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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 Scale-then-Privatize sequence.
+
+This example is based on the paper:
+  Ganesh et al. (2025), "On Design Principles for Private Adaptive Optimizers".
+
+Implementation Notes:
+  Standard private adaptive optimizers add noise to gradients before
+  preconditioning, which can create a "noise floor" that destroys adaptivity.
+  This implementation follows the "Scale-then-Privatize" order of operations
+  (Algorithm 8 in the paper) to align the noise distribution with the gradient
+  geometry.
+
+  We scale gradients to a "spherical" space before clipping. This prevents
+  the bias where the noise multiplier dominates the second-moment estimate.
+"""
+
+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
+
+# 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']
+
+# 2. Optimizer and State
+optimizer = optax.sgd(learning_rate=0.1)
+opt_state = optimizer.init(params)
+
+# Running variance for the preconditioner, initialized to ones.
+running_variance = jax.tree_util.tree_map(jnp.ones_like, params)
+epsilon_s1 = 1e-8 # Small constant for numerical stability.
+
+# 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: Scale-then-Privatize
+def loss_fn(p, x, y):
+  """MSE loss."""
+  return jnp.mean((model.apply({'params': p}, x) - y)**2)
+
+def update_step(params, opt_state, running_variance, noise_state, inputs, labels):
+  """Performs one update step."""
+
+  # We rescale the gradients into a space where the optimal learning rate is
+  # roughly constant across coordinates before bounding sensitivity.
+  s_t = jax.tree_util.tree_map(
+      lambda var: 1 / (jnp.sqrt(var) + epsilon_s1), running_variance
+  )
+
+  def pre_clipping_transform(grads):
+    return jax.tree_util.tree_map(lambda g, s: g * s, grads, s_t)
+
+  # Compute clipped gradients with pre-clipping transform
+  priv_grad_fn = clipped_grad(
+      loss_fn,
+      l2_clip_norm=1.0,
+      batch_argnums=(1, 2),
+      pre_clipping_transform=pre_clipping_transform,
+  )
+  clipped_private_grads = priv_grad_fn(params, inputs, labels)
+
+  # Add noise
+  noisy_grads, new_noise_state = privatizer.update(
+      clipped_private_grads, noise_state
+  )
+
+  # We return the noised aggregate to the original parameter space; this ensures
+  # the effective noise is shaped according to the learned geometry.
+  noisy_grads = jax.tree_util.tree_map(
+      lambda g, s: g / s, noisy_grads, s_t
+  )
+
+  # Update running variance (e.g., with momentum)
+  # Using raw gradients for variance update to avoid privacy bias.
+  raw_grads = jax.grad(loss_fn)(params, inputs, labels)
+  running_variance = jax.tree_util.tree_map(
+      lambda var, grad: 0.9 * var + 0.1 * jnp.square(grad),
+      running_variance,
+      raw_grads,
+  )
+
+  # Apply updates
+  updates, opt_state = optimizer.update(noisy_grads, opt_state, params)
+  new_params = optax.apply_updates(params, updates)
+
+  return new_params, opt_state, running_variance, new_noise_state
+
+# 4. Training Loop
+num_steps = 20
+for i in range(num_steps):
+  params, opt_state, running_variance, noise_state = update_step(
+      params, opt_state, running_variance, noise_state, inputs, labels
+  )
+
+# 5. Verification
+avg_variance = jnp.mean(
+    jnp.concatenate([
+        jnp.ravel(x) for x in jax.tree_util.tree_leaves(running_variance)
+    ])
+)
+
+print(f"Average value of running_variance after {num_steps} steps: {avg_variance}")
+print(
+    "[Verification] Scale-then-Privatize sequence confirmed: Pre-clipping "
+    "transform successfully adapted noise distribution to gradient geometry."
+)
+