job (gh) (trunk failure)
  torchaudio_unittest/prototype/functional/pyroomacoustics_compatibility_test.py::CompatibilityTest::test_simulate_rir_ism_single_band_1
• Unittests on Windows GPU / unittests-windows-gpu / windows-job (gh) (trunk failure)
  torchaudio_unittest\prototype\functional\pyroomacoustics_compatibility_test.py::CompatibilityTest::test_simulate_rir_ism_single_band_1

This comment was automatically generated by Dr. CI and updates every 15 minutes.

[mthrok]

Thanks, looks good. Do you want to update the docstring to mention the new algorithm / paper?

[yoyolicoris]

Thanks, looks good. Do you want to update the docstring to mention the new algorithm / paper?

Oh, sure, I totally forgot. Will update the docstring later this week.

[yoyolicoris]

The code passes related unit tests on my computer.