diff --git a/cautious_benchmark.png b/cautious_benchmark.png
new file mode 100644
index 000000000..67effebf3
Binary files /dev/null and b/cautious_benchmark.png differ
diff --git a/docs/api/contrib.rst b/docs/api/contrib.rst
index c30e3143e..99333aa7d 100644
--- a/docs/api/contrib.rst
+++ b/docs/api/contrib.rst
@@ -13,6 +13,8 @@ are not supported by the main library.
     ademamix
     adopt
     simplified_ademamix
+    cautious
+    CautiousState
     cocob
     COCOBState
     dadapt_adamw
diff --git a/optax/contrib/__init__.py b/optax/contrib/__init__.py
index 35c032ef0..a86c8c27b 100644
--- a/optax/contrib/__init__.py
+++ b/optax/contrib/__init__.py
@@ -26,6 +26,8 @@
 from optax.contrib._ademamix import simplified_ademamix
 from optax.contrib._adopt import adopt
 from optax.contrib._adopt import scale_by_adopt
+from optax.contrib._cautious import cautious
+from optax.contrib._cautious import CautiousState
 from optax.contrib._cocob import cocob
 from optax.contrib._cocob import COCOBState
 from optax.contrib._cocob import scale_by_cocob
diff --git a/optax/contrib/_cautious.py b/optax/contrib/_cautious.py
new file mode 100644
index 000000000..725d9229f
--- /dev/null
+++ b/optax/contrib/_cautious.py
@@ -0,0 +1,146 @@
+# Copyright 2024 DeepMind Technologies Limited. All Rights Reserved.
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#     http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+# ==============================================================================
+"""Cautious optimizer wrapper.
+
+Reference:
+  Liang et al., `Cautious Optimizers: Improving Training with One Line of Code
+  <https://arxiv.org/abs/2411.16085>`_, 2024.
+"""
+
+from typing import NamedTuple, Optional
+
+import jax
+import jax.numpy as jnp
+from optax._src import base
+
+
+class CautiousState(NamedTuple):
+  """State for the :func:`optax.contrib.cautious` wrapper."""
+
+  base_optimizer_state: base.OptState
+
+
+def cautious(
+    base_optimizer: base.GradientTransformation,
+    eps: float = 1e-8,
+) -> base.GradientTransformationExtraArgs:
+  r"""Cautious wrapper: mask updates that disagree with the current gradient.
+
+  Wraps an arbitrary ``base_optimizer`` and, on every step, zeroes the
+  coordinates of the proposed update that would move *against* the current
+  gradient (i.e. would locally *increase* the loss), then rescales the
+  surviving coordinates so the average update magnitude is preserved.
+
+  Concretely, let :math:`u_t` be the update proposed by the base optimizer
+  (using Optax's additive convention ``params <- params + u_t``) and
+  :math:`g_t` the current gradient. The cautious mask keeps only the
+  descent-aligned coordinates:
+
+  .. math::
+
+    \phi_t = \mathbb{1}\!\left[u_t \odot g_t < 0\right],
+
+  and rescales them per parameter tensor so the mean magnitude is unchanged:
+
+  .. math::
+
+    \tilde{u}_t = \phi_t \odot u_t \cdot \frac{n}{\sum \phi_t + \varepsilon},
+
+  where :math:`n` is the number of elements of the tensor. The mask
+  condition :math:`u_t \odot g_t < 0` is exactly the paper's alignment
+  condition :math:`(-u_t) \odot g_t > 0` re-expressed in Optax's additive
+  update convention (Optax updates are the negative of the paper's, since
+  Optax *adds* the update while the paper *subtracts* it).
+
+  This single-line modification provably preserves the Hamiltonian / Lyapunov
+  descent of the base optimizer: the cautious update always satisfies
+  :math:`\langle \tilde{u}_t, g_t \rangle \le 0`, so it never points uphill,
+  whereas a momentum-based base optimizer can. Empirically the authors report
+  up to a 1.47x sample-efficiency gain when wrapping AdamW for LLM and ViT
+  pre-training, at the cost of one elementwise mask.
+
+  Because the mask needs *both* the raw gradient and the base optimizer's
+  proposed update, ``cautious`` is implemented as a wrapper (like
+  :func:`optax.contrib.schedule_free`) rather than a chainable
+  ``scale_by_*`` transform.
+
+  Args:
+    base_optimizer: The optimizer to wrap (e.g. ``optax.adamw(1e-3)``,
+      ``optax.lion(1e-4)``, or any :class:`optax.GradientTransformation`).
