consolidate the code example

docs/gradient-checkpointing.md

@@ -360,40 +360,55 @@ You may consider offloading to CPU memory instead of recomputing when checkpoint
 
 ```{code-cell}
 from jax.ad_checkpoint import checkpoint
-    policy = jax.checkpoint_policies.offload_dot_with_no_batch_dims(
-        "device", "pinned_host")
 
-    @functools.partial(checkpoint, policy=policy)
-    def f(x):
-      x = jnp.einsum('ij,jk->ik', x, x, precision=lax.Precision.HIGHEST)
-      x = jnp.sin(x)
-      x = jnp.einsum('ij,jk->ik', x, x, precision=lax.Precision.HIGHEST)
-      x = jnp.sin(x)
-      x = jnp.einsum('ij,jk->ik', x, x, precision=lax.Precision.HIGHEST)
-      x = jnp.sin(x)
-      x = jnp.sum(x)
-      return x
+def checkpoint_offload_dot_with_no_batch_dims(self):
+  policy = jax.checkpoint_policies.offload_dot_with_no_batch_dims(
+      "device", "pinned_host")
+
+  @functools.partial(checkpoint, policy=policy)
+  def f(x):
+    x = jnp.einsum('ij,jk->ik', x, x, precision=lax.Precision.HIGHEST)
+    x = jnp.sin(x)
+    x = jnp.einsum('ij,jk->ik', x, x, precision=lax.Precision.HIGHEST)
+    x = jnp.sin(x)
+    x = jnp.einsum('ij,jk->ik', x, x, precision=lax.Precision.HIGHEST)
+    x = jnp.sin(x)
+    x = jnp.sum(x)
+    return x
 ```
 
 One of JAX's checkpoint policies allows specified checkpoint names to be offloaded to CPUs. This policy is implemented through `jax.checkpoint_policies.save_and_offload_only_these_names`, which has four arguments: `names_which_can_be_saved`, `names_which_can_be_offloaded`, the offloading source, and destination. Names listed in `names_which_can_be_saved` are kept on the device, names listed in `names_which_can_be_offloaded` are moved to CPU memory, and other names or operations without names are recomputed. For example, if we have checkpoint names `y`, `z`, and `w`, `y` can be saved on the device, `z` can be offloaded to CPU memory, and `w` can be recomputed.
 
 ```{code-cell}
 from jax.ad_checkpoint import checkpoint, checkpoint_name
+from jax._src import test_util as jtu
 
-def g(self):
+def checkpoint_names_saved_offloaded_recomputed(self):
+  mesh = jtu.create_mesh((2,), ("x",))
+  shape = (256, 128)
+  np_inp = np.arange(math.prod(shape), dtype=np.float32).reshape(shape)
+  s = NamedSharding(mesh, P("x"))
+  inp = jax.device_put(np_inp, s)
 
-    policy = jax.checkpoint_policies.save_and_offload_only_these_names(
-        names_which_can_be_saved=["y"], names_which_can_be_offloaded=["z"],
-        offload_src='device', offload_dst='pinned_host')
+  policy = jax.checkpoint_policies.save_and_offload_only_these_names(
+      names_which_can_be_saved=["y"], names_which_can_be_offloaded=["z"],
+      offload_src='device', offload_dst='pinned_host')
 
-    @functools.partial(checkpoint, policy=policy)
-    def f(x):
+  @functools.partial(checkpoint, policy=policy)
+  def f(x):
+    def g(ys, _):
+      y, _ = ys
       y = checkpoint_name(jnp.sin(y), "y")
       z = checkpoint_name(jnp.sin(y), "z")
+      z = z.T
       w = checkpoint_name(jnp.sin(z), "w")
-      return jnp.sum(w)
+      return (w.T, jnp.sum(w)), None
+    _, scan_out = jax.lax.scan(g, (x, np.array(1, dtype=np.float32)), [np_inp])[0]
+    return scan_out
 ```
 
+The code defines a function `f` that which applies checkpointing with a custom policy. This policy determines which computations can be saved or offloaded during execution. Inside `f`, there is a nested function `g` that performs the core computations. The `jax.lax.scan` function is used to apply `g` repeatedly over the input data.
+
 #### List of policies
 
 The policies are: