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Inductor CPU backend debugging and profiling#

Created On: Jul 01, 2023 | Last Updated: Jan 08, 2025 | Last Verified: Nov 05, 2024

Authors: Xuan Liao, Haozhe Zhu, Jiong Gong, Weihan Wang

Overview#

PyTorch 2.0 introduced the compilation API called torch.compile. This new feature offers a significant speedup over eager mode execution through graph-level optimization powered by the default Inductor backend.

This tutorial is intended to provide an in-depth introduction on the debugging and performance profiling on Inductor CPU backend by delving into the intricacies of torch.compile.

Meanwhile, you may also find related tutorials about torch.compile around basic usage, comprehensive troubleshooting and GPU-specific knowledge like GPU performance profiling.

We will start debugging with a motivating example that triggers compilation issues and accuracy problems by demonstrating the process of debugging to pinpoint the problems.

By enabling logging and exploring the underlying generated code, you can learn how to narrow down the failure step by step and finally figure out the route cause.

Following that, we will proceed to discuss how to profile the compiled code and, through a performance comparison with eager mode, elaborate on the reasons why torch.compile can provide an additional performance boost compared to its eager counterpart.

Debugging#

Here is a simple example to run the torch.compile using Inductor and compare its result with eager mode:

importtorch
deffoo1(x1 , x2 ):
 a = torch.neg (x1 )
 b = torch.maximum (x2 , a)
 y = torch.cat ([b], dim=0)
 return y
x1 = torch.randint (256, (1, 8), dtype=torch.uint8 )
x2 = torch.randint (256, (8390, 8), dtype=torch.uint8 )
compiled_foo1 = torch.compile (foo1)
result = compiled_foo1(x1 , x2 )

The correct implementation of neg in the cpp codegen is as follows:

defneg1(x):
 return f"decltype({x})(-{x})"

In order to demonstrate the debugging, we will modify the function to a wrong one later.

Get more logging information#

No debugging information would be provided if you run this simple example by default. In order to get more useful debugging and logging information, we usually add a TORCH_COMPILE_DEBUG environment variable like below:

TORCH_COMPILE_DEBUG=1pythonxx.py

This would print more debug information in the output logs and also dump the intermediate IRs generated during the codegen process. You can find the dumped file paths in the log like below:

torch._inductor.debug:[WARNING]model___20debugtrace:/tmp/torchinductor_root/rx/crxfi2ybd7yp5sbj2pnhw33wfhtdw7wumvrobyp5sjvdui5ktjc2.debug

In this directory, the following files are saved for debugging purposes:

File

Description

fx_graph_runnable.py

Executable FX graph, after decomposition, before pattern match

fx_graph_transformed.py

Transformed FX graph, after pattern match

ir_pre_fusion.txt

Inductor IR before fusion

ir_post_fusion.txt

Inductor IR after fusion

output_code.py

Generated Python code for graph, with C++/Triton kernels

Note that fx_graph_runnable.py and output_code.py are both runnable and editable in order to make debugging easier. Here are the main parts of code extracted from the files and we correlate the C++ generated line with the FX code line.

fx_graph_runnable:

defforward1(self, arg0_1, arg1_1):
 neg = torch.ops.aten.neg.default(arg0_1); arg0_1 = None
 maximum = torch.ops.aten.maximum.default(arg1_1, neg); arg1_1 = neg = None
 clone = torch.ops.aten.clone.default(maximum); maximum = None
 return (clone,)

C++ kernel in output_code:

importtorch
fromtorch._inductor.async_compileimport AsyncCompile
async_compile = AsyncCompile()
cpp_fused_cat_maximum_neg_0 = async_compile.cpp('''
#include "/tmp/torchinductor_root/gv/cgv6n5aotqjo5w4vknjibhengeycuattfto532hkxpozszcgxr3x.h"
extern "C" void kernel(const unsigned char* in_ptr0,
 const unsigned char* in_ptr1,
 unsigned char* out_ptr0)
{
 {
 #pragma GCC ivdep
 for(long i0=static_cast<long>(0L); i0<static_cast<long>(8390L); i0+=static_cast<long>(1L))
 {
 #pragma GCC ivdep
 for(long i1=static_cast<long>(0L); i1<static_cast<long>(8L); i1+=static_cast<long>(1L))
 {
 auto tmp0 = in_ptr0[static_cast<long>(i1 + (8L*i0))];
 auto tmp1 = in_ptr1[static_cast<long>(i1)];
 // Corresponding FX code line: neg = torch.ops.aten.neg.default(arg0_1); arg0_1 = None
 auto tmp2 = decltype(tmp1)(-tmp1);
 // Corresponding FX code line: maximum = torch.ops.aten.maximum.default(arg1_1, neg); arg1_1 = neg = None
 auto tmp3 = max_propagate_nan(tmp0, tmp2);
 // Corresponding FX code line: clone = torch.ops.aten.clone.default(maximum); maximum = None
 out_ptr0[static_cast<long>(i1 + (8L*i0))] = tmp3;
 }
 }
 }
}''')

