Auto-scheduling Sparse Matrix Multiplication on CPU with Custom Sketch Rule

Author: Chengfan Jia

This is a tutorial on how to use the auto-scheduler to tune a sparse matrix multiplication for CPUs.

Auto-scheduler is designed to explore the schedule with best performance for a given computation declaration automatically. While sometimes, we may have a demand to try some special ops which may not been well-supported by auto-scheduler’s default sketch rules and result in poor performance. Fortunately, auto-scheduler currently allows user to provide a CustomSketch to cover these cases.

We use sparse matrix multiplication as an example in this tutorial to demonstrate how to implement and plug a custom sketch rule to the auto-scheduler’s search policy.

Note that this tutorial will not run on Windows or recent versions of macOS. To get it to run, you will need to wrap the body of this tutorial in a if __name__ == "__main__": block.

import os

import numpy as np
import tvm
import tvm.testing
from tvm import te, auto_scheduler, runtime, topi
from tvm.auto_scheduler import _ffi_api
from tvm.topi.utils import get_const_tuple
from tvm.topi.sparse.utils import random_bsr_matrix

Define the computation

To begin with, let us define the computation of a sparse matmul with several relu and bias add. The function should return the list of input/output tensors. From these tensors, the auto-scheduler can get the whole computational graph.

@auto_scheduler.register_workload
def sparse_dense(M, N, K, w_data_shape, w_indices_shape, w_indptr_shape, dtype):
    X = te.placeholder(shape=(M, K), dtype=dtype)
    W_data = te.placeholder(shape=w_data_shape, dtype=dtype)
    W_indices = te.placeholder(shape=w_indices_shape, dtype="int32")
    W_indptr = te.placeholder(shape=w_indptr_shape, dtype="int32")
    B = te.placeholder(shape=(M, N), dtype=dtype)

    out = topi.nn.sparse_dense(topi.nn.relu(X), W_data, W_indices, W_indptr)
    out = te.compute((M, N), lambda i, j: out[i, j] + B[i, j], name="BiasAdd")
    out = topi.nn.relu(out)

    return [X, W_data, W_indices, W_indptr, B, out]

Special step for sparse workload

During schedule tuning, auto-scheduler will use random inputs to measure the performance of a generated schedule. While we cannot directly use a random array as the input of a sparse op, for the “indices” and “indptr” array are meaningful for the computation.

To solve this problem, we register these as special buffers, and load them when process program measuring. See the tvm.auto_scheduler.measure.py for more details.

# Define the basic shapes of this sparse computation
M = 128
K = 256
N = 512
BS_R = 16
BS_C = 1
density = 0.6

# Generate the test data with numpy
X_np = np.random.randn(M, K).astype("float32")
X_np = np.maximum(np.zeros((M, K), dtype="float32"), X_np)  # Relu
W_sp_np = random_bsr_matrix(N, K, BS_R, BS_C, density=density, dtype="float32")
W_np = W_sp_np.todense()
Y_np = X_np @ W_np.T  # Process the matrix multiplication
B_np = np.random.randn(M, N).astype("float32")
Y_np = Y_np + B_np  # Bias add
Y_np = np.maximum(np.zeros((M, N), dtype="float32"), Y_np)  # Relu

Create the search task

We then create a search task with M=N=K=512 and dtype=”float32” If your machine supports avx instructions, you can

• replace “llvm” below with “llvm -mcpu=core-avx2” to enable AVX2

• replace “llvm” below with “llvm -mcpu=skylake-avx512” to enable AVX-512

target = tvm.target.Target("llvm")

# Register the sparse data to task inputs
prefix = "sparse_dense_bsr_%d_%d_%d_%d_%d_%d_" % (
    N,
    K,
    BS_R,
    BS_C,
    W_sp_np.indices.shape[0],
    W_sp_np.indptr.shape[0],
)
task = tvm.auto_scheduler.SearchTask(
    func=sparse_dense,
    args=(M, N, K, W_sp_np.data.shape, W_sp_np.indices.shape, W_sp_np.indptr.shape, "float32"),
    target=target,
    task_inputs={
        prefix + "W_data": runtime.ndarray.array(W_sp_np.data),
        prefix + "W_indices": runtime.ndarray.array(W_sp_np.indices),
        prefix + "W_indptr": runtime.ndarray.array(W_sp_np.indptr),
    },
    task_inputs_save_to_file=True,
)

