# Control autodiff’s saved values with jax.checkpoint (aka jax.remat)#

import jax
import jax.numpy as jnp


## TL;DR#

Use the jax.checkpoint decorator (aliased as jax.remat) with jax.grad to control which intermediates are saved on the forward pass versus recomputed on the backward pass, trading off memory and FLOPs.

Don’t miss the practical notes for a discussion about how jax.checkpoint interacts with jax.jit.

Without using jax.checkpoint, the forward pass of jax.grad(f)(x) saves, for use on the backward pass, the values of Jacobian coefficients and other intermediates. We call these saved values residuals:

def g(W, x):
y = jnp.dot(W, x)
return jnp.sin(y)

def f(W1, W2, W3, x):
x = g(W1, x)
x = g(W2, x)
x = g(W3, x)
return x

W1 = jnp.ones((5, 4))
W2 = jnp.ones((6, 5))
W3 = jnp.ones((7, 6))
x = jnp.ones(4)

# Inspect the 'residual' values to be saved on the forward pass
# if we were to evaluate jax.grad(f)(W1, W2, W3, x)

f32[5,4] from the argument 'W1'
f32[6,5] from the argument 'W2'
f32[7,6] from the argument 'W3'
f32 from the argument 'x'
f32 output of sin from <ipython-input-4-f510dde58e22>:3 (g)
f32 output of cos from <ipython-input-4-f510dde58e22>:3 (g)
f32 output of sin from <ipython-input-4-f510dde58e22>:3 (g)
f32 output of cos from <ipython-input-4-f510dde58e22>:3 (g)
f32 output of cos from <ipython-input-4-f510dde58e22>:3 (g)


By applying jax.checkpoint to sub-functions, as a decorator or at specific application sites, we force JAX not to save any of that sub-function’s residuals. Instead, only the inputs of a jax.checkpoint-decorated function might be saved, and any residuals consumed on the backward pass are re-computed from those inputs as needed:

def f2(W1, W2, W3, x):
x = jax.checkpoint(g)(W1, x)
x = jax.checkpoint(g)(W2, x)
x = jax.checkpoint(g)(W3, x)
return x


f32[5,4] from the argument 'W1'
f32[6,5] from the argument 'W2'
f32[7,6] from the argument 'W3'
f32 from the argument 'x'
f32 output of sin from <ipython-input-4-f510dde58e22>:3 (g)
f32 output of sin from <ipython-input-4-f510dde58e22>:3 (g)


Here the values of two sin applications are saved because they are arguments in subsequent applications of the jax.checkpoint-decorated g function, and inputs to a jax.checkpoint-decorated function may be saved. But no values of cos applications are saved.

To control which values are saveable without having to edit the definition of the function to be differentiated, you can use a rematerialization policy. Here is an example that saves only the results of dot operations with no batch dimensions (since they are often FLOP-bound, and hence worth saving rather than recomputing):

f3 = jax.checkpoint(f, policy=jax.checkpoint_policies.dots_with_no_batch_dims_saveable)

f32[5,4] from the argument 'W1'
f32[6,5] from the argument 'W2'
f32[7,6] from the argument 'W3'
f32 from the argument 'x'
f32 output of dot_general from <ipython-input-4-f510dde58e22>:2 (g)
f32 output of dot_general from <ipython-input-4-f510dde58e22>:2 (g)
f32 output of dot_general from <ipython-input-4-f510dde58e22>:2 (g)


You can also use policies to refer to intermediate values you name using jax.ad_checkpoint.checkpoint_name:

from jax.ad_checkpoint import checkpoint_name

def f4(W1, W2, W3, x):
x = checkpoint_name(g(W1, x), name='a')
x = checkpoint_name(g(W2, x), name='b')
x = checkpoint_name(g(W3, x), name='c')
return x

f4 = jax.checkpoint(f4, policy=jax.checkpoint_policies.save_only_these_names('a'))

f32[5,4] from the argument 'W1'
f32[6,5] from the argument 'W2'
f32[7,6] from the argument 'W3'
f32 from the argument 'x'
f32 named 'a' from <ipython-input-7-fc0ed1c14b8d>:4 (f4)