Determine component of error#

When encountering errors or accuracy problems, a straightforward solution to find the bug is to narrow down the problem. The first thing to do is to determine the component where the error occurs. Luckily, it can be simply achieved by changing the backend of torch.compile.

Code

Description

torch.compile(fn, backend="eager")

Enable Dynamo

torch.compile(fn, backend="aot_eager")

Enable Dynamo + AOT Autograd

torch.compile(fn, backend="inductor")

Enable Dynamo + AOT Autograd + Inductor

If the model can successfully run when the backend is set to eager or aot_eager while it fails with inductor, we can narrow down the failure to Inductor.

Compilation error#

As we know, the evolved chain of graph-level optimization is like:

torch.neg(Python)->torch.ops.aten.neg.default(withinFXgraph)->ops.neg(withinIRnode)->tmp2=-tmp1(withinC++kernel)

If you encounter a compilation error, there is something wrong when compiling C++ kernels in the output code. This type of error indicates that bugs are introduced when lowering IR nodes to output code. The root cause of compilation error is usually shown in the traceback log.

For example, the neg function is modified like this:

defneg2(x):
 return f"-{x}"

The logging gives the following compile error with a rather clear reason.

 torch._dynamo.exc.BackendCompilerFailed: backend='inductor' raised:
 CppCompileError: C++ compile error
 /tmp/torchinductor_root/xg/cxga5tk3b4lkwoxyigrtocjp5s7vc5cg2ikuscf6bk6pjqip2bhx.cpp: In function ‘void kernel(const unsigned char*, const unsigned char*, unsigned char*)’:
 /tmp/torchinductor_root/xg/cxga5tk3b4lkwoxyigrtocjp5s7vc5cg2ikuscf6bk6pjqip2bhx.cpp:17:57: error: no matching function for call to ‘max_propagate_nan(unsigned char&, int&)’
 17 | auto tmp3 = max_propagate_nan(tmp0, tmp2);
 | ^
 In file included from /tmp/torchinductor_root/xg/cxga5tk3b4lkwoxyigrtocjp5s7vc5cg2ikuscf6bk6pjqip2bhx.cpp:2:
 /tmp/torchinductor_root/gv/cgv6n5aotqjo5w4vknjibhengeycuattfto532hkxpozszcgxr3x.h:27:17: note: candidate: ‘template<class scalar_t> scalar_t max_propagate_nan(scalar_t, scalar_t)’
 27 | inline scalar_t max_propagate_nan(scalar_t a, scalar_t b) {
 | ^~~~~~~~~~~~~~~~~
 /tmp/torchinductor_root/gv/cgv6n5aotqjo5w4vknjibhengeycuattfto532hkxpozszcgxr3x.h:27:17: note: template argument deduction/substitution failed:
/tmp/torchinductor_root/xg/cxga5tk3b4lkwoxyigrtocjp5s7vc5cg2ikuscf6bk6pjqip2bhx.cpp:17:57: note: deduced conflicting types for parameter ‘scalar_t’ (‘unsigned char’ and ‘int’)
 17 | auto tmp3 = max_propagate_nan(tmp0, tmp2);
 | ^

Let us also see the corresponding C++ kernel in output code and IR node.

