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2026-07-13 13:18:33 +08:00

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C++

// Copyright (c) Microsoft Corporation.
// SPDX-License-Identifier: Apache-2.0
// DeepSpeed Team
#include "z3.h"
#include "deepcompile.h"
#include <ATen/native/cuda/Resize.h>
namespace dc {
const size_t TIMEOUT_SYMMETRIC_MEMORY_BARRIER = 60000;
class Z3CustomOpExecutor : public CustomOpExecutor {
public:
Z3CustomOpExecutor(c10::intrusive_ptr<c10d::ProcessGroup> process_group,
std::shared_ptr<DSParamRegistry> param_registry,
std::shared_ptr<DoubleBufferedReduceBucket> reduce_buckets,
std::vector<long> ds_ids,
ncclComm_t nccl_comm,
at::cuda::CUDAStream ag_stream,
at::cuda::CUDAStream rs_stream,
at::cuda::CUDAStream copy_stream,
at::cuda::CUDAStream offload_stream,
at::cuda::CUDAStream reload_stream,
bool pre_div_reduce)
: CustomOpExecutor(process_group,
param_registry,
reduce_buckets,
ds_ids,
nccl_comm,
rs_stream,
copy_stream,
pre_div_reduce),
ag_stream_(ag_stream),
offload_stream_(offload_stream),
reload_stream_(reload_stream)
{
for (long ds_id : ds_ids_) {
ag_comm_done_events_[ds_id] =
std::make_shared<at::cuda::CUDAEvent>(cudaEventDisableTiming);
ag_comp_done_events_[ds_id] =
std::make_shared<at::cuda::CUDAEvent>(cudaEventDisableTiming);
param_use_count_[ds_id] = 0;
}
}
~Z3CustomOpExecutor() {}
void endBackward() override
{
CustomOpExecutor::endBackward();
if (param_updated_) {
for (auto& it : has_acc_grad_) {
it.second = false;
param_registry_->setValid(it.first, false);
}
}
for (auto& it : reload_buffers_) {
it.second.record_stream(at::cuda::getCurrentCUDAStream());
}
reload_buffers_.clear();
}
void launchAllGather(at::Tensor output_buf,
long ds_id,
c10::intrusive_ptr<c10d::symmetric_memory::SymmetricMemory> symm_mem)
{
const DSParam& param = param_registry_->getParam(ds_id);
at::Tensor ds_tensor = param.getDSTensor();
if (ds_tensor.scalar_type() != output_buf.scalar_type()) {
at::cuda::CUDAStreamGuard guard(ag_stream_);
ds_tensor = ds_tensor.to(output_buf.scalar_type(), true, true);
}
if (symm_mem == nullptr) {
// Fast path: assume uniform shard sizes (ZeRO-3 partitions are padded to uniform size)
const int world_size = process_group_->getSize();
const int64_t shard_elems = ds_tensor.numel();
// Perform all-gather directly into the pre-allocated padded output buffer
// NCCL requires contiguous storage; use .contiguous() explicitly
ncclResult_t result = ncclAllGather(ds_tensor.contiguous().data_ptr(),
output_buf.data_ptr(),
shard_elems,
get_nccl_data_type(ds_tensor.scalar_type()),
nccl_comm_,
ag_stream_);
if (result != ncclSuccess) { throw std::runtime_error("NCCL AllGather failed"); }
} else {
at::cuda::CUDAStreamGuard guard(ag_stream_);
int world_size = process_group_->getSize();
int rank = process_group_->getRank();
at::Tensor local_buf =
symm_mem->get_buffer(rank, ds_tensor.sizes(), ds_tensor.scalar_type(), 0);
local_buf.copy_(ds_tensor, true);
symm_mem->barrier(0, TIMEOUT_SYMMETRIC_MEMORY_BARRIER);
auto chunks = output_buf.flatten().chunk(world_size);
for (int step = 0; step < world_size; step++) {
int remote_rank = (rank - step + world_size) % world_size;
auto src_buf = symm_mem->get_buffer(
remote_rank, ds_tensor.sizes(), ds_tensor.scalar_type(), 0);
chunks[remote_rank].copy_(src_buf.flatten(), true);
}
symm_mem->barrier(0, TIMEOUT_SYMMETRIC_MEMORY_BARRIER);
}
param_registry_->registerGatheredParam(ds_id, output_buf);
param_registry_->setValid(ds_id, true);
}
at::Tensor allgatherParam(long ds_id,
std::optional<at::ScalarType> dtype,
c10::intrusive_ptr<c10d::symmetric_memory::SymmetricMemory> symm_mem)
{
const DSParam& param = param_registry_->getParam(ds_id);
const at::Tensor& ds_tensor = param.getDSTensor();
const int world_size = process_group_->getSize();
const int64_t true_numel = static_cast<int64_t>(productDim(param.getShape()));
const int64_t padded_per_rank = (true_numel + world_size - 1) / world_size;
const int64_t padded_numel = static_cast<int64_t>(world_size) * padded_per_rank;
at::ScalarType target_dtype = dtype ? dtype.value() : ds_tensor.scalar_type();
if (param_registry_->isValid(ds_id)) {
// Return a view sliced to the true size with the original shape
//
// Persistent params are gathered in their original dtype which may
// be different from the requested.
