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pytorch-image-models/tests/test_optim.py

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""" Optimzier Tests
These tests were adapted from PyTorch' optimizer tests.
"""
import math
import pytest
import functools
from copy import deepcopy
import torch
from torch.testing._internal.common_utils import TestCase
from torch.autograd import Variable
from timm.scheduler import PlateauLRScheduler
from timm.optim import create_optimizer_v2
# HACK relying on internal PyTorch test functionality for comparisons that I don't want to write
torch_tc = TestCase()
def _test_basic_cases_template(weight, bias, input, constructor, scheduler_constructors):
weight = Variable(weight, requires_grad=True)
bias = Variable(bias, requires_grad=True)
input = Variable(input)
optimizer = constructor(weight, bias)
schedulers = []
for scheduler_constructor in scheduler_constructors:
schedulers.append(scheduler_constructor(optimizer))
# to check if the optimizer can be printed as a string
optimizer.__repr__()
def fn():
optimizer.zero_grad()
y = weight.mv(input)
if y.is_cuda and bias.is_cuda and y.get_device() != bias.get_device():
y = y.cuda(bias.get_device())
loss = (y + bias).pow(2).sum()
loss.backward()
return loss
initial_value = fn().item()
for _i in range(200):
for scheduler in schedulers:
if isinstance(scheduler, PlateauLRScheduler):
val_loss = fn()
scheduler.step(val_loss)
else:
scheduler.step()
optimizer.step(fn)
assert fn().item() < initial_value
def _test_state_dict(weight, bias, input, constructor):
weight = Variable(weight, requires_grad=True)
bias = Variable(bias, requires_grad=True)
input = Variable(input)
def fn_base(optimizer, weight, bias):
optimizer.zero_grad()
i = input_cuda if weight.is_cuda else input
loss = (weight.mv(i) + bias).pow(2).sum()
loss.backward()
return loss
optimizer = constructor(weight, bias)
fn = functools.partial(fn_base, optimizer, weight, bias)
# Prime the optimizer
for _i in range(20):
optimizer.step(fn)
# Clone the weights and construct new optimizer for them
weight_c = Variable(weight.data.clone(), requires_grad=True)
bias_c = Variable(bias.data.clone(), requires_grad=True)
optimizer_c = constructor(weight_c, bias_c)
fn_c = functools.partial(fn_base, optimizer_c, weight_c, bias_c)
# Load state dict
state_dict = deepcopy(optimizer.state_dict())
state_dict_c = deepcopy(optimizer.state_dict())
optimizer_c.load_state_dict(state_dict_c)
# Run both optimizations in parallel
for _i in range(20):
optimizer.step(fn)
optimizer_c.step(fn_c)
#assert torch.equal(weight, weight_c)
#assert torch.equal(bias, bias_c)
torch_tc.assertEqual(weight, weight_c)
torch_tc.assertEqual(bias, bias_c)
# Make sure state dict wasn't modified
torch_tc.assertEqual(state_dict, state_dict_c)
# Make sure state dict is deterministic with equal but not identical parameters
torch_tc.assertEqual(optimizer.state_dict(), optimizer_c.state_dict())
# Make sure repeated parameters have identical representation in state dict
optimizer_c.param_groups.extend(optimizer_c.param_groups)
torch_tc.assertEqual(optimizer.state_dict()['param_groups'][-1], optimizer_c.state_dict()['param_groups'][-1])