C++ kernel:

include"/tmp/torchinductor_root/gv/cgv6n5aotqjo5w4vknjibhengeycuattfto532hkxpozszcgxr3x.h"
extern"C"voidkernel(constunsignedchar*in_ptr0,
constunsignedchar*in_ptr1,
unsignedchar*out_ptr0)
{
{
#pragma GCC ivdep
for(longi0=static_cast<long>(0L);i0<static_cast<long>(8390L);i0+=static_cast<long>(1L))
{
#pragma GCC ivdep
for(longi1=static_cast<long>(0L);i1<static_cast<long>(8L);i1+=static_cast<long>(1L))
{
autotmp0=in_ptr0[static_cast<long>(i1+(8L*i0))];
autotmp1=in_ptr1[static_cast<long>(i1)];
autotmp2=-tmp1;
autotmp3=max_propagate_nan(tmp0,tmp2);
out_ptr0[static_cast<long>(i1+(8L*i0))]=tmp3;
}
}
}
}

IR node:

buf0:SchedulerNode(ComputedBuffer)
buf0.writes=[MemoryDep('buf0',c0,{c0:67120})]
buf0.unmet_dependencies=[]
buf0.met_dependencies=
[MemoryDep('arg0_1',c1,{c0:8390,c1:8}),
MemoryDep('arg1_1',c0,{c0:67120})]
buf0.users=[NodeUser(node=OUTPUT,can_inplace=False)]
buf0.group.device=cpu
buf0.group.iteration=((8390,8),())
buf0.sizes=([8390,8],[])
classbuf0_loop_body:
var_ranges={z0:8390,z1:8}
index0=8*z0+z1
index1=z1
defbody(self,ops):
get_index=self.get_index('index0')
load=ops.load('arg1_1',get_index)
get_index_1=self.get_index('index1')
load_1=ops.load('arg0_1',get_index_1)
neg=ops.neg(load_1)
maximum=ops.maximum(load,neg)
get_index_2=self.get_index('index0')
store=ops.store('buf0',get_index_2,maximum,None)
returnstore

According to the traceback logging, the compilation error is caused by the data type inconsistency of max_propagate_nan’s inputs. By checking the C++ kernel, we know that tmp2 is no longer long after doing - as tmp0 is long. We can easily match - and max_propagate_nan in C++ kernel with ops.neg and ops.maximum in IR node respectively.

Now we successfully find that the root cause is the implementation of ops.neg in cpp codegen, which silently changes the data type when doing neg.

Accuracy debugging#

Otherwise, if the model runs with other errors or accuracy problem, you can use the PyTorch debugging tool called Minifier.

The core idea of Minifier is to keep removing the nodes and inputs of graph until finding the minimal graph with problem. It helps to automatically generate a minified problematic graph through 4 strategies: truncating suffix, delta debugging, eliminating dead code and removing unused inputs.

We will now show the debugging process for the accuracy problem with the help of Minifer. The accuracy problem refers to the case where the outputs of backends eager and inductor are different.

For instance, we modify the example like this:

fromtorch._dynamo.utilsimport same
deffoo2(x1 , x2 ):
 a = torch.neg (x1 )
 b = torch.maximum (x2 , a)
 y = torch.cat ([b], dim=0)
 return y
x1 = torch.randn ((1, 8), dtype=torch.float32 )
x2 = torch.randn ((8390, 8), dtype=torch.float32 )
expected_result = foo2(x1 , x2 )
compiled_foo2 = torch.compile (foo2)
actual_result = compiled_foo2(x1 , x2 )
assert same(expected_result , actual_result ) == True

And also modify the neg function:

defneg3(x):
 return f"decltype({x})(2 * {x})"

An accuracy problem would be raised as follows:

torch._dynamo.utils:[ERROR]Accuracyfailed:allclosenotwithintol=0.0001
Traceback(mostrecentcalllast):
File"test_script.py",line18,in<module>
assertsame(expected_result,actual_result)==True
AssertionError

To debug an accuracy problem with Minifier, two environment variables are needed:

TORCHDYNAMO_REPRO_AFTER="aot"TORCHDYNAMO_REPRO_LEVEL=4pythonxx.py

Which gives us logging information that demonstrates the steps of minifying:

Startedoffwith6nodes
Tryinggranularity2
Strategy:Truncatesuffix(G:2)(6nodes,2inputs)
SUCCESS:Wentfrom6to4nodes
Tryinggranularity4
Strategy:Removeunusedinputs(G:4)(4nodes,2inputs)
SUCCESS:Wentfrom4to3nodes

After running, we get the final minified graph with the target node neg:

defforward2(self, arg0_1):
 neg = torch.ops.aten.neg.default(arg0_1); arg0_1 = None
 return (neg,)

For more usage details about Minifier, please refer to Troubleshooting.