auto base = param_registry_->getGatheredParam(ds_id);
return base.flatten()
.to(target_dtype)
.index({torch::indexing::Slice(0, true_numel)})
.view(param.getShape());
}
at::Tensor output_buf;
if (param_registry_->hasGatheredParam(ds_id)) {
auto existing = param_registry_->getGatheredParam(ds_id);
if (existing.defined() && existing.numel() == padded_numel) { output_buf = existing; }
}
if (!output_buf.defined()) {
at::cuda::CUDAStreamGuard guard(ag_stream_);
output_buf = torch::empty({padded_numel}, ds_tensor.options().dtype(target_dtype));
}
assert(hasKey(ag_comp_done_events_, ds_id));
ag_comp_done_events_[ds_id]->record();
ag_comp_done_events_[ds_id]->block(ag_stream_);
launchAllGather(output_buf, ds_id, symm_mem);
ag_comm_done_events_[ds_id]->record(ag_stream_);
// Return a view of the gathered padded buffer matching the true param shape
return output_buf.flatten()
.index({torch::indexing::Slice(0, true_numel)})
.view(param.getShape());
}
void prefetchParamsFused(const std::vector<long>& ds_ids,
const std::optional<std::vector<at::ScalarType>> dtypes,
c10::intrusive_ptr<c10d::symmetric_memory::SymmetricMemory> symm_mem)
{
std::vector<std::tuple<long, std::optional<at::ScalarType>>> invalid_params;
for (int i = 0; i < ds_ids.size(); i++) {
if (!param_registry_->isValid(ds_ids[i])) {
auto dtype = dtypes ? dtypes.value()[i] : std::optional<at::ScalarType>();
invalid_params.push_back(std::make_tuple(ds_ids[i], dtype));
}
}
std::unordered_map<long, at::Tensor> output_bufs;
for (const auto& [ds_id, dtype] : invalid_params) {
const DSParam& param = param_registry_->getParam(ds_id);
const at::Tensor& ds_tensor = param.getDSTensor();
const int world_size = process_group_->getSize();
const int64_t shard_elems = ds_tensor.numel();
const int64_t padded_numel = static_cast<int64_t>(world_size) * shard_elems;
if (param_registry_->hasGatheredParam(ds_id)) {
auto existing = param_registry_->getGatheredParam(ds_id);
if (existing.defined() && existing.numel() == padded_numel) {
output_bufs[ds_id] = existing;
continue;
}
}
auto target_dtype = dtype ? dtype.value() : ds_tensor.scalar_type();
output_bufs[ds_id] =
torch::empty({padded_numel}, ds_tensor.options().dtype(target_dtype));
}
for (const auto& [ds_id, _] : invalid_params) {
ag_comp_done_events_[ds_id]->record();
ag_comp_done_events_[ds_id]->block(ag_stream_);
}
ncclGroupStart();
for (const auto& [ds_id, _] : invalid_params) {
assert(hasKey(output_bufs, ds_id));
launchAllGather(output_bufs.at(ds_id), ds_id, symm_mem);
}
ncclGroupEnd();
for (const auto& [ds_id, _] : invalid_params) {
ag_comm_done_events_[ds_id]->record(ag_stream_);
}
}
void releaseParam(long ds_id, long n_users)
{
const DSParam& param = param_registry_->getParam(ds_id);
assert(hasKey(param_use_count_, ds_id));
if (param_use_count_[ds_id] == 0) { param_use_count_[ds_id] = n_users; }
param_use_count_[ds_id]--;
if (param_use_count_[ds_id] == 0 && !param.isPersistent()) {
at::Tensor gathered_param = param_registry_->getGatheredParam(ds_id);
if (gathered_param.defined()) { // gathered param is undefined while profiling
auto storage = gathered_param.storage();
if (storage.nbytes() > 0) {
// Required so the caching allocator defers reuse for consumer-stream kernels
// queued behind wait_allgather.