When playing around with these toy examples, we can get a closer look at what’s going on using the print_fwd_bwd utility defined in this notebook:

from jax.tree_util import tree_flatten, tree_unflatten

from rich.console import Console
from rich.table import Table
import rich.text

def print_fwd_bwd(f, *args, **kwargs) -> None:
args, in_tree = tree_flatten((args, kwargs))

def f_(*args):
args, kwargs = tree_unflatten(in_tree, args)
return f(*args, **kwargs)

fwd = jax.make_jaxpr(lambda *args: jax.vjp(f_, *args))(*args).jaxpr

y, f_vjp = jax.vjp(f_, *args)
res, in_tree = tree_flatten(f_vjp)

def g_(*args):
*res, y = args
f_vjp = tree_unflatten(in_tree, res)
return f_vjp(y)

bwd = jax.make_jaxpr(g_)(*res, y).jaxpr

"[bold green]backward computation:")
rich.text.Text.from_ansi(str(bwd)))
console = Console(width=240, force_jupyter=True)
console.print(table)

def _renderable_repr(self):
return self.html
rich.jupyter.JupyterRenderable._repr_html_ = _renderable_repr

# no use of jax.checkpoint:
print_fwd_bwd(f, W1, W2, W3, x)


forward computation:                                                        backward computation:

{ lambda ; a:f32[5,4] b:f32[6,5] c:f32[7,6] d:f32. let                   { lambda ; a:f32 b:f32 c:f32[7,6] d:f32 e:f32 f:f32[6,5] g:f32 h:f32
e:f32 = dot_general[dimension_numbers=((, ), ([], []))] a d        i:f32[5,4] j:f32. let
f:f32 = sin e                                                            k:f32 = mul j a
g:f32 = cos e                                                            l:f32 = dot_general[dimension_numbers=((, ), ([], []))] k c
h:f32 = dot_general[dimension_numbers=((, ), ([], []))] b f        m:f32[7,6] = dot_general[dimension_numbers=(([], []), ([], []))] k b
i:f32 = sin h                                                            n:f32 = mul l d
j:f32 = cos h                                                            o:f32 = dot_general[dimension_numbers=((, ), ([], []))] n f
k:f32 = dot_general[dimension_numbers=((, ), ([], []))] c i        p:f32[6,5] = dot_general[dimension_numbers=(([], []), ([], []))] n e
l:f32 = sin k                                                            q:f32 = mul o g
m:f32 = cos k                                                            r:f32 = dot_general[dimension_numbers=((, ), ([], []))] q i
in (l, m, i, c, j, f, b, g, d, a) }                                           s:f32[5,4] = dot_general[dimension_numbers=(([], []), ([], []))] q h
in (s, p, m, r) }

# using jax.checkpoint with policy=jax.checkpoint_policies.dots_with_no_batch_dims_saveable:
print_fwd_bwd(f3, W1, W2, W3, x)


forward computation:                                                        backward computation:

{ lambda ; a:f32[5,4] b:f32[6,5] c:f32[7,6] d:f32. let                   { lambda ; a:f32 b:f32 c:f32 d:f32[5,4] e:f32[6,5] f:f32[7,6] g:f32 h:f32. let
e:f32 = dot_general[dimension_numbers=((, ), ([], []))] a d        i:f32[5,4] j:f32[6,5] k:f32[7,6] l:f32 = remat2[
f:f32 = sin e                                                              differentiated=True
g:f32 = dot_general[dimension_numbers=((, ), ([], []))] b f          jaxpr={ lambda ; m:f32 n:f32 o:f32 p:f32[5,4] q:f32[6,5] r:f32[7,6]
h:f32 = sin g                                                                  s:f32 t:f32. let
i:f32 = dot_general[dimension_numbers=((, ), ([], []))] c h              u:f32 = sin m
j:f32 = sin i                                                                  v:f32 = cos m
in (j, e, g, i, a, b, c, d) }                                                       w:f32 = sin n
x:f32 = cos n
y:f32 = cos o
z:f32 = mul t y
ba:f32 = dot_general[dimension_numbers=((, ), ([], []))] z r
bb:f32 = mul ba x
bc:f32 = dot_general[dimension_numbers=((, ), ([], []))] bb q
bd:f32 = mul bc v
be:f32 = dot_general[dimension_numbers=((, ), ([], []))] bd p
bf:f32[5,4] = dot_general[dimension_numbers=(([], []), ([], []))] bd s
bg:f32[6,5] = dot_general[dimension_numbers=(([], []), ([], []))] bb u
bh:f32[7,6] = dot_general[dimension_numbers=(([], []), ([], []))] z w
in (bf, bg, bh, be) }
policy=<function dot_with_no_batch_dims at 0x7f5e469b1700>
prevent_cse=True
] a b c d e f g h
in (i, j, k, l) }


## Let’s think step by step#

You might want to first (re)read the Autodiff Cookbook Part 1.