# Inspect the computational graph
print("Computational DAG:")
print(task.compute_dag)

Computational DAG:
placeholder = PLACEHOLDER [33]
placeholder = PLACEHOLDER [4916, 16, 1]
placeholder = PLACEHOLDER [4916]
placeholder = PLACEHOLDER [128, 256]
compute(i0, i1) = max(placeholder[i0, i1], 0f)
compute(i, nb_j, j) += (placeholder[(placeholder[nb_j] + elem_idx), j, c]*compute[i, (placeholder[(placeholder[nb_j] + elem_idx)] + c)])
compute(m, n) = compute[m, floordiv(n, 16), floormod(n, 16)]
placeholder = PLACEHOLDER [128, 512]
BiasAdd(i, j) = (compute[i, j] + placeholder[i, j])
compute(i0, i1) = max(BiasAdd[i0, i1], 0f)

Write the custom sketch for sparse dense op

Before tuning, we will need to define the CustomSketchRule for the sparse dense op.

CustomSketchRule consists of two parts: the condition function and the apply function.

• condition function: describe when to apply this sketch rule. For example, we can only apply the rule to the sparse ops by matching their name and tag.

• apply function: describe how to generate the initial sketch. You can implement it using auto-scheduler provided loop state APIs.

def meet_condition_func(search_policy, state, stage_id):
    state = auto_scheduler.loop_state.State(state, search_policy.search_task.compute_dag)
    if state.stages[stage_id].op.tag in [
        "sparse_dense_sp_rhs_bsrmm",
        "sparse_dense_sp_rhs_bsrmm_block",
    ]:
        return auto_scheduler.PreloadCustomSketchRule.APPLY_AND_SKIP_REST
    else:
        return auto_scheduler.PreloadCustomSketchRule.PASS


def apply_func(search_policy, state, stage_id):
    ret = []
    s0 = auto_scheduler.loop_state.State(state, search_policy.search_task.compute_dag)
    if s0.stages[stage_id].op.tag == "sparse_dense_sp_rhs_bsrmm_block":
        return [s0.state_object, stage_id - 1]

    sparse_dense = s0.stages[stage_id].op
    sparse_dense_block = s0.stages[stage_id - 1].op
    assert sparse_dense.tag == "sparse_dense_sp_rhs_bsrmm"
    assert sparse_dense_block.tag == "sparse_dense_sp_rhs_bsrmm_block"

    # Set the default consumer of compute block
    consumer = sparse_dense

    # If sparse dense has a single elementwise consumer
    # We can compute inline the sparse_dense output stage
    consumers = _ffi_api.SearchPolicyUtilsGetConsumers(
        search_policy.search_task, s0.state_object, stage_id
    )
    if len(consumers) == 1:
        consumer_id = int(consumers.items()[0][0])
        if _ffi_api.SearchPolicyUtilsIsElementwiseMatch(
            search_policy.search_task, s0.state_object, stage_id, consumer_id
        ):
            consumer = s0.stages[consumer_id].op
            s0.compute_inline(sparse_dense)

    i, nb_j, j, row_offset, c = s0[sparse_dense_block].iters
    m, n = s0[consumer].iters
    i0, i1, i2 = s0.split(sparse_dense_block, i, [None, None])
    m0, m1 = s0.follow_split(consumer, m, len(s0.transform_steps) - 1, 1)
    j0, j1 = s0.split(sparse_dense_block, nb_j, [None])
    n0, n1 = s0.follow_split(consumer, n, len(s0.transform_steps) - 1, 1)
    s0.reorder(sparse_dense_block, [i0, j0, i1, j1, row_offset, i2, j, c])
    s0.reorder(consumer, [m0, n0, m1, n1])
    s0.compute_at(sparse_dense_block, consumer, n0)

    ret.append([s0.state_object, stage_id - 2])

    return ret

Next, we set parameters for the auto-scheduler with the custom sketch plugged in.

• num_measure_trials is the number of measurement trials we can use during the search. We only make 10 trials in this tutorial for a fast demonstration. In practice, 1000 is a good value for the search to converge. You can do more trials according to your time budget.

• In addition, we use RecordToFile to dump measurement records into a file sparse_dense.json. The measurement records can be used to query the history best, resume the search, and do more analyses later.

• see auto_scheduler.TuningOptions for more parameters

• Here, we need to create a auto_scheduler.SketchPolicy object, and add the custom sketch rule as a init_search_callbacks.

log_file = "sparse_dense.json"
tune_option = auto_scheduler.TuningOptions(
    num_measure_trials=10,
    measure_callbacks=[auto_scheduler.RecordToFile(log_file)],
    verbose=2,
)

search_policy = auto_scheduler.SketchPolicy(
    task,
    program_cost_model=auto_scheduler.XGBModel(),
    init_search_callbacks=[
        auto_scheduler.PreloadCustomSketchRule(meet_condition_func, apply_func, "SparseDense")
    ],
)

Run the search

Now we get all inputs ready. We can kick off the search and let the auto-scheduler do its magic. After some measurement trials, we can load the best schedule from the log file and apply it.

# Run auto-tuning (search)
# Notice: We do not run the tuning in our webpage server since it takes too long.
# Uncomment the following line to run it by yourself.
task.tune(tune_option, search_policy)

# Apply the best schedule
sch, args = task.apply_best(log_file)

We can lower the schedule to see the IR after auto-scheduling. The auto-scheduler correctly performs optimizations including multi-level tiling, layout transformation, parallelization, vectorization, unrolling, and operator fusion.

print("Lowered TIR:")
print(tvm.lower(sch, args, simple_mode=True))

Lowered TIR:
# from tvm.script import ir as I
# from tvm.script import tir as T

@I.ir_module
class Module:
    @T.prim_func
    def main(placeholder: T.Buffer((128, 256), "float32"), placeholder_1: T.Buffer((4916, 16, 1), "float32"), placeholder_2: T.Buffer((4916,), "int32"), placeholder_3: T.Buffer((33,), "int32"), placeholder_4: T.Buffer((128, 512), "float32"), compute: T.Buffer((128, 512), "float32")):
        T.func_attr({"from_legacy_te_schedule": T.bool(True), "global_symbol": "main", "tir.noalias": T.bool(True)})
        for i0_outer_i1_outer_fused in T.parallel(128):
            compute_1 = T.allocate([512], "float32", "global")
            compute_2 = T.Buffer((512,), data=compute_1)
            for i_inner_init, j_init in T.grid(32, 16):
                compute_2[i_inner_init * 16 + j_init] = T.float32(0)
            for elem_idx, i_inner, j in T.grid(T.Let(placeholder_5[cse_var_1 + 1] - placeholder_5[cse_var_1], where={cse_var_1: i0_outer_i1_outer_fused % 32}), 32, 16):
                cse_var_1 = T.int32()
                placeholder_5 = T.Buffer((33,), "int32", data=placeholder_3.data)
                cse_var_2: T.int32 = i0_outer_i1_outer_fused % 32
                if T.likely(elem_idx < placeholder_5[cse_var_2 + 1] - placeholder_5[cse_var_2]):
                    placeholder_6 = T.Buffer((78656,), data=placeholder_1.data)
                    placeholder_7 = T.Buffer((32768,), data=placeholder.data)
                    placeholder_8 = T.Buffer((4916,), "int32", data=placeholder_2.data)
                    cse_var_3: T.int32 = i_inner * 16 + j
                    compute_2[cse_var_3] = compute_2[cse_var_3] + placeholder_6[placeholder_5[cse_var_2] * 16 + elem_idx * 16 + j] * T.max(placeholder_7[i0_outer_i1_outer_fused // 32 * 8192 + i_inner * 256 + placeholder_8[placeholder_5[cse_var_2] + elem_idx]], T.float32(0))
            for i0_inner in range(32):
                cse_var_4: T.int32 = i0_outer_i1_outer_fused // 32 * 16384 + i0_inner * 512 + i0_outer_i1_outer_fused % 32 * 16
                compute_3 = T.Buffer((65536,), data=compute.data)
                placeholder_5 = T.Buffer((65536,), data=placeholder_4.data)
                compute_3[cse_var_4:cse_var_4 + 16] = T.max(compute_2[i0_inner * 16:i0_inner * 16 + 16] + placeholder_5[cse_var_4:cse_var_4 + 16], T.Broadcast(T.float32(0), 16))

Check correctness and evaluate performance

We build the binary and check its correctness and performance.

func = tvm.build(sch, args, target)

dev = tvm.cpu()

X_tvm = tvm.nd.array(X_np, device=dev)
W_data_tvm = tvm.nd.array(W_sp_np.data, device=dev)
W_indices_tvm = tvm.nd.array(W_sp_np.indices, device=dev)
W_indptr_tvm = tvm.nd.array(W_sp_np.indptr, device=dev)
B_tvm = tvm.nd.array(B_np, device=dev)
Y_tvm = tvm.nd.empty(Y_np.shape, device=dev)

func(X_tvm, W_data_tvm, W_indices_tvm, W_indptr_tvm, B_tvm, Y_tvm)

# Check results
tvm.testing.assert_allclose(Y_np, Y_tvm.numpy(), atol=1e-4, rtol=1e-4)

# Evaluate execution time.
evaluator = func.time_evaluator(func.entry_name, dev, min_repeat_ms=500)
print(
    "Execution time of this operator: %.3f ms"
    % (
        np.median(evaluator(X_tvm, W_data_tvm, W_indices_tvm, W_indptr_tvm, B_tvm, Y_tvm).results)
        * 1000
    )
)

Execution time of this operator: 1.663 ms

Note

Tuning result example

----------------------------------------------------------------------
Lowered TIR:
primfn(placeholder_5: handle, placeholder_6: handle, placeholder_7: handle, placeholder_8: handle, placeholder_9: handle, compute_1: handle) -> ()
  attr = {"global_symbol": "main", "tir.noalias": True}
  buffers = {placeholder_2: Buffer(placeholder_10: Pointer(float32), float32, [9831, 16, 1], []),
             placeholder_4: Buffer(placeholder_11: Pointer(int32), int32, [33], []),
             placeholder_3: Buffer(placeholder_12: Pointer(float32), float32, [512, 512], []),
             compute: Buffer(compute_2: Pointer(float32), float32, [512, 512], []),
             placeholder_1: Buffer(placeholder_13: Pointer(float32), float32, [512, 512], []),
             placeholder: Buffer(placeholder_14: Pointer(int32), int32, [9831], [])}
  buffer_map = {placeholder_7: placeholder, placeholder_9: placeholder_1, placeholder_6: placeholder_2, compute_1: compute, placeholder_5: placeholder_3, placeholder_8: placeholder_4} {
  for (i0.outer.i1.outer.fused: int32, 0, 1024) "parallel" {
    attr [compute_3: Pointer(float32)] "storage_scope" = "global";
    allocate(compute_3, float32, [256]) {
      for (nb_j.inner: int32, 0, 2) {
        for (i.inner.init: int32, 0, 8) {
          for (j.init: int32, 0, 16) {
            compute_3[(((i.inner.init*32) + (nb_j.inner*16)) + j.init)] = 0f32
          }
        }
        for (elem_idx: int32, 0, ((int32*)placeholder_11[(((floormod(i0.outer.i1.outer.fused, 16)*2) + nb_j.inner) + 1)] - (int32*)placeholder_11[((floormod(i0.outer.i1.outer.fused, 16)*2) + nb_j.inner)])) {
          for (i.inner: int32, 0, 8) {
            for (j: int32, 0, 16) {
              compute_3[(((i.inner*32) + (nb_j.inner*16)) + j)] = ((float32*)compute_3[(((i.inner*32) + (nb_j.inner*16)) + j)] + ((float32*)placeholder_10[((((int32*)placeholder_11[((floormod(i0.outer.i1.outer.fused, 16)*2) + nb_j.inner)]*16) + (elem_idx*16)) + j)]*max((float32*)placeholder_12[(((floordiv(i0.outer.i1.outer.fused, 16)*4096) + (i.inner*512)) + (int32*)placeholder_14[((int32*)placeholder_11[((floormod(i0.outer.i1.outer.fused, 16)*2) + nb_j.inner)] + elem_idx)])], 0f32)))
            }
          }
        }
      }
      for (i0.inner: int32, 0, 8) {
        compute_2[ramp((((floordiv(i0.outer.i1.outer.fused, 16)*4096) + (i0.inner*512)) + (floormod(i0.outer.i1.outer.fused, 16)*32)), 1, 32)] = max(((float32x32*)compute_3[ramp((i0.inner*32), 1, 32)] + (float32x32*)placeholder_13[ramp((((floordiv(i0.outer.i1.outer.fused, 16)*4096) + (i0.inner*512)) + (floormod(i0.outer.i1.outer.fused, 16)*32)), 1, 32)]), broadcast(0f32, 32))
      }
    }
  }
}