gathered_param.record_stream(at::cuda::getCurrentCUDAStream());
at::native::resize_bytes_cuda(storage.unsafeGetStorageImpl(), 0);
}
const auto options = gathered_param.options();
at::Tensor empty_buffer = torch::empty({0}, options);
gathered_param.set_data(empty_buffer);
}
param_registry_->unregisterGatheredParam(ds_id);
}
}
at::Tensor waitAllgather(at::Tensor v, long ds_id)
{
assert(hasKey(ag_comm_done_events_, ds_id));
ag_comm_done_events_[ds_id]->block(at::cuda::getCurrentCUDAStream());
return v;
}
void flushReduceBucket(at::ScalarType scalar_type) override
{
if (!hasKey(reduce_tasks_, scalar_type)) { return; }
blockCopyEvents(scalar_type);
// Calculate temporary buffer size for accumulated gradients or
// communication/storage dtype mismatches.
int64_t tmp_recv_numel = 0;
for (const ReduceTask& t : reduce_tasks_.at(scalar_type)) {
auto recv_buf = param_registry_->getParam(t.getDSId()).getGradBuffer();
int64_t recv_numel = recv_buf.numel();
bool use_tmp_recv = recv_numel > 0 && (has_acc_grad_.at(t.getDSId()) ||
recv_buf.scalar_type() != scalar_type);
if (use_tmp_recv) { tmp_recv_numel += recv_numel; }
}
// Allocate temporary buffer if needed
at::Tensor tmp_recv_buf = at::Tensor();
if (tmp_recv_numel > 0) {
at::cuda::CUDAStreamGuard guard(rs_stream_);
tmp_recv_buf = torch::empty({tmp_recv_numel},
at::TensorOptions().dtype(scalar_type).device(at::kCUDA));
}
applyPreDivision(scalar_type);
// NCCL ReduceScatter operation
ncclGroupStart();
int64_t offset = 0;
for (const ReduceTask& t : reduce_tasks_.at(scalar_type)) {
auto recv_buf = param_registry_->getParam(t.getDSId()).getGradBuffer();
bool acc_grad = has_acc_grad_.at(t.getDSId());
int64_t recv_numel = recv_buf.numel();
bool use_tmp_recv =
recv_numel > 0 && (acc_grad || recv_buf.scalar_type() != scalar_type);
if (use_tmp_recv) {
recv_buf =
tmp_recv_buf.index({torch::indexing::Slice(offset, offset + recv_numel)});
}
ncclResult_t result = ncclReduceScatter(t.getSendBuf().data_ptr(),
recv_buf.data_ptr(),
recv_numel,
get_nccl_data_type(scalar_type),
getReductionOp(),
nccl_comm_,
rs_stream_);
if (result != ncclSuccess) { throw std::runtime_error("NCCL ReduceScatter failed"); }
if (use_tmp_recv) { offset += recv_numel; }
}
ncclGroupEnd();
// Move temporary receive results into the ZeRO grad buffer.
