feat(train): align native SmolVLA recipe

This commit is contained in:
Logic
2026-07-31 10:11:04 +08:00
parent 2ac926d427
commit ebac9860fe
7 changed files with 277 additions and 29 deletions
+119 -20
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@@ -71,6 +71,19 @@ from hydra.utils import instantiate
log = logging.getLogger(__name__) log = logging.getLogger(__name__)
SMOLVLA_NATIVE_TRAINING_PRESET = {
'lr': 1e-4,
'betas': (0.9, 0.95),
'eps': 1e-8,
'weight_decay': 1e-10,
'grad_clip': 10.0,
'warmup_steps': 1000,
'scheduler_type': 'cosine',
'scheduler_decay_steps': 30000,
'scheduler_decay_lr': 2.5e-6,
}
# 注册列表长度解析器(用于配置中如 ${len:${data.camera_names}} # 注册列表长度解析器(用于配置中如 ${len:${data.camera_names}}
if not OmegaConf.has_resolver("len"): if not OmegaConf.has_resolver("len"):
OmegaConf.register_new_resolver("len", lambda x: len(x)) OmegaConf.register_new_resolver("len", lambda x: len(x))
@@ -154,7 +167,7 @@ def get_lr_schedule_with_warmup(optimizer, warmup_steps, max_steps, scheduler_ty
Args: Args:
optimizer: PyTorch 优化器 optimizer: PyTorch 优化器
warmup_steps: 预热步数 warmup_steps: 预热步数
max_steps: 总训练步数 max_steps: 余弦衰减步数
scheduler_type: 预热后的调度器类型 ('cosine''constant') scheduler_type: 预热后的调度器类型 ('cosine''constant')
min_lr: 最小学习率(用于余弦衰减) min_lr: 最小学习率(用于余弦衰减)
@@ -167,16 +180,24 @@ def get_lr_schedule_with_warmup(optimizer, warmup_steps, max_steps, scheduler_ty
min_lr_ratio = min_lr / base_lr if base_lr > 0 else 0.0 min_lr_ratio = min_lr / base_lr if base_lr > 0 else 0.0
def lr_lambda(step): def lr_lambda(step):
# 预热阶段:从 0 线性增加到 1 # LeRobot CosineDecayWithWarmupSchedulerConfig 的线性预热:
# 从一个很小的非零 LR 开始,避免首步完全为 0。
if step < warmup_steps: if step < warmup_steps:
return float(step) / float(max(1, warmup_steps)) if step <= 0:
return 1.0 / float(max(1, warmup_steps + 1))
frac = 1.0 - float(step) / float(max(1, warmup_steps))
return (1.0 / float(max(1, warmup_steps + 1)) - 1.0) * frac + 1.0
# 预热后阶段 # 预热后阶段
if scheduler_type == 'cosine': if scheduler_type == 'cosine':
# 从 1 到 min_lr_ratio 的余弦退火 # 与 LeRobot SmolVLA/PI0 的 CosineDecayWithWarmupSchedulerConfig 对齐:
progress = float(step - warmup_steps) / float(max(1, max_steps - warmup_steps)) # 1) 余弦衰减步数是固定的 num_decay_stepsSmolVLA 默认 30k),
cosine_decay = 0.5 * (1.0 + math.cos(math.pi * progress)) # 2) 超过 decay_steps 后 clamp 到 decay_lr,不能继续 cos() 进入下一周期;
return max(min_lr_ratio, cosine_decay) # 否则 150k 训练会显示成“正弦波”式反复升降。
decay_steps = max(1, int(max_steps))
clamped_step = min(max(int(step), 0), decay_steps)
cosine_decay = 0.5 * (1.0 + math.cos(math.pi * clamped_step / decay_steps))
return (1.0 - min_lr_ratio) * cosine_decay + min_lr_ratio
else: else:
# 恒定学习率 # 恒定学习率
return 1.0 return 1.0
@@ -184,6 +205,46 @@ def get_lr_schedule_with_warmup(optimizer, warmup_steps, max_steps, scheduler_ty
return LambdaLR(optimizer, lr_lambda) return LambdaLR(optimizer, lr_lambda)
def _is_smolvla_native_agent_config(agent_cfg) -> bool:
target = str(agent_cfg.get('_target_', '')) if hasattr(agent_cfg, 'get') else ''
return target == 'roboimi.vla.agent_smolvla_native.SmolVLANativeAgent'
def resolve_training_recipe(cfg):
"""Return optimizer/scheduler knobs, using LeRobot SmolVLA defaults for smolvla_native."""