Performance profiling#

Within this section, we will demonstrate the process of conducting performance analysis for a model that has been compiled using the Inductor CPU backend. In the example below, we benchmark a Hugging Face Transformer model MobileBertForQuestionAnswering with both the eager mode and the Inductor graph mode. The execution time and the speedup ratio of Inductor are printed after the benchmark. We use Intel(R) Xeon(R) Platinum 8358 CPU @ 2.60GHz and run benchmark on the first socket to demonstrate the optimization within this section. We set following environment variable as a best practice to benchmark on Intel(R) CPU.

exportKMP_BLOCKTIME=1
exportKMP_SETTINGS=1
exportKMP_AFFINITY=granularity=fine,compact,1,0
exportLD_PRELOAD=${CONDA_PREFIX:-"$(dirname$(whichconda))/../"}/lib/libiomp5.so:${CONDA_PREFIX:-"$(dirname$(whichconda))/../"}/lib/libjemalloc.so
exportMALLOC_CONF="oversize_threshold:1,background_thread:true,metadata_thp:auto,dirty_decay_ms:-1,muzzy_decay_ms:-1"
numactl-C0-31-m0pythonbench.py
# bench.py
fromtransformersimport MobileBertForQuestionAnswering
# Initialize an eager model
model = MobileBertForQuestionAnswering.from_pretrained("csarron/mobilebert-uncased-squad-v2")
seq_length = 128
bs = 128
vocab_size = model.config.vocab_size
input = torch.randint (0, vocab_size, (bs, seq_length), dtype=torch.int64 )
input_dict = {"input_ids": input}
# Initialize the inductor model
compiled_model = torch.compile (model)
with torch.no_grad ():
 compiled_model(**input_dict)
NUM_ITERS=50
importtimeit
with torch.no_grad ():
 # warmup
 for _ in range(10):
 model(**input_dict)
 eager_t = timeit.timeit("model(**input_dict)", number=NUM_ITERS, globals=globals())
with torch.no_grad ():
 # warmup
 for _ in range(10):
 compiled_model(**input_dict)
 inductor_t = timeit.timeit("compiled_model(**input_dict)", number=NUM_ITERS, globals=globals())
# print(f"eager use: {eager_t * 1000 / NUM_ITERS} ms/iter")
# print(f"inductor use: {inductor_t * 1000 / NUM_ITERS} ms/iter")
# print(f"speed up ratio: {eager_t / inductor_t}")

Output:

eageruse:802.1023553796113ms/iter
inductoruse:339.95180135127157ms/iter
speedupratio:2.359459053287382

In our own testing, we find the Inductor CPU backend speed up the model by around 2.355x.

Next, let’s dive deep into the performance at the operation level to understand where the speed-up comes from. Pytorch Profiler is a good tool to help us. Inductor CPU backend has the support to report the time of the fusion kernels to the profiler with the enable_kernel_profile configuration option:

fromtorch._inductorimport config
config.cpp.enable_kernel_profile = True

Following the steps in Pytorch Profiler We are able to get the profiling table and trace files.

# bench.py
fromtorch.profilerimport profile , schedule , ProfilerActivity
RESULT_DIR = "./prof_trace"
my_schedule = schedule (
 skip_first=10,
 wait=5,
 warmup=5,
 active=1,
 repeat=5)
deftrace_handler(p ):
 output = p.key_averages ().table(sort_by="self_cpu_time_total", row_limit=20)
 # print(output)
 p.export_chrome_trace (f"{RESULT_DIR}/{p .step_num}.json")
for _ in range(10):
 model(**input_dict) # compiled_model(**input_dict) to get inductor model profiling
total = 0
with profile (
 activities=[ProfilerActivity.CPU ],
 schedule =my_schedule,
 on_trace_ready=trace_handler
) as p :
 for _ in range(50):
 model(**input_dict) # compiled_model(**input_dict) to get inductor model profiling
 p.step ()

We get the following performance profiling table for the eager-mode model (omitting some columns):