{
at::cuda::CUDAStreamGuard guard(rs_stream_);
int64_t offset = 0;
for (const ReduceTask& t : reduce_tasks_.at(scalar_type)) {
auto recv_buf = param_registry_->getParam(t.getDSId()).getGradBuffer();
bool acc_grad = has_acc_grad_.at(t.getDSId());
int64_t recv_numel = recv_buf.numel();
bool use_tmp_recv =
recv_numel > 0 && (acc_grad || recv_buf.scalar_type() != scalar_type);
if (use_tmp_recv) {
auto reduced_slice =
tmp_recv_buf.index({torch::indexing::Slice(offset, offset + recv_numel)});
if (reduced_slice.scalar_type() != recv_buf.scalar_type()) {
reduced_slice = reduced_slice.to(recv_buf.scalar_type());
}
if (acc_grad) {
recv_buf.add_(reduced_slice);
} else {
recv_buf.copy_(reduced_slice, true);
}
offset += recv_numel;
}
has_acc_grad_[t.getDSId()] = true;
}
}
performCleanup(scalar_type);
// Record stream for temporary buffer to prevent early deallocation
if (tmp_recv_numel > 0) { tmp_recv_buf.record_stream(rs_stream_); }
}
at::Tensor offloadTensor(at::Tensor tensor, long id)
{
if (!hasKey(offload_events_, id)) {
offload_events_[id] = std::make_shared<at::cuda::CUDAEvent>(cudaEventDisableTiming);
offload_comp_done_events_[id] =
std::make_shared<at::cuda::CUDAEvent>(cudaEventDisableTiming);
const auto options = at::TensorOptions().pinned_memory(true).device(torch::kCPU);
offload_buffers_[id] = at::empty_like(tensor, options);
}
offload_comp_done_events_[id]->record();
offload_comp_done_events_[id]->block(offload_stream_);
{
at::cuda::CUDAStreamGuard guard(offload_stream_);
offload_buffers_.at(id).copy_(tensor, true);
}
tensor.record_stream(offload_stream_);
offload_events_[id]->record(offload_stream_);
assert(hasKey(offload_buffers_, id));
return offload_buffers_.at(id);
}
at::Tensor reloadTensor(at::Tensor tensor, long id)
{
if (!hasKey(reload_events_, id)) {
reload_events_[id] = std::make_shared<at::cuda::CUDAEvent>(cudaEventDisableTiming);
}
assert(hasKey(offload_buffers_, id));
offload_events_[id]->block(reload_stream_);
at::Tensor ten;
{
at::cuda::CUDAStreamGuard guard(reload_stream_);
assert(hasKey(offload_buffers_, id));
at::Tensor buf = offload_buffers_.at(id);
const auto options = at::TensorOptions().device(torch::kCUDA);
ten = at::empty_like(buf, options);
ten.copy_(buf, true);
reload_buffers_[id] = ten;
}
reload_events_[id]->record(reload_stream_);
return ten;
}
at::Tensor waitOffload(at::Tensor tensor, long id)
{
assert(hasKey(offload_events_, id));
offload_events_[id]->block(at::cuda::getCurrentCUDAStream());
assert(hasKey(offload_buffers_, id));
return offload_buffers_.at(id);
}
at::Tensor waitReload(at::Tensor tensor, long id)
{
assert(hasKey(reload_events_, id));
reload_events_[id]->block(at::cuda::getCurrentCUDAStream());
assert(hasKey(reload_buffers_, id));
auto ten = reload_buffers_.at(id);
// We can't release here because the tensor is still being used
// We will need "freeReloadedTensor" after the last user of the tensor to call
// ".record_stream". As it is a bit complicated, we clear the buffer and do at the end of
// the backward pass for now. reload_buffers_.erase(id);
return ten;
}
void offloadParameter(at::Tensor tensor, long ds_id) { param_registry_->offload(ds_id); }
void reloadParameter(at::Tensor tensor, long ds_id) { param_registry_->reload(ds_id); }
bool hasReloadBuffer(long id) { return hasKey(reload_buffers_, id); }
bool hasParam(long ds_id) const { return hasKey(has_acc_grad_, ds_id); }
private:
at::cuda::CUDAStream ag_stream_;
at::cuda::CUDAStream offload_stream_;
at::cuda::CUDAStream reload_stream_;
std::unordered_map<long, std::shared_ptr<at::cuda::CUDAEvent>> ag_comp_done_events_;
std::unordered_map<long, std::shared_ptr<at::cuda::CUDAEvent>> ag_comm_done_events_;
std::unordered_map<long, std::shared_ptr<at::cuda::CUDAEvent>> offload_events_;
std::unordered_map<long, std::shared_ptr<at::cuda::CUDAEvent>> offload_comp_done_events_;
