diff --git a/tests/test_sgdw.py b/tests/test_sgdw.py
new file mode 100644
index 00000000..a7abcc0e
--- /dev/null
+++ b/tests/test_sgdw.py
@@ -0,0 +1,34 @@
+"""SGDW decay is a scale of the parameter, as in arXiv 1711.05101."""
+import torch
+import torch_optimizer as optim
+
+
+def _step(theta, grad, lr, weight_decay, momentum=0.0):
+    param = torch.nn.Parameter(torch.as_tensor(theta, dtype=torch.float64).clone())
+    opt = optim.SGDW(
+        [param], lr=lr, weight_decay=weight_decay, momentum=momentum
+    )
+    param.grad = torch.as_tensor(grad, dtype=torch.float64).clone()
+    opt.step()
+    return param.detach()
+
+
+def test_zero_gradient_scales_the_parameter():
+    # θ = 4, μ = 0.1, λ = 0.2. The old update subtracted λμ and landed
+    # on 3.98. The paper scale (1 − λμ) θ is 3.92.
+    got = _step([4.0, -2.0], [0.0, 0.0], lr=0.1, weight_decay=0.2)
+    assert torch.allclose(got, torch.tensor([3.92, -1.96], dtype=torch.float64))
+
+
+def test_decay_precedes_the_gradient_step():
+    # Scale first, then subtract μ g: 4*(1-0.02) - 0.1*1.5 = 3.77.
+    got = _step([4.0], [1.5], lr=0.1, weight_decay=0.2)
+    assert torch.allclose(got, torch.tensor([3.77], dtype=torch.float64))
+
+
+def test_no_decay_matches_plain_sgd():
+    theta = torch.tensor([1.5, -0.25, 3.0], dtype=torch.float64)
+    grad = torch.tensor([0.4, -1.0, 0.0], dtype=torch.float64)
+    got = _step(theta, grad, lr=0.1, weight_decay=0.0, momentum=0.9)
+    # one momentum step from an empty buffer stores g and subtracts μ g
+    assert torch.allclose(got, theta - 0.1 * grad)
diff --git a/torch_optimizer/sgdw.py b/torch_optimizer/sgdw.py
index 62e7d998..24dd7f7c 100644
--- a/torch_optimizer/sgdw.py
+++ b/torch_optimizer/sgdw.py
@@ -113,10 +113,11 @@ def step(self, closure: OptLossClosure = None) -> OptFloat:
                     else:
                         d_p = buf
 
-                # Apply momentum
-                p.data.add_(d_p, alpha=-group["lr"])
-
-                # Apply weight decay
+                # Decoupled weight decay scales the parameter. The paper
+                # (Loshchilov and Hutter, arXiv 1711.05101) writes
+                # θ ← (1 − λ μ) θ, then subtracts the step. Adding −λ μ to
+                # every coordinate subtracts a constant instead.
                 if weight_decay != 0:
-                    p.data.add_(weight_decay, alpha=-group["lr"])
+                    p.data.mul_(1 - group["lr"] * weight_decay)
+                p.data.add_(d_p, alpha=-group["lr"])
         return loss