if _is_smolvla_native_agent_config(cfg.agent):
preset = SMOLVLA_NATIVE_TRAINING_PRESET
scheduler_steps = int(cfg.train.get(
'scheduler_decay_steps',
cfg.train.get('lr_scheduler_steps', cfg.train.max_steps),
))
return {
'lr': float(preset['lr']),
'betas': tuple(preset['betas']),
'eps': float(preset['eps']),
'weight_decay': float(preset['weight_decay']),
'grad_clip': float(preset['grad_clip']),
'warmup_steps': int(preset['warmup_steps']),
'scheduler_type': str(preset['scheduler_type']),
'scheduler_steps': scheduler_steps,
'min_lr': float(preset['scheduler_decay_lr']),
'preset_name': 'lerobot_smolvla',
}
return {
'lr': float(cfg.train.lr),
'betas': (0.9, 0.999),
'eps': 1e-8,
'weight_decay': float(cfg.train.get('weight_decay', 1e-5)),
'grad_clip': float(cfg.train.get('grad_clip', 1.0)),
'warmup_steps': int(cfg.train.get('warmup_steps', 500)),
'scheduler_type': str(cfg.train.get('scheduler_type', 'cosine')),
'scheduler_steps': int(cfg.train.max_steps),
'min_lr': float(cfg.train.get('min_lr', 1e-6)),
'preset_name': 'config',
}
def _instantiate_dataset(cfg, dataset_image_resize_shape, episode_indices=None): def _instantiate_dataset(cfg, dataset_image_resize_shape, episode_indices=None):
kwargs = {'image_resize_shape': dataset_image_resize_shape} kwargs = {'image_resize_shape': dataset_image_resize_shape}
if episode_indices is not None: if episode_indices is not None:
@@ -256,7 +317,7 @@ def build_train_val_datasets(cfg, dataset_image_resize_shape):
return dataset, train_dataset, val_dataset, None return dataset, train_dataset, val_dataset, None
def build_training_optimizer(agent, lr, weight_decay): def build_training_optimizer(agent, lr, weight_decay, betas=(0.9, 0.999), eps=1e-8):
"""为训练脚本构建优化器,优先复用任意 head 自带的参数分组。""" """为训练脚本构建优化器,优先复用任意 head 自带的参数分组。"""
trainable_params = [param for param in agent.parameters() if param.requires_grad] trainable_params = [param for param in agent.parameters() if param.requires_grad]
noise_pred_net = getattr(agent, 'noise_pred_net', None) noise_pred_net = getattr(agent, 'noise_pred_net', None)
@@ -264,7 +325,7 @@ def build_training_optimizer(agent, lr, weight_decay):
use_head_groups = callable(get_optim_groups) use_head_groups = callable(get_optim_groups)
if not use_head_groups: if not use_head_groups:
return AdamW(trainable_params, lr=lr, weight_decay=weight_decay) return AdamW(trainable_params, lr=lr, weight_decay=weight_decay, betas=betas, eps=eps)
head_groups = [] head_groups = []
grouped_param_ids = set() grouped_param_ids = set()
@@ -306,7 +367,7 @@ def build_training_optimizer(agent, lr, weight_decay):
if grouped_param_ids != all_trainable_param_ids: if grouped_param_ids != all_trainable_param_ids:
raise ValueError('Optimizer parameter groups must include each trainable parameter exactly once') raise ValueError('Optimizer parameter groups must include each trainable parameter exactly once')
return AdamW(optim_groups, lr=lr, weight_decay=weight_decay) return AdamW(optim_groups, lr=lr, weight_decay=weight_decay, betas=betas, eps=eps)
def _init_swanlab(cfg): def _init_swanlab(cfg):
@@ -353,7 +414,12 @@ def _init_swanlab(cfg):
init_kwargs['experiment_name'] = run_name init_kwargs['experiment_name'] = run_name
try: try:
swanlab.init(**init_kwargs) run = swanlab.init(**init_kwargs)
if run is not None:
try:
setattr(swanlab, "_roboimi_swanlab_run", run)
except Exception:
pass
except Exception as exc: except Exception as exc:
raise RuntimeError( raise RuntimeError(
f"SwanLab logging is enabled, but SwanLab init/login failed: {exc}" f"SwanLab logging is enabled, but SwanLab init/login failed: {exc}"
@@ -362,6 +428,24 @@ def _init_swanlab(cfg):
return swanlab return swanlab
def _log_swanlab_init_details(swanlab_module):
if swanlab_module is None:
return
run = getattr(swanlab_module, "_roboimi_swanlab_run", None)
if run is None:
run = getattr(swanlab_module, "run", None)
if run is None:
return
url = getattr(run, "url", None)
swanlog_dir = getattr(run, "swanlog_dir", None)
log.info(
"🦢 SwanLab initialized%s%s",
f" | url={url}" if url else "",
f" | swanlog_dir={swanlog_dir}" if swanlog_dir else "",
)
def _log_to_swanlab(swanlab_module, payload, step=None): def _log_to_swanlab(swanlab_module, payload, step=None):
if swanlab_module is None: if swanlab_module is None:
return return
@@ -440,6 +524,7 @@ def _run_training(cfg: DictConfig):
log.info(f"🚀 开始 VLA 训练 (设备: {cfg.train.device})") log.info(f"🚀 开始 VLA 训练 (设备: {cfg.train.device})")
_configure_cuda_runtime(cfg) _configure_cuda_runtime(cfg)
swanlab_module = _init_swanlab(cfg) swanlab_module = _init_swanlab(cfg)
_log_swanlab_init_details(swanlab_module)
try: try:
action_mse_val_freq_epochs = int(cfg.train.get('action_mse_val_freq_epochs', 0) or 0) action_mse_val_freq_epochs = int(cfg.train.get('action_mse_val_freq_epochs', 0) or 0)
explicit_val_episode_indices = cfg.train.get('val_episode_indices', None) explicit_val_episode_indices = cfg.train.get('val_episode_indices', None)
@@ -609,21 +694,35 @@ def _run_training(cfg: DictConfig):
# ========================================================================= # =========================================================================
# 4. 设置优化器与学习率调度器 # 4. 设置优化器与学习率调度器
# ========================================================================= # =========================================================================
weight_decay = float(cfg.train.get('weight_decay', 1e-5)) training_recipe = resolve_training_recipe(cfg)
grad_clip = float(cfg.train.get('grad_clip', 1.0)) weight_decay = training_recipe['weight_decay']
grad_clip = training_recipe['grad_clip']
optimizer = build_training_optimizer(agent, lr=cfg.train.lr, weight_decay=weight_decay) train_lr = training_recipe['lr']
log.info(f"🔧 优化器: AdamW (学习率={cfg.train.lr}, weight_decay={weight_decay})") optimizer = build_training_optimizer(
agent,
lr=train_lr,
weight_decay=weight_decay,
betas=training_recipe['betas'],
eps=training_recipe['eps'],
)
log.info(
"🔧 优化器: AdamW (preset=%s, 学习率=%s, betas=%s, eps=%s, weight_decay=%s)",
training_recipe['preset_name'],
train_lr,
training_recipe['betas'],
training_recipe['eps'],
weight_decay,
)
# 设置带预热的学習率调度器 # 设置带预热的学習率调度器
warmup_steps = int(cfg.train.get('warmup_steps', 500)) warmup_steps = training_recipe['warmup_steps']
scheduler_type = cfg.train.get('scheduler_type', 'cosine') scheduler_type = training_recipe['scheduler_type']
min_lr = float(cfg.train.get('min_lr', 1e-6)) min_lr = training_recipe['min_lr']
scheduler = get_lr_schedule_with_warmup( scheduler = get_lr_schedule_with_warmup(
optimizer, optimizer,
warmup_steps=warmup_steps, warmup_steps=warmup_steps,
max_steps=cfg.train.max_steps, max_steps=training_recipe['scheduler_steps'],
scheduler_type=scheduler_type, scheduler_type=scheduler_type,
min_lr=min_lr min_lr=min_lr
) )
+4 -1
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@@ -192,7 +192,10 @@ class SmolVLANativeAgent(nn.Module):
return [fallback_or(item) for item in task] return [fallback_or(item) for item in task]
def _tokenize_tasks(self, task, batch_size: int, device: torch.device) -> dict: def _tokenize_tasks(self, task, batch_size: int, device: torch.device) -> dict:
tasks = [str(text) + '\n' for text in self._resolve_task(task, batch_size)] tasks = []
for text in self._resolve_task(task, batch_size):
text = str(text)
tasks.append(text if text.endswith('\n') else f'{text}\n')
old_padding_side = getattr(self.tokenizer, 'padding_side', None) old_padding_side = getattr(self.tokenizer, 'padding_side', None)
if old_padding_side is not None: if old_padding_side is not None:
self.tokenizer.padding_side = 'right' self.tokenizer.padding_side = 'right'
+3 -3
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@@ -7,11 +7,11 @@ tokenizer_name: ${agent.model_config.vlm_model_name}
action_dim: 16 action_dim: 16
obs_dim: 16 obs_dim: 16
normalization_type: "min_max" normalization_type: "gaussian"
chunk_size: 16 chunk_size: 32
pred_horizon: ${agent.chunk_size} pred_horizon: ${agent.chunk_size}
obs_horizon: 2 obs_horizon: 2
n_action_steps: 8 n_action_steps: 16
num_action_steps: ${agent.n_action_steps} num_action_steps: ${agent.n_action_steps}
action_chunk_start: 0 action_chunk_start: 0
camera_names: ${data.camera_names} camera_names: ${data.camera_names}
+12 -2
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@@ -416,12 +416,15 @@ class SmolVLANativeAgentTest(unittest.TestCase):
self.assertEqual(cfg.agent._target_, 'roboimi.vla.agent_smolvla_native.SmolVLANativeAgent') self.assertEqual(cfg.agent._target_, 'roboimi.vla.agent_smolvla_native.SmolVLANativeAgent')
self.assertEqual(cfg.agent.action_dim, 16) self.assertEqual(cfg.agent.action_dim, 16)
self.assertEqual(cfg.agent.obs_dim, 16) self.assertEqual(cfg.agent.obs_dim, 16)
self.assertEqual(cfg.agent.normalization_type, 'gaussian')
self.assertEqual(list(cfg.agent.camera_names), list(cfg.data.camera_names)) self.assertEqual(list(cfg.agent.camera_names), list(cfg.data.camera_names))
self.assertEqual(cfg.agent.num_cams, len(cfg.data.camera_names)) self.assertEqual(cfg.agent.num_cams, len(cfg.data.camera_names))
self.assertEqual(cfg.agent.chunk_size, 16) self.assertEqual(cfg.agent.chunk_size, 32)
self.assertEqual(cfg.agent.n_action_steps, 8) self.assertEqual(cfg.agent.n_action_steps, 16)
self.assertEqual(cfg.agent.model_config.max_state_dim, 32) self.assertEqual(cfg.agent.model_config.max_state_dim, 32)
self.assertEqual(cfg.agent.model_config.max_action_dim, 32) self.assertEqual(cfg.agent.model_config.max_action_dim, 32)
self.assertEqual(cfg.agent.model_config.chunk_size, 32)
self.assertEqual(cfg.agent.model_config.n_action_steps, 16)
self.assertIsNone(cfg.agent.dataset_image_resize_shape) self.assertIsNone(cfg.agent.dataset_image_resize_shape)
self.assertIsNone(cfg.agent.eval_image_resize_shape) self.assertIsNone(cfg.agent.eval_image_resize_shape)
self.assertEqual(list(cfg.agent.model_config.resize_imgs_with_padding), [512, 512]) self.assertEqual(list(cfg.agent.model_config.resize_imgs_with_padding), [512, 512])
@@ -430,6 +433,13 @@ class SmolVLANativeAgentTest(unittest.TestCase):
self.assertNotIn('tokenizer_name', cfg.agent.model_config) self.assertNotIn('tokenizer_name', cfg.agent.model_config)
self.assertEqual(cfg.agent.model_config.vlm_model_name, 'HuggingFaceTB/SmolVLM2-500M-Video-Instruct') self.assertEqual(cfg.agent.model_config.vlm_model_name, 'HuggingFaceTB/SmolVLM2-500M-Video-Instruct')
def test_tokenize_tasks_preserves_existing_single_newline(self):
agent = _make_agent(model_config={'resize_imgs_with_padding': None, 'pad_language_to': 'max_length', 'tokenizer_max_length': 12})
agent._tokenize_tasks(['already newline\n', 'needs newline'], batch_size=2, device=torch.device('cpu'))