std::unordered_map<long, std::shared_ptr<at::cuda::CUDAEvent>> reload_events_;
std::unordered_map<long, at::Tensor> offload_buffers_;
std::unordered_map<long, at::Tensor> reload_buffers_;
std::unordered_map<long, long> param_use_count_;
};
namespace {
at::cuda::CUDAStream get_ag_stream()
{
static at::cuda::CUDAStream ag_stream = at::cuda::getStreamFromPool(true);
return ag_stream;
}
at::cuda::CUDAStream get_rs_stream()
{
static at::cuda::CUDAStream rs_stream = at::cuda::getStreamFromPool(true);
return rs_stream;
}
at::cuda::CUDAStream get_copy_stream()
{
static at::cuda::CUDAStream copy_stream = at::cuda::getStreamFromPool(true);
return copy_stream;
}
at::cuda::CUDAStream get_offload_stream()
{
static at::cuda::CUDAStream offload_stream = at::cuda::getStreamFromPool(true);
return offload_stream;
}
at::cuda::CUDAStream get_reload_stream()
{
static at::cuda::CUDAStream reload_stream = at::cuda::getStreamFromPool(true);
return reload_stream;
}
} // namespace
void register_graph_z3(long graph_id, const std::vector<long>& ds_ids)
{
executors[graph_id] = std::make_shared<Z3CustomOpExecutor>(process_group,
param_registry,
reduce_buckets,
ds_ids,
nccl_comm,
get_ag_stream(),
get_rs_stream(),
get_copy_stream(),
get_offload_stream(),
get_reload_stream(),
pre_div_reduce);
}
void register_z3_param(long ds_id,
const std::vector<int64_t>& ds_shape,
at::Tensor ds_tensor,
at::Tensor grad_buffer,
bool persistent,
std::optional<at::ScalarType> expected_grad_dtype)
{
param_registry->registerParam(
ds_id, ds_shape, ds_tensor, grad_buffer, true, 0, persistent, expected_grad_dtype);
if (persistent) { param_registry->registerGatheredParam(ds_id, ds_tensor); }
// Validate that padded shard sizes are uniform across ranks at registration time
// DeepSpeed pads parameters to ensure even division, so we check the padded size
// which should be uniform across all ranks for correct allgather behavior
const int64_t local_count = ds_tensor.numel();
const int world_size = process_group->getSize();
// Calculate padded size (aligned to world_size)
// Use ds_shape to compute the full (unpartitioned) parameter size
int64_t total_numel = 1;
for (const auto dim : ds_shape) { total_numel *= dim; }
const int64_t padded_per_rank = (total_numel + world_size - 1) / world_size;
// For verification: all ranks should have the same padded size
auto count_options = at::TensorOptions().dtype(at::kLong).device(at::kCUDA);
at::Tensor local_padded_tensor = torch::tensor({padded_per_rank}, count_options);
std::vector<at::Tensor> all_padded_counts(world_size);
for (int i = 0; i < world_size; ++i) {
all_padded_counts[i] = torch::empty_like(local_padded_tensor);
}
// Build lvalue buffers for output and input as required by ProcessGroup::allgather
// The first argument must be a single-element vector containing a vector of WORLD_SIZE tensors
std::vector<std::vector<at::Tensor>> output_tensors(1);
output_tensors[0] = all_padded_counts;
std::vector<at::Tensor> input_tensors = {local_padded_tensor};
process_group->allgather(output_tensors, input_tensors)->wait();
// Verify all ranks agree on the padded size
for (int i = 0; i < world_size; ++i) {
int64_t padded_count = all_padded_counts[i].to(torch::kCPU).item<int64_t>();
if (padded_count != padded_per_rank) {
throw std::runtime_error(
"ZeRO-3 registration error: inconsistent padded shard sizes across ranks. "
"This is an internal error - please report this issue.");
}
}
}
at::Tensor allgather_param(at::Tensor param_tensor,
long graph_id,
long ds_id,
std::optional<at::ScalarType> dtype)
{
auto executor = getExecutor<Z3CustomOpExecutor>(graph_id, executors);
if (sync_before_allgather) { c10::cuda::device_synchronize(); }
auto ret = executor->allgatherParam(ds_id, dtype, symm_mem);
if (sync_after_allgather) { c10::cuda::device_synchronize(); }
return ret;
}
void set_persistent(long ds_id)