# Check that state dict can be loaded even when we cast parameters
# to a different type and move to a different device.
if not torch.cuda.is_available():
return
input_cuda = Variable(input.data.float().cuda())
weight_cuda = Variable(weight.data.float().cuda(), requires_grad=True)
bias_cuda = Variable(bias.data.float().cuda(), requires_grad=True)
optimizer_cuda = constructor(weight_cuda, bias_cuda)
fn_cuda = functools.partial(fn_base, optimizer_cuda, weight_cuda, bias_cuda)
state_dict = deepcopy(optimizer.state_dict())
state_dict_c = deepcopy(optimizer.state_dict())
optimizer_cuda.load_state_dict(state_dict_c)
# Make sure state dict wasn't modified
torch_tc.assertEqual(state_dict, state_dict_c)
for _i in range(20):
optimizer.step(fn)
optimizer_cuda.step(fn_cuda)
torch_tc.assertEqual(weight, weight_cuda)
torch_tc.assertEqual(bias, bias_cuda)
# validate deepcopy() copies all public attributes
def getPublicAttr(obj):
return set(k for k in obj.__dict__ if not k.startswith('_'))
assert getPublicAttr(optimizer) == getPublicAttr(deepcopy(optimizer))
def _test_basic_cases(constructor, scheduler_constructors=None):
if scheduler_constructors is None:
scheduler_constructors = []
_test_state_dict(
torch.randn(10, 5),
torch.randn(10),
torch.randn(5),
constructor
)
_test_basic_cases_template(
torch.randn(10, 5),
torch.randn(10),
torch.randn(5),
constructor,
scheduler_constructors
)
# non-contiguous parameters
_test_basic_cases_template(
torch.randn(10, 5, 2)[..., 0],
torch.randn(10, 2)[..., 0],
torch.randn(5),
constructor,
scheduler_constructors
)
# CUDA
if not torch.cuda.is_available():
return
_test_basic_cases_template(
torch.randn(10, 5).cuda(),
torch.randn(10).cuda(),
torch.randn(5).cuda(),
constructor,
scheduler_constructors
)
def _test_model(optimizer, params, device=torch.device('cpu')):
weight = torch.tensor(
[[-0.2109, -0.4976], [-0.1413, -0.3420], [-0.2524, 0.6976]],
device=device, requires_grad=True)
bias = torch.tensor([-0.1085, -0.2979, 0.6892], device=device, requires_grad=True)
weight2 = torch.tensor([[-0.0508, -0.3941, -0.2843]], device=device, requires_grad=True)
bias2 = torch.tensor([-0.0711], device=device, requires_grad=True)
input = torch.tensor([0.1, 0.2, 0.3, 0.4, 0.5, 0.6], device=device).reshape(3, 2)
model = torch.nn.Sequential(torch.nn.Linear(2, 3),
torch.nn.Sigmoid(),
torch.nn.Linear(3, 1),
torch.nn.Sigmoid())
model.to(device)
pretrained_dict = model.state_dict()
pretrained_dict['0.weight'] = weight
pretrained_dict['0.bias'] = bias
pretrained_dict['2.weight'] = weight2
pretrained_dict['2.bias'] = bias2
model.load_state_dict(pretrained_dict)
optimizer = create_optimizer_v2(model, opt=optimizer, **params)
prev_loss = float('inf')
for i in range(20):
optimizer.zero_grad()
output = model(input)
loss = output.sum()
loss.backward()
loss = loss.item()
assert loss < prev_loss
prev_loss = loss
optimizer.step()
def rosenbrock(tensor):
x, y = tensor
return (1 - x) ** 2 + 100 * (y - x ** 2) ** 2
def drosenbrock(tensor):
x, y = tensor
return torch.tensor((-400 * x * (y - x ** 2) - 2 * (1 - x), 200 * (y - x ** 2)))
def _test_rosenbrock(constructor, scheduler_constructors=None):
if scheduler_constructors is None:
scheduler_constructors = []
params_t = torch.tensor([1.5, 1.5])
params = Variable(params_t, requires_grad=True)
optimizer = constructor([params])
schedulers = []
for scheduler_constructor in scheduler_constructors:
schedulers.append(scheduler_constructor(optimizer))
solution = torch.tensor([1, 1])
initial_dist = params.data.dist(solution)
def eval(params, w):
# Depending on w, provide only the x or y gradient
optimizer.zero_grad()
loss = rosenbrock(params)
loss.backward()
grad = drosenbrock(params.data)
# NB: We torture test the optimizer by returning an
# uncoalesced sparse tensor
if w:
i = torch.LongTensor([[0, 0]])
x = grad[0]
v = torch.tensor([x / 4., x - x / 4.])
else:
i = torch.LongTensor([[1, 1]])
y = grad[1]
v = torch.tensor([y - y / 4., y / 4.])
x = torch.sparse.DoubleTensor(i, v, torch.Size([2])).to(dtype=v.dtype)
with torch.no_grad():
params.grad = x.to_dense()
return loss
for i in range(2000):
# Do cyclic coordinate descent
w = i % 2
optimizer.step(functools.partial(eval, params, w))
for scheduler in schedulers:
if isinstance(scheduler, PlateauLRScheduler):
scheduler.step(rosenbrock(params))
else:
scheduler.step()
torch_tc.assertLessEqual(params.data.dist(solution), initial_dist)
def _build_params_dict(weight, bias, **kwargs):
return [{'params': [weight]}, dict(params=[bias], **kwargs)]
def _build_params_dict_single(weight, bias, **kwargs):
return [dict(params=bias, **kwargs)]
#@pytest.mark.parametrize('optimizer', ['sgd', 'momentum'])
# FIXME momentum variant frequently fails in GitHub runner, but never local after many attempts
@pytest.mark.parametrize('optimizer', ['sgd'])
def test_sgd(optimizer):
_test_basic_cases(
lambda weight, bias: create_optimizer_v2([weight, bias], optimizer, lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict(weight, bias, lr=1e-2),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=1e-2),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=1e-2), optimizer)
)
# _test_basic_cases(
# lambda weight, bias: create_optimizer_v2([weight, bias], optimizer, lr=1e-3),
# [lambda opt: StepLR(opt, gamma=0.9, step_size=10)]
# )
# _test_basic_cases(
# lambda weight, bias: create_optimizer_v2([weight, bias], optimizer, lr=1e-3),
# [lambda opt: WarmUpLR(opt, warmup_factor=0.4, warmup_iters=4, warmup_method="linear")]
# )
# _test_basic_cases(
# lambda weight, bias: optimizer([weight, bias], lr=1e-3),
# [lambda opt: WarmUpLR(opt, warmup_factor=0.4, warmup_iters=4, warmup_method="constant")]
# )
# _test_basic_cases(
# lambda weight, bias: optimizer([weight, bias], lr=1e-3),
# [lambda opt: StepLR(opt, gamma=0.9, step_size=10),
# lambda opt: WarmUpLR(opt, warmup_factor=0.4, warmup_iters=4)]
# )
# _test_basic_cases(
# lambda weight, bias: optimizer([weight, bias], lr=1e-3),
# [lambda opt: StepLR(opt, gamma=0.9, step_size=10),
# lambda opt: ReduceLROnPlateau(opt)]
# )
# _test_basic_cases(
# lambda weight, bias: optimizer([weight, bias], lr=1e-3),
# [lambda opt: StepLR(opt, gamma=0.99, step_size=10),
# lambda opt: ExponentialLR(opt, gamma=0.99),
# lambda opt: ReduceLROnPlateau(opt)]
# )
_test_basic_cases(
lambda weight, bias: create_optimizer_v2([weight, bias], optimizer, lr=3e-3, momentum=1)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2([weight, bias], optimizer, lr=3e-3, momentum=1, weight_decay=.1)
)
_test_rosenbrock(
lambda params: create_optimizer_v2(params, optimizer, lr=1e-3)
)
_test_model(optimizer, dict(lr=1e-3))
@pytest.mark.parametrize('optimizer', ['adamw', 'adam', 'nadam', 'adamax'])
def test_adam(optimizer):
_test_basic_cases(
lambda weight, bias: create_optimizer_v2([weight, bias], optimizer, lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict(weight, bias, lr=3e-3),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=3e-3),
optimizer,
lr=1e-3)
)
_test_rosenbrock(
lambda params: create_optimizer_v2(params, optimizer, lr=5e-2)
)
_test_model(optimizer, dict(lr=5e-2))
@pytest.mark.parametrize('optimizer', ['adabelief'])
def test_adabelief(optimizer):
_test_basic_cases(
lambda weight, bias: create_optimizer_v2([weight, bias], optimizer, lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict(weight, bias, lr=3e-3),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=3e-3),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=3e-3), optimizer)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2([weight, bias], optimizer, lr=1e-3, weight_decay=1)
)
_test_rosenbrock(
lambda params: create_optimizer_v2(params, optimizer, lr=5e-2)
)
_test_model(optimizer, dict(lr=5e-2))
@pytest.mark.parametrize('optimizer', ['radam', 'radabelief'])
def test_rectified(optimizer):
_test_basic_cases(
lambda weight, bias: create_optimizer_v2([weight, bias], optimizer, lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict(weight, bias, lr=3e-3),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=3e-3),
optimizer,
lr=1e-3)
)
_test_rosenbrock(
lambda params: create_optimizer_v2(params, optimizer, lr=1e-3)
)
_test_model(optimizer, dict(lr=1e-3))
@pytest.mark.parametrize('optimizer', ['adadelta', 'adagrad'])
def test_adaother(optimizer):
_test_basic_cases(
lambda weight, bias: create_optimizer_v2([weight, bias], optimizer, lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict(weight, bias, lr=3e-3),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=3e-3),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=3e-3), optimizer)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2([weight, bias], optimizer, lr=1e-3, weight_decay=1)
)
_test_rosenbrock(
lambda params: create_optimizer_v2(params, optimizer, lr=1e-1)
)
_test_model(optimizer, dict(lr=5e-2))
@pytest.mark.parametrize('optimizer', ['adafactor'])
def test_adafactor(optimizer):
_test_basic_cases(
lambda weight, bias: create_optimizer_v2([weight, bias], optimizer, lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict(weight, bias, lr=3e-3),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=3e-3),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(_build_params_dict_single(weight, bias), optimizer)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2([weight, bias], optimizer, lr=1e-3, weight_decay=1)
)
_test_rosenbrock(
lambda params: create_optimizer_v2(params, optimizer, lr=5e-2)
)
_test_model(optimizer, dict(lr=5e-2))
@pytest.mark.parametrize('optimizer', ['lamb', 'lambc'])
def test_lamb(optimizer):
_test_basic_cases(
lambda weight, bias: create_optimizer_v2([weight, bias], optimizer, lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict(weight, bias, lr=1e-3),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=1e-3),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=1e-3), optimizer)
)
_test_rosenbrock(
lambda params: create_optimizer_v2(params, optimizer, lr=1e-3)
)
_test_model(optimizer, dict(lr=1e-3))
@pytest.mark.parametrize('optimizer', ['lars', 'larc', 'nlars', 'nlarc'])
def test_lars(optimizer):
_test_basic_cases(
lambda weight, bias: create_optimizer_v2([weight, bias], optimizer, lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict(weight, bias, lr=1e-3),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=1e-3),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=1e-3), optimizer)
)
_test_rosenbrock(
lambda params: create_optimizer_v2(params, optimizer, lr=1e-3)
)
_test_model(optimizer, dict(lr=1e-3))
@pytest.mark.parametrize('optimizer', ['madgrad', 'madgradw'])
def test_madgrad(optimizer):
_test_basic_cases(
lambda weight, bias: create_optimizer_v2([weight, bias], optimizer, lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict(weight, bias, lr=3e-3),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=3e-3),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=3e-3), optimizer)
)
_test_rosenbrock(
lambda params: create_optimizer_v2(params, optimizer, lr=1e-2)
)
_test_model(optimizer, dict(lr=1e-2))
@pytest.mark.parametrize('optimizer', ['novograd'])
def test_novograd(optimizer):
_test_basic_cases(
lambda weight, bias: create_optimizer_v2([weight, bias], optimizer, lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict(weight, bias, lr=3e-3),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=3e-3),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=3e-3), optimizer)
)
_test_rosenbrock(
lambda params: create_optimizer_v2(params, optimizer, lr=1e-3)
)
_test_model(optimizer, dict(lr=1e-3))
@pytest.mark.parametrize('optimizer', ['rmsprop', 'rmsproptf'])
def test_rmsprop(optimizer):
_test_basic_cases(
lambda weight, bias: create_optimizer_v2([weight, bias], optimizer, lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict(weight, bias, lr=3e-3),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=3e-3),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=3e-3), optimizer)
)
_test_rosenbrock(
lambda params: create_optimizer_v2(params, optimizer, lr=1e-2)
)
_test_model(optimizer, dict(lr=1e-2))
@pytest.mark.parametrize('optimizer', ['adamp'])
def test_adamp(optimizer):
_test_basic_cases(
lambda weight, bias: create_optimizer_v2([weight, bias], optimizer, lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict(weight, bias, lr=3e-3),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=3e-3),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=3e-3), optimizer)
)
_test_rosenbrock(
lambda params: create_optimizer_v2(params, optimizer, lr=5e-2)
)
_test_model(optimizer, dict(lr=5e-2))
@pytest.mark.parametrize('optimizer', ['sgdp'])
def test_sgdp(optimizer):
_test_basic_cases(
lambda weight, bias: create_optimizer_v2([weight, bias], optimizer, lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict(weight, bias, lr=3e-3),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=3e-3),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=3e-3), optimizer)
)
_test_rosenbrock(
lambda params: create_optimizer_v2(params, optimizer, lr=1e-3)
)
_test_model(optimizer, dict(lr=1e-3))
@pytest.mark.parametrize('optimizer', ['lookahead_sgd', 'lookahead_momentum'])
def test_lookahead_sgd(optimizer):
_test_basic_cases(
lambda weight, bias: create_optimizer_v2([weight, bias], optimizer, lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict(weight, bias, lr=3e-3),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=3e-3),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=3e-3), optimizer)
)
_test_rosenbrock(
lambda params: create_optimizer_v2(params, optimizer, lr=1e-3)
)
@pytest.mark.parametrize('optimizer', ['lookahead_adamw', 'lookahead_adam'])
def test_lookahead_adam(optimizer):
_test_basic_cases(
lambda weight, bias: create_optimizer_v2([weight, bias], optimizer, lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict(weight, bias, lr=3e-3),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=3e-3),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=3e-3), optimizer)
)
_test_rosenbrock(
lambda params: create_optimizer_v2(params, optimizer, lr=5e-2)
)
@pytest.mark.parametrize('optimizer', ['lookahead_radam'])
def test_lookahead_radam(optimizer):
_test_basic_cases(
lambda weight, bias: create_optimizer_v2([weight, bias], optimizer, lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict(weight, bias, lr=3e-3),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=3e-3),
optimizer,
lr=1e-3)
)
_test_basic_cases(
lambda weight, bias: create_optimizer_v2(
_build_params_dict_single(weight, bias, lr=3e-3), optimizer)
)
_test_rosenbrock(
lambda params: create_optimizer_v2(params, optimizer, lr=1e-4)
)