self.assertEqual(agent.tokenizer.calls[-1]['texts'], ['already newline\n', 'needs newline\n'])
if __name__ == '__main__': if __name__ == '__main__':
unittest.main() unittest.main()
+139 -3
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@@ -39,6 +39,17 @@ class FakeLoader:
return iter(()) return iter(())
class FakeTqdm:
def __init__(self, iterable, **_kwargs):
self.iterable = iterable
def __iter__(self):
return iter(self.iterable)
def set_postfix(self, *_args, **_kwargs):
return None
class FakeScheduler: class FakeScheduler:
def state_dict(self): def state_dict(self):
return {} return {}
@@ -46,13 +57,18 @@ class FakeScheduler:
def load_state_dict(self, state_dict): def load_state_dict(self, state_dict):
return None return None
def step(self):
return None
class RecordingAdamW: class RecordingAdamW:
created = [] created = []
def __init__(self, params, lr, weight_decay): def __init__(self, params, lr, weight_decay, betas=(0.9, 0.999), eps=1e-8):
self.lr = lr self.lr = lr
self.weight_decay = weight_decay self.weight_decay = weight_decay
self.betas = betas
self.eps = eps
self.param_groups = self._normalize_param_groups(params, lr, weight_decay) self.param_groups = self._normalize_param_groups(params, lr, weight_decay)
RecordingAdamW.created.append(self) RecordingAdamW.created.append(self)
@@ -79,6 +95,12 @@ class RecordingAdamW:
def load_state_dict(self, state_dict): def load_state_dict(self, state_dict):
return None return None
def zero_grad(self):
return None
def step(self):
return None
class RecordingTransformerHead(nn.Module): class RecordingTransformerHead(nn.Module):
def __init__(self): def __init__(self):
@@ -324,7 +346,7 @@ class TrainVLATransformerOptimizerTest(unittest.TestCase):
mock.patch.object(module, 'get_lr_schedule_with_warmup', return_value=FakeScheduler()), \ mock.patch.object(module, 'get_lr_schedule_with_warmup', return_value=FakeScheduler()), \
mock.patch.object(module, 'AdamW', RecordingAdamW), \ mock.patch.object(module, 'AdamW', RecordingAdamW), \
mock.patch.object(module.torch, 'save', return_value=None), \ mock.patch.object(module.torch, 'save', return_value=None), \
mock.patch.object(module, 'tqdm', side_effect=lambda iterable, **kwargs: iterable): mock.patch.object(module, 'tqdm', side_effect=lambda iterable, **kwargs: FakeTqdm(iterable, **kwargs)):
module.main(cfg) module.main(cfg)
finally: finally:
os.chdir(previous_cwd) os.chdir(previous_cwd)
@@ -404,7 +426,7 @@ class TrainVLATransformerOptimizerTest(unittest.TestCase):
mock.patch.object(module, 'get_lr_schedule_with_warmup', return_value=FakeScheduler()), \ mock.patch.object(module, 'get_lr_schedule_with_warmup', return_value=FakeScheduler()), \
mock.patch.object(module, 'AdamW', RecordingAdamW), \ mock.patch.object(module, 'AdamW', RecordingAdamW), \
mock.patch.object(module.torch, 'save', return_value=None), \ mock.patch.object(module.torch, 'save', return_value=None), \
mock.patch.object(module, 'tqdm', side_effect=lambda iterable, **kwargs: iterable): mock.patch.object(module, 'tqdm', side_effect=lambda iterable, **kwargs: FakeTqdm(iterable, **kwargs)):
module.main(cfg) module.main(cfg)
finally: finally:
os.chdir(previous_cwd) os.chdir(previous_cwd)
@@ -464,3 +486,117 @@ class TrainVLASmolVLAOptimizerTest(unittest.TestCase):
self.assertIn('noise_pred_net.proj.weight', optimizer_names) self.assertIn('noise_pred_net.proj.weight', optimizer_names)
self.assertNotIn('condition_encoder.vlm.weight', optimizer_names) self.assertNotIn('condition_encoder.vlm.weight', optimizer_names)