{
param_registry->setPersistent(ds_id, true);
// Allocate buffer here
// Memory fragmentation will be more severe if we allocate in forward/backward
for (auto& it : executors) {
if (it.second->hasParam(ds_id)) {
auto executor = getExecutor<Z3CustomOpExecutor>(it.first, executors);
auto dtype = param_registry->getParam(ds_id).getDtype();
executor->allgatherParam(ds_id, dtype, symm_mem);
}
}
}
void prefetch_params_fused(long graph_id,
const std::vector<at::Tensor>& params,
const std::vector<long>& ds_ids,
const std::optional<std::vector<at::ScalarType>>& dtypes)
{
auto executor = getExecutor<Z3CustomOpExecutor>(graph_id, executors);
executor->prefetchParamsFused(ds_ids, dtypes, symm_mem);
}
void prefetch_params_fused_meta(long graph_id,
const std::vector<at::Tensor>& params,
const std::vector<long>& ds_ids,
const std::optional<std::vector<at::ScalarType>>& dtypes)
{
}
// for profiling
void invalidate_gathered_param(long ds_id)
{
const DSParam& param = param_registry->getParam(ds_id);
if (param.isPersistent()) { return; }
param_registry->unregisterGatheredParam(ds_id);
param_registry->registerGatheredParam(ds_id, at::Tensor());
}
void clear_all_gathered_params()
{
for (const auto& it : param_registry->getParams()) {
long ds_id = it.first;
const DSParam& param = param_registry->getParam(ds_id);
if (param.isPersistent()) { continue; }
if (param_registry->hasGatheredParam(ds_id)) {
param_registry->unregisterGatheredParam(ds_id);
}
}
}
at::Tensor allgather_param_meta(at::Tensor param_tensor,
long graph_id,
long ds_id,
std::optional<at::ScalarType> dtype)
{
const DSParam& param = param_registry->getParam(ds_id);
auto options = param.getDSTensor().options().device(c10::kMeta);
at::Tensor output_buf = torch::empty(param.getShape(), options.dtype(dtype));
return output_buf;
}
at::Tensor release_param(at::Tensor dummy, long graph_id, long ds_id, long n_users)
{
auto executor = getExecutor<Z3CustomOpExecutor>(graph_id, executors);
executor->releaseParam(ds_id, n_users);
return dummy;
}
at::Tensor release_param_meta(at::Tensor dummy, long graph_id, long ds_id, long n_users)
{
return dummy;
}
at::Tensor wait_allgather(at::Tensor v, long graph_id, long ds_id)
{
auto executor = getExecutor<Z3CustomOpExecutor>(graph_id, executors);
executor->waitAllgather(v, ds_id);
return v;
}
at::Tensor wait_allgather_meta(at::Tensor v, long graph_id, long ds_id) { return v; }
at::Tensor offload_tensor(at::Tensor tensor, long graph_id, long id)
{
auto executor = getExecutor<Z3CustomOpExecutor>(graph_id, executors);
return executor->offloadTensor(tensor, id);
}
at::Tensor reload_tensor(at::Tensor tensor, long graph_id, long id)
{
auto executor = getExecutor<Z3CustomOpExecutor>(graph_id, executors);
return executor->reloadTensor(tensor, id);
}
at::Tensor wait_offload(at::Tensor tensor, long graph_id, long id)
{
auto executor = getExecutor<Z3CustomOpExecutor>(graph_id, executors);
return executor->waitOffload(tensor, id);
}
at::Tensor wait_reload(at::Tensor tensor, long graph_id, long id)
{
auto executor = getExecutor<Z3CustomOpExecutor>(graph_id, executors);
if (profile && !executor->hasReloadBuffer(id)) { return tensor; }
return executor->waitReload(tensor, id);
}
at::Tensor test_call(at::Tensor a)
{
std::cout << "test_call" << std::endl;
return a;
}
void reload_parameter(at::Tensor tensor, long graph_id, long ds_id)
{
auto executor = getExecutor<Z3CustomOpExecutor>(graph_id, executors);
executor->reloadParameter(tensor, ds_id);
}
void offload_parameter(at::Tensor tensor, long graph_id, long ds_id)
{
auto executor = getExecutor<Z3CustomOpExecutor>(graph_id, executors);
executor->offloadParameter(tensor, ds_id);
}
void reload_parameter_meta(at::Tensor param_tensor, long graph_id, long ds_id) {}
void offload_parameter_meta(at::Tensor tensor, long graph_id, long ds_id) {}
} // namespace dc