-------------------------------------------------------------
NameCPUtotal%CPUtotal# of Calls
-------------------------------------------------------------
aten::addmm45.73%370.814ms362
aten::add19.89%161.276ms363
aten::copy_14.97%121.416ms488
aten::mul9.02%73.154ms194
aten::clamp_min8.81%71.444ms96
aten::bmm5.46%44.258ms48
ProfilerStep*100.00%810.920ms1
aten::div2.89%23.447ms24
aten::_softmax1.00%8.087ms24
aten::linear46.48%376.888ms362
aten::clone2.77%22.430ms98
aten::t0.31%2.502ms362
aten::view0.14%1.161ms850
aten::transpose0.17%1.377ms386
aten::index_select0.12%952.000us3
aten::expand0.12%986.000us458
aten::matmul8.31%67.420ms48
aten::cat0.09%703.000us1
aten::as_strided0.08%656.000us963
aten::relu8.86%71.864ms96
-------------------------------------------------------------
SelfCPUtimetotal:810.920ms

Similarly, we also get the table for the compiled model with Inductor (omitting some columns):

-----------------------------------------------------------------------------------
NameCPUtotal%CPUtotal# of Calls
-----------------------------------------------------------------------------------
mkl::_mkl_linear68.79%231.573ms362
aten::bmm8.02%26.992ms48
ProfilerStep*100.00%336.642ms1
graph_0_cpp_fused_constant_pad_nd_embedding_00.27%915.000us1
aten::empty0.27%911.000us362
graph_0_cpp_fused__mkl_linear_add_mul_relu_1510.27%901.000us1
graph_0_cpp_fused__mkl_linear_add_mul_relu_2260.27%899.000us1
graph_0_cpp_fused__mkl_linear_add_mul_relu_3610.27%898.000us1
graph_0_cpp_fused__mkl_linear_add_mul_relu_1210.27%895.000us1
graph_0_cpp_fused__mkl_linear_add_mul_relu_310.27%893.000us1
graph_0_cpp_fused__mkl_linear_add_mul_relu_760.26%892.000us1
graph_0_cpp_fused__mkl_linear_add_mul_relu_2560.26%892.000us1
graph_0_cpp_fused__mkl_linear_add_mul_relu_3460.26%892.000us1
graph_0_cpp_fused__mkl_linear_add_mul_relu_2410.26%891.000us1
graph_0_cpp_fused__mkl_linear_add_mul_relu_3160.26%891.000us1
graph_0_cpp_fused__mkl_linear_add_mul_relu_910.26%890.000us1
graph_0_cpp_fused__mkl_linear_add_mul_relu_1060.26%890.000us1
graph_0_cpp_fused__mkl_linear_add_mul_relu_2110.26%890.000us1
graph_0_cpp_fused__mkl_linear_add_mul_relu_610.26%889.000us1
graph_0_cpp_fused__mkl_linear_add_mul_relu_2860.26%889.000us1
-----------------------------------------------------------------------------------
SelfCPUtimetotal:336.642ms

From the profiling table of the eager model, we can see the most time consumption ops are [aten::addmm, aten::add, aten::copy_, aten::mul, aten::clamp_min, aten::bmm]. Comparing with the inductor model profiling table, we notice an mkl::_mkl_linear entry and multiple fused kernels in the form graph_0_cpp_fused_*. They are the major optimizations that the inductor model is doing. Let us discuss them separately.

(1) Regarding mkl::_mkl_linear: You may notice the number of calls to this kernel is 362, which is exactly the same as aten::linear in the eager model profiling table. The CPU total of aten::linear is 376.888ms, while it is 231.573ms for mkl::_mkl_linear. This suggests a ~1.63x for the "linear" part. The speedup mainly comes from packing the weight tensor to block memory format and invoking cblas_sgemm_compute within the Inductor CPU backend to have a better cache behavior during GEMM computation.

(2) Regarding other memory-intensive ops: The end-to-end latency for the eager/inductor model is 802/339ms in our testing. So we can roughly infer that the speed up for the other memory-intensive ops is around 3.94x. Let’s read the generated code to understand how the inductor achieves this impressive optimization. You can find the generated code by searching cpp_fused__mkl_linear_add_mul_relu_151 in output_code.py