self.assertNotIn('condition_encoder.vlm.bias', optimizer_names) self.assertNotIn('condition_encoder.vlm.bias', optimizer_names)
def test_smolvla_native_training_preset_overrides_optimizer_scheduler_and_grad_clip(self):
module = TrainVLATransformerOptimizerTest()._load_train_vla_module()
class _NativeAgent(nn.Module):
def __init__(self):
super().__init__()
self.model = nn.Linear(2, 2)
def to(self, device):
return self
def get_normalization_stats(self):
return {}
agent = _NativeAgent()
cfg = TrainVLATransformerOptimizerTest()._make_cfg()
cfg.agent = AttrDict(_target_='roboimi.vla.agent_smolvla_native.SmolVLANativeAgent')
cfg.train.lr = 9e-4
cfg.train.max_steps = 1
cfg.train.weight_decay = 0.123
cfg.train.grad_clip = 1.0
cfg.train.warmup_steps = 7
cfg.train.scheduler_type = 'constant'
cfg.train.min_lr = 1e-7
scheduler_calls = []
clip_calls = []
def fake_instantiate(config_node, **_kwargs):
if config_node is cfg.data:
return FakeDataset()
if config_node is cfg.agent:
return agent
raise AssertionError(f'unexpected instantiate config: {config_node!r}')
def fake_scheduler(*args, **kwargs):
scheduler_calls.append(kwargs)
return FakeScheduler()
def fake_clip(parameters, max_norm):
clip_calls.append(float(max_norm))
return torch.tensor(0.0)
class OneBatchLoader:
def __len__(self):
return 1
def __iter__(self):
batch = {
'observation.front': torch.zeros(1, 1, 1, 2, 2),
'observation.state': torch.zeros(1, 1, 2),
'action': torch.zeros(1, 1, 2),
}
return iter([batch])
def fake_compute_loss(_batch):
return agent.model.weight.sum() * 0.0 + torch.tensor(1.0, requires_grad=True)
agent.compute_loss = fake_compute_loss
with tempfile.TemporaryDirectory() as tempdir:
previous_cwd = os.getcwd()
try:
os.chdir(tempdir)
with mock.patch.object(module, 'instantiate', side_effect=fake_instantiate), \
mock.patch.object(module, 'DataLoader', side_effect=lambda *args, **kwargs: OneBatchLoader()), \
mock.patch.object(module, 'get_lr_schedule_with_warmup', side_effect=fake_scheduler), \
mock.patch.object(module, 'AdamW', RecordingAdamW), \
mock.patch.object(module.torch.nn.utils, 'clip_grad_norm_', side_effect=fake_clip), \
mock.patch.object(module.torch, 'save', return_value=None), \
mock.patch.object(module, 'tqdm', side_effect=lambda iterable, **kwargs: FakeTqdm(iterable, **kwargs)):
module.main(cfg)
finally:
os.chdir(previous_cwd)
optimizer = RecordingAdamW.created[-1]
self.assertEqual(optimizer.lr, 1e-4)
self.assertEqual(optimizer.weight_decay, 1e-10)
self.assertEqual(optimizer.betas, (0.9, 0.95))
self.assertEqual(optimizer.eps, 1e-8)
self.assertEqual(scheduler_calls[-1], {
'warmup_steps': 1000,
'max_steps': 1,
'scheduler_type': 'cosine',
'min_lr': 2.5e-6,
})
self.assertEqual(clip_calls, [10.0])
def test_cosine_scheduler_spans_requested_training_steps_then_clamps(self):
module = TrainVLATransformerOptimizerTest()._load_train_vla_module()
param = nn.Parameter(torch.tensor(1.0))
optimizer = torch.optim.SGD([param], lr=1e-4)
base_lr = optimizer.param_groups[0]['lr']
scheduler = module.get_lr_schedule_with_warmup(
optimizer,
warmup_steps=1000,
max_steps=150000,
scheduler_type='cosine',
min_lr=2.5e-6,
)
lr_lambda = scheduler.lr_lambdas[0]
observed = {}
for step in (0, 1, 1000, 30000, 40000, 150000, 160000):
observed[step] = base_lr * lr_lambda(step)
self.assertGreater(observed[1], observed[0])
self.assertLess(observed[1000], 1e-4)
self.assertGreater(observed[30000], observed[40000])
self.assertGreater(observed[40000], observed[150000])
self.assertAlmostEqual(observed[150000], 2.5e-6, places=12)
self.assertAlmostEqual(observed[160000], 2.5e-6, places=12)