+    eps: Small constant in the rescaling denominator. With the default
+      ``1e-8`` the wrapper reduces *exactly* to ``base_optimizer`` when every
+      coordinate agrees with the gradient (the mean-preserving normalization).
+      The original paper uses ``eps=1``, which additionally damps the update
+      when only a few coordinates survive; pass ``eps=1.0`` to match it.
+
+  Returns:
+    A :class:`optax.GradientTransformationExtraArgs`.
+
+  Examples:
+    >>> import optax
+    >>> import jax
+    >>> import jax.numpy as jnp
+    >>> def f(x): return jnp.sum(x ** 2)  # simple quadratic objective
+    >>> base = optax.adamw(learning_rate=0.1)
+    >>> solver = optax.contrib.cautious(base)
+    >>> params = jnp.array([1., 2., 3.])
+    >>> print('Objective function: ', f(params))
+    Objective function:  14.0
+    >>> opt_state = solver.init(params)
+    >>> for _ in range(5):
+    ...   grad = jax.grad(f)(params)
+    ...   updates, opt_state = solver.update(grad, opt_state, params)
+    ...   params = optax.apply_updates(params, updates)
+    ...   print('Objective function: {:.2E}'.format(f(params)))
+    Objective function: 1.28E+01
+    Objective function: 1.17E+01
+    Objective function: 1.07E+01
+    Objective function: 9.69E+00
+    Objective function: 8.77E+00
+
+  References:
+    Liang et al, `Cautious Optimizers: Improving Training with One Line of Code
+    <https://arxiv.org/abs/2411.16085>`_, 2024.
+  """
+  base_optimizer = base.with_extra_args_support(base_optimizer)
+
+  def init_fn(params: base.Params) -> CautiousState:
+    return CautiousState(base_optimizer_state=base_optimizer.init(params))
+
+  def update_fn(
+      updates: base.Updates,
+      state: CautiousState,
+      params: Optional[base.Params] = None,
+      **extra_args,
+  ):
+    # ``updates`` are the raw gradients fed to the wrapper.
+    grads = updates
+    base_updates, new_base_state = base_optimizer.update(
+        grads, state.base_optimizer_state, params, **extra_args
+    )
+
+    def _mask_leaf(update_leaf, grad_leaf):
+      # Keep coordinates where the update opposes the gradient (descent in the
+      # additive Optax convention ``params <- params + update``).
+      keep = (update_leaf * grad_leaf < 0).astype(update_leaf.dtype)
+      # Per-tensor mean-preserving rescale.
+      scale = keep.size / (jnp.sum(keep) + eps)
+      return update_leaf * keep * scale
+
+    cautious_updates = jax.tree.map(_mask_leaf, base_updates, grads)
+    return cautious_updates, CautiousState(base_optimizer_state=new_base_state)
+
+  # pyrefly: ignore[bad-argument-type]
+  return base.GradientTransformationExtraArgs(init_fn, update_fn)
diff --git a/optax/contrib/_cautious_test.py b/optax/contrib/_cautious_test.py
new file mode 100644
index 000000000..4d493f6ad
--- /dev/null
+++ b/optax/contrib/_cautious_test.py
@@ -0,0 +1,222 @@
+# Copyright 2024 DeepMind Technologies Limited. All Rights Reserved.
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#     http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+# ==============================================================================
+"""Tests for the cautious optimizer wrapper."""
+
+from absl.testing import absltest
+from absl.testing import parameterized
+import jax
+import jax.numpy as jnp
+import numpy as np
+import optax
+from optax import contrib
+from optax._src import test_utils
+from optax.contrib._cautious import cautious, CautiousState
+
+
+class CautiousTest(parameterized.TestCase):
+
+  def test_state_structure(self):
+    """The wrapper state holds the base optimizer state."""
+    params = jnp.ones((4,))
+    base = optax.adam(1e-2)
+    tx = cautious(base)
+    state = tx.init(params)
+    self.assertIsInstance(state, CautiousState)
+    # The inner state matches what the base optimizer would produce.
+    test_utils.assert_trees_all_equal_structs(
+        state.base_optimizer_state, base.init(params)
+    )
+
+  def test_fully_aligned_reduces_to_base(self):
+    """When every coordinate agrees with the gradient, cautious == base.
+
+    For SGD on a convex quadratic with no momentum, the update direction is
+    always ``-lr * grad``, so ``update * grad < 0`` holds for every coordinate
+    and the mask is all-ones. With ``eps`` tiny, the rescale is ~1, so the
+    cautious updates must equal the base updates.