cpp_fused__mkl_linear_add_mul_relu_151 = async_compile.cpp('''
#include <ATen/record_function.h>
#include "/tmp/torchinductor_root/lr/clrlgu27q4ggd472umdzwsu6qcpqxcuusjxqvx2hwitjbujiiz7z.h"
extern "C" void kernel(float* in_out_ptr0,
 const float* in_ptr0,
 const float* in_ptr1,
 const float* in_ptr2,
 const float* in_ptr3)
{
 RECORD_FUNCTION("graph_0_cpp_fused__mkl_linear_add_mul_relu_151", c10::ArrayRef<c10::IValue>({}));
 #pragma omp parallel num_threads(32)
 {
 {
 #pragma omp for
 for(long i0=static_cast<long>(0L); i0<static_cast<long>(16384L); i0+=static_cast<long>(1L))
 {
 for(long i1=static_cast<long>(0L); i1<static_cast<long>(512L); i1+=static_cast<long>(8L))
 {
 auto tmp0 = at::vec::Vectorized<float>::loadu(in_ptr0 + static_cast<long>(i1 + (512L*i0)));
 auto tmp1 = at::vec::Vectorized<float>::loadu(in_ptr1 + static_cast<long>(i1));
 auto tmp3 = at::vec::Vectorized<float>::loadu(in_out_ptr0 + static_cast<long>(i1 + (512L*i0)));
 auto tmp5 = at::vec::Vectorized<float>::loadu(in_ptr2 + static_cast<long>(i1));
 auto tmp7 = at::vec::Vectorized<float>::loadu(in_ptr3 + static_cast<long>(i1));
 auto tmp2 = tmp0 + tmp1;
 auto tmp4 = tmp2 + tmp3;
 auto tmp6 = tmp4 * tmp5;
 auto tmp8 = tmp6 + tmp7;
 tmp8.store(in_out_ptr0 + static_cast<long>(i1 + (512L*i0)));
 }
 }
 }
 }
}''')

From the generated code above, we can see this kernel has done a typical Loop Fusion on [add, add, mul, add]. This is a memory-bound bottle neck preventing good performance. To get a more intuitive feeling about this optimization, we can infer the sizes and stride of the inputs and further benchmark this [add, add, mul, add] pattern.

# bench.py
deffunc(arg_0 , arg_1 , arg_2 , arg_3 , arg_4 ):
 add_0 = arg_0 + arg_1
 add_1 = add_0 + arg_2
 mul_1 = add_1 * arg_3
 add_2 = mul_1 + arg_4
 arg_2 = add_2
 return arg_2
arg_0 = torch.rand (16384, 512)
arg_1 = torch.rand (1, 512)
arg_2 = torch.zeros (16384, 512)
arg_3 = torch.rand (1, 512)
arg_4 = torch.rand (1, 512)
input = (arg_0 , arg_1 , arg_2 , arg_3 , arg_4 )
inductor_func = torch.compile (func)
with torch.no_grad ():
 inductor_func(*input)
importtimeit
NUM_ITERS=100
with torch.no_grad ():
 # warmup
 for _ in range(10):
 func(*input)
 eager_t = timeit.timeit("func(*input)", number=NUM_ITERS, globals=globals())
with torch.no_grad ():
 # warmup
 for _ in range(10):
 inductor_func(*input)
 inductor_t = timeit.timeit("inductor_func(*input)", number=NUM_ITERS, globals=globals())
# print(f"eager use: {eager_t * 1000 / NUM_ITERS} ms/iter")
# print(f"inductor use: {inductor_t * 1000 / NUM_ITERS} ms/iter")
# print(f"speed up ratio: {eager_t / inductor_t}")

Output:

eageruse:5.780875144992024ms/iter
inductoruse:0.9588955780491233ms/iter
speedupratio:6.0286805751604735

This is just an example. The profiling table shows all element-wise op are fused within the inductor automatically in this model. You can read more kernels in output_code.py

Conclusion#

The document gives an in-depth tutorial for the Inductor CPU backend.

With motivating examples, we walk through the process of debugging and profiling. The main idea is to narrow down the problem.

We demonstrate step by step the way to delve deeper the issue and find the root cause of failures, with the help of debugging logging and the tool Minifier. Firstly determine which component the failure occurs in and then try to generate the smallest snippet of code that can reproduce the failure.

When the performance with Inductor is better than that of eager mode, we provide a solid analytical method for performance profiling. We show how to find the time-consuming hotspot with PyTorch Profiler and figure out the operator-level or kernel-level reason to explain the phenomenon.

Total running time of the script: (10 minutes 58.871 seconds)