diff --git a/examples/dp2_delayed_preconditioner_example.py b/examples/dp2_delayed_preconditioner_example.py
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+# 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.
+# ==============================================================================
+"""Example of a private adaptive optimizer with a Delayed Preconditioner (DP^2).
+
+This example is based on the paper:
+  Li et al. (2023), "Differentially Private Adaptive Optimization with Delayed
+  Preconditioners" (ICLR 2023).
+
+Implementation Notes:
+  This implementation demonstrates the "alternating-phase protocol" of DP^2.
+  The core idea is to introduce a trade-off between the staleness of the
+  preconditioner and noise reduction. The preconditioner (second-moment
+  estimate) is kept "stale" for a fixed number of steps (`delay_s`). During
+  this period, noised gradients are accumulated.
+
+  When the delay period is over, the preconditioner is updated using the
+  *average* of the accumulated gradients. By averaging over multiple steps,
+  the signal from the gradients is amplified relative to the noise, leading
+  to a more stable and reliable preconditioner update. This helps to prevent
+  the noise from overwhelming the adaptive learning rate calculation, a common
+  problem in standard private adaptive optimizers.
+"""
+
+import jax
+import jax.numpy as jnp
+import flax.linen as nn
+from jax_privacy.clipping import clipped_grad
+from jax_privacy import noise_addition
+import optax
+import numpy as np
+
+# 1. Model and Data Setup
+class SimpleModel(nn.Module):
+  """A simple linear model."""
+  @nn.compact
+  def __call__(self, x):
+    return nn.Dense(features=1)(x)
+
+# Dummy data
+inputs = jnp.array([[1.0, 2.0], [3.0, 4.0]])
+labels = jnp.array([[0.5], [1.5]])
+
+model = SimpleModel()
+key = jax.random.PRNGKey(42)
+params = model.init(key, inputs)['params']
+epsilon_s1 = 1e-8
+
+# 2. Optimizer and DP^2 State
+optimizer = optax.sgd(learning_rate=0.1)
+opt_state = optimizer.init(params)
+
+# DP^2 parameters
+delay_s = 5
+preconditioner_v = jax.tree_util.tree_map(jnp.ones_like, params)
+gradient_accumulator = jax.tree_util.tree_map(jnp.zeros_like, params)
+
+# Privatizer for noise addition
+stddev = 1.0 * 0.1 # l2_clip_norm * noise_multiplier
+privatizer = noise_addition.gaussian_privatizer(
+    stddev=stddev,
+    prng_key=jax.random.PRNGKey(0),
+)
+noise_state = privatizer.init(params)
+
+# 3. Core Logic: DP^2 Update Step
+def loss_fn(p, x, y):
+  """MSE loss."""
+  return jnp.mean((model.apply({'params': p}, x) - y)**2)
+
+def update_step(
+    step_idx, params, opt_state, preconditioner_v, gradient_accumulator, noise_state
+):
+  """Performs one update step with delayed preconditioning."""
+  
+  # Scale gradients using the (potentially stale) preconditioner
+  s_t = jax.tree_util.tree_map(
+      lambda var: 1 / (jnp.sqrt(var) + epsilon_s1), preconditioner_v
+  )
+  def pre_clipping_transform(grads):
+    return jax.tree_util.tree_map(lambda g, s: g * s, grads, s_t)
+
+  # Compute and clip gradients
+  priv_grad_fn = clipped_grad(
+      loss_fn,
+      l2_clip_norm=1.0,
+      batch_argnums=(1, 2),
+      pre_clipping_transform=pre_clipping_transform,
+  )
+  clipped_grads = priv_grad_fn(params, inputs, labels)
+
+  # Add noise
+  noised_grads, new_noise_state = privatizer.update(
+      clipped_grads, noise_state
+  )
+  
+  # Rescale noised gradients back to original space
+  noised_grads_rescaled = jax.tree_util.tree_map(
+      lambda g, s: g / s, noised_grads, s_t
+  )
+
+  # Apply updates to parameters
+  updates, new_opt_state = optimizer.update(noised_grads_rescaled, opt_state, params)
+  new_params = optax.apply_updates(params, updates)
+
+  # --- DP^2 Preconditioner Update Logic ---
+  # Accumulate the noised (but not rescaled) gradients
+  new_gradient_accumulator = jax.tree_util.tree_map(
+      lambda acc, g: acc + g, gradient_accumulator, noised_grads
+  )
+
+  def update_preconditioner(operand):
+    """Update preconditioner with averaged gradients and reset accumulator."""
+    p_v, grad_acc = operand
+    # By delaying the update, we ensure that the second-moment estimate is
+    # derived from a higher signal-to-noise ratio average, preventing the
+    # noise from dominating the adaptive learning rates.
+    avg_grad = jax.tree_util.tree_map(lambda x: x / delay_s, grad_acc)
+    new_p_v = jax.tree_util.tree_map(
+        lambda v, g: 0.9 * v + 0.1 * jnp.square(g), p_v, avg_grad
+    )
+    new_grad_acc = jax.tree_util.tree_map(jnp.zeros_like, grad_acc)
+    return new_p_v, new_grad_acc
+
+  def keep_stale(operand):
+    """Keep preconditioner stale and continue accumulating."""
+    return operand
+
+  # "Snap" update: conditionally update the preconditioner
+  new_preconditioner_v, new_gradient_accumulator = jax.lax.cond(
+      (step_idx + 1) % delay_s == 0,
+      update_preconditioner,
+      keep_stale,
+      (preconditioner_v, new_gradient_accumulator),
+  )
+
+  return (
+      new_params,
+      new_opt_state,
+      new_preconditioner_v,
+      new_gradient_accumulator,
+      new_noise_state,
+  )
+
+# 4. Training Loop & Verification
+v_initial = preconditioner_v
+snapshots = {}
+num_steps = 20
+
+for i in range(num_steps):
+  (
+      params,
+      opt_state,
+      preconditioner_v,
+      gradient_accumulator,
+      noise_state,
+  ) = update_step(
+      i, params, opt_state, preconditioner_v, gradient_accumulator, noise_state
+  )
+  if i == 3: # Step 4 (0-indexed)
+    snapshots['v_step4'] = preconditioner_v
+  if i == 4: # Step 5 (0-indexed)
+    snapshots['v_step5'] = preconditioner_v
+
+# 5. Correctness Verification
+def flatten_pytree(pt):
+  return np.concatenate([np.ravel(x) for x in jax.tree_util.tree_leaves(pt)])
+
+v_initial_flat = flatten_pytree(v_initial)
+v_step4_flat = flatten_pytree(snapshots['v_step4'])
+v_step5_flat = flatten_pytree(snapshots['v_step5'])
+
+# Assert that the preconditioner was stale until the update step
+np.testing.assert_allclose(v_initial_flat, v_step4_flat, rtol=1e-6)
+assert not np.allclose(v_step4_flat, v_step5_flat), "Preconditioner did not update at step 5."
+
+print(
+    "[Verification] DP2 Protocol confirmed: Preconditioner remained stale for"
+    " steps 1-4 and updated successfully at step 5."
+)