+    """
+    params = jnp.array([1.0, -2.0, 3.0, 0.5])
+    grad = jnp.array([0.3, -0.7, 1.2, -0.1])
+
+    base = optax.sgd(1e-1)
+    base_state = base.init(params)
+    base_updates, _ = base.update(grad, base_state, params)
+
+    tx = cautious(base, eps=1e-8)
+    state = tx.init(params)
+    caut_updates, _ = tx.update(grad, state, params)
+
+    test_utils.assert_trees_all_close(
+        caut_updates, base_updates, rtol=1e-6, atol=1e-6)
+
+  def test_masks_misaligned_coordinates(self):
+    """Coordinates where the update agrees with the gradient sign are zeroed.
+
+    We construct a base update by hand (via SGD with momentum) so that the
+    momentum term points *with* the gradient on some coordinates (uphill in
+    Optax's additive convention) and verify those are masked out.
+    """
+    # Hand-rolled: use a base "optimizer" that just returns a fixed update so
+    # we can control alignment precisely.
+    fixed_update = jnp.array([-1.0, 1.0, -2.0, 2.0])
+    grad = jnp.array([1.0, 1.0, 1.0, -1.0])
+    # update * grad: [-1, +1, -2, -2] -> keep where < 0: [T, F, T, T]
+    expected_keep = jnp.array([1.0, 0.0, 1.0, 1.0])
+
+    def _const_update(updates, state, params=None, **kw):
+      del updates, params, kw
+      return fixed_update, state
+
+    const_opt = optax.GradientTransformation(
+        lambda p: optax.EmptyState(), _const_update
+    )
+    tx = cautious(const_opt, eps=1e-8)
+    state = tx.init(jnp.zeros(4))
+    caut_updates, _ = tx.update(grad, state, jnp.zeros(4))
+
+    # The masked-out coordinate (index 1) must be exactly zero.
+    self.assertEqual(float(caut_updates[1]), 0.0)
+    # Surviving coordinates keep their sign.
+    kept = caut_updates != 0
+    np.testing.assert_array_equal(kept.astype(jnp.float32), expected_keep)
+
+  def test_rescaling_preserves_mean_magnitude(self):
+    """The surviving updates are rescaled by n / (num_kept + eps)."""
+    fixed_update = jnp.array([-1.0, 1.0, -1.0, 1.0])
+    grad = jnp.array([1.0, 1.0, 1.0, 1.0])
+    # update * grad: [-1, +1, -1, +1] -> keep [T, F, T, F], num_kept = 2, n = 4
+    # scale = 4 / 2 = 2. Surviving coords were -1, -1 -> become -2, -2.
+
+    def _const_update(updates, state, params=None, **kw):
+      del updates, params, kw
+      return fixed_update, state
+
+    const_opt = optax.GradientTransformation(
+        lambda p: optax.EmptyState(), _const_update
+    )
+    tx = cautious(const_opt, eps=1e-8)
+    state = tx.init(jnp.zeros(4))
+    caut_updates, _ = tx.update(grad, state, jnp.zeros(4))
+
+    expected = jnp.array([-2.0, 0.0, -2.0, 0.0])
+    test_utils.assert_trees_all_close(
+        caut_updates, expected, rtol=1e-6, atol=1e-6)
+
+  def test_descent_guarantee(self):
+    """The cautious update never points uphill: <update, grad> <= 0.
+
+    This is the core theoretical property. We exercise it on a momentum
+    optimizer (which *can* overshoot) over many random steps and verify the
+    inner product of the cautious update with the gradient is always <= 0.
+    """
+    key = jax.random.PRNGKey(0)
+    params = jax.random.normal(key, (32,))
+    base = optax.sgd(1e-1, momentum=0.95)  # momentum can overshoot
+    tx = cautious(base)
+    state = tx.init(params)
+
+    worst = -jnp.inf
+    for i in range(50):
+      key, sub = jax.random.split(key)
+      # Noisy, sign-flipping gradients to stress the momentum term.
+      grad = jax.random.normal(sub, (32,)) + 0.3 * jnp.sin(i * params)
+      updates, state = tx.update(grad, state, params)
+      inner = float(jnp.vdot(updates, grad))
+      worst = max(worst, inner)
+      params = optax.apply_updates(params, updates)
+
+    # Allow a tiny positive tolerance for floating point noise.
+    self.assertLessEqual(worst, 1e-5)
+
+  def test_pytree_params(self):
+    """Works with dict (pytree) parameters and masks each leaf independently."""
+    params = {'w': jnp.ones((3,)), 'b': jnp.zeros((2,))}
+    tx = cautious(optax.adam(1e-2))
+    state = tx.init(params)
+    grads = {'w': jnp.array([1.0, -1.0, 1.0]), 'b': jnp.array([0.5, -0.5])}
+    updates, _ = tx.update(grads, state, params)
+    jax.tree.map(
+        lambda u: self.assertTrue(jnp.all(jnp.isfinite(u))), updates
+    )
+
+  @parameterized.parameters(
+      {'base_name': 'adam'},
+      {'base_name': 'adamw'},
+      {'base_name': 'lion'},
+      {'base_name': 'sgd'},
+  )
+  def test_wraps_common_optimizers(self, base_name):
+    """cautious() should descend a quadratic when wrapping common optimizers."""
+    params = jnp.array([3.0, -2.0, 1.0, 4.0])
+    base = getattr(optax, base_name)(learning_rate=1e-1)
+    tx = cautious(base)
+    state = tx.init(params)
+
+    def loss(p):
+      return jnp.sum(p ** 2)
+
+    initial = loss(params)
+    for _ in range(100):
+      grad = jax.grad(loss)(params)
+      updates, state = tx.update(grad, state, params)
+      params = optax.apply_updates(params, updates)
+    self.assertLess(loss(params), initial)
+
+  def test_jit_compatible(self):
+    """The wrapped optimizer can be jitted end-to-end."""
+    params = jnp.array([1.0, 2.0, 3.0])
+    tx = cautious(optax.adam(1e-2))
+
+    @jax.jit
+    def step(params, state):
+      grad = jax.grad(lambda p: jnp.sum(p**2))(params)
+      updates, state = tx.update(grad, state, params)
+      return optax.apply_updates(params, updates), state
+
+    state = tx.init(params)
+    for _ in range(5):
+      params, state = step(params, state)
+    self.assertTrue(jnp.all(jnp.isfinite(params)))
+
+  def test_eps_one_matches_paper(self):
+    """eps=1.0 reproduces the paper's n / (num_kept + 1) damping."""
+    fixed_update = jnp.array([-1.0, -1.0, -1.0, 1.0])
+    grad = jnp.array([1.0, 1.0, 1.0, 1.0])
+    # keep [T, T, T, F], num_kept = 3, n = 4, scale = 4 / (3 + 1) = 1.0
+
+    def _const_update(updates, state, params=None, **kw):
+      del updates, params, kw
+      return fixed_update, state
+
+    const_opt = optax.GradientTransformation(
+        lambda p: optax.EmptyState(), _const_update
+    )
+    tx = cautious(const_opt, eps=1.0)
+    state = tx.init(jnp.zeros(4))
+    caut_updates, _ = tx.update(grad, state, jnp.zeros(4))
+    expected = jnp.array([-1.0, -1.0, -1.0, 0.0])  # scale 1.0
+    test_utils.assert_trees_all_close(
+        caut_updates, expected, rtol=1e-6, atol=1e-6)
+
+
+class CautiousExportTest(absltest.TestCase):
+
+  def test_exported_from_contrib(self):
+    self.assertIs(contrib.cautious, cautious)
+    self.assertIs(contrib.CautiousState, CautiousState)
+
+
+if __name__ == '__main__':
+  absltest.main()
diff --git a/optax/contrib/_common_test.py b/optax/contrib/_common_test.py
index 510ef3159..cafab7c7e 100644
--- a/optax/contrib/_common_test.py
+++ b/optax/contrib/_common_test.py
@@ -96,6 +96,14 @@
         'wrapper_name': 'reduce_on_plateau',
         'wrapper_kwargs': {},
     },
+    {
+        # Momentum so the cautious mask is actually exercised (momentum can
+        # disagree with the current gradient); converges on all test targets.
+        'opt_name': 'sgd',
+        'opt_kwargs': {'learning_rate': 1e-3, 'momentum': 0.9},
+        'wrapper_name': 'cautious',
+        'wrapper_kwargs': {},
+    },
 ]
 
 # Adding here instantiations of wrappers with any base optimizer
diff --git a/optax/tree_utils/_tree_math_test.py b/optax/tree_utils/_tree_math_test.py
index bb73854a1..d6a92b700 100644
--- a/optax/tree_utils/_tree_math_test.py
+++ b/optax/tree_utils/_tree_math_test.py
@@ -126,7 +126,7 @@ def test_tree_add_scale_dtype(self):
   def test_tree_vdot(self):
     expected = jnp.vdot(self.array_a, self.array_b)
     got = tu.tree_vdot(self.array_a, self.array_b)
-    np.testing.assert_allclose(expected, got)
+    np.testing.assert_allclose(expected, got, rtol=1e-6)
 
     expected = 15.0
     got = tu.tree_vdot(self.tree_a_dict, self.tree_b_dict)