### Fundamentals of jax.checkpoint#

In both jax.linearize and jax.vjp there is flexibility in how and when some values are computed. Different choices can trade off memory use against FLOPs. JAX provides control over these choices with jax.checkpoint.

One such choice is whether to perform Jacobian coefficient computations on the forward pass, as soon as the inputs are available, or on the backward pass, just before the coefficients are needed. Consider the example of sin_vjp:

def sin_vjp(x):
y = jnp.sin(x)
cos_x = jnp.cos(x)
return y, lambda y_bar: cos_x * y_bar


Another valid implementation would compute the value of jnp.cos(x) on the backward pass rather than on the forward pass:

def sin_vjp2(x):
y = jnp.sin(x)
return y, lambda y_bar: jnp.cos(x) * y_bar


For this particular function, the amount of memory used by the two versions is the same, though we’ve reduced the FLOPs for the primal computation (i.e. the forward pass) and increased the FLOPs for the cotangent computation (i.e. the backward pass).

There’s another choice when it comes to function composition. Recall our VJP rule for a composition of two functions:

def f(x):
y = g(x)
z = h(y)
return z

def f_vjp(x):
y, g_vjp = jax.vjp(g, x)
z, h_vjp = jax.vjp(h, y)
def f_bwd(z_bar):
y_bar, = h_vjp(z_bar)
x_bar, = g_vjp(y_bar)
return x_bar
return z, f_bwd


An alternative is:

def f_vjp_checkpoint(x):
y = g(x)
z, h_vjp = jax.vjp(h, y)
def f_bwd2(z_bar):
y_bar, = h_vjp(z_bar)
_, g_vjp = jax.vjp(g, x)
x_bar, = g_vjp(y_bar)
return x_bar
return z, f_bwd2


In words, this alternative implementation doesn’t compute g_vjp, or the residual values in its closure, on the forward pass. Instead it only computes them in the backward pass f_bwd2. That means f_vjp_checkpoint requires less memory: if g and h each required similar amounts of memory for their residuals, each much larger than x, then the function produced by f_vjp_checkpoint(x) requires half the memory as that of f_vjp(x)!

The cost we pay is redundant work: in f_bwd2 we must re-evaluate g(x) as part of jax.vjp(g, x) just to discard its value (in the underscore variable on the line _, g_vjp = jax.vjp(g, x)).

We can get this VJP behavior in autodiff � without having to write VJP functions directly � by instead using jax.checkpoint in an alternative definition of the original function f:

def f_checkpoint(x):
y = jax.checkpoint(g)(x)
z = h(y)
return z


In other words, we apply jax.checkpoint to g, the first stage of f, rather than to f itself. This way, when we evaluate jax.grad(f_checkpoint)(x), we’d get a computation like:

1. run the forward pass of g, discarding residual values;

2. run the forward pass of h, saving residuals;

3. run the backward pass of h, consuming residuals from step 2;

4. re-run the forward pass of g, saving residuals;

5. run the backward pass of g, consuming residuals from step 4.

That is, by evaluating jax.grad(f_checkpoint)(x) we’d get the same computation as:

def f_checkpoint_grad(x):
y = g(x)                  # step 1
_, h_vjp = jax.vjp(h)(y)  # step 2
y_bar, = h_vjp(1.0)       # step 3
_, g_vjp = jax.vjp(g, x)  # step 4
x_bar, = g_vjp(y_bar)     # step 5
return x_bar


In general, jax.checkpoint(foo) is a new function which has the same input-output behavior as foo, but behaves differently under autodiff, particularly under jax.linearize and jax.vjp (and their wrappers, like jax.grad) but not jax.jvp. When differentiated, only the input to a jax.checkpoint-differentiated function is stored on the forward pass; on the backward pass, residuals (i.e. intermediates from foo and its Jacobian coefficient values needed for the backward pass) are recomputed.

Notice that if f = lambda x: h(g(x)) is the function we want to differentiate, i.e. if we want to apply jax.grad(f), we don’t get any memory savings by applying jax.checkpoint to f itself. That’s because evaluating jax.grad(jax.checkpoint(f))(x) would lead to a computation like:

1. run the forward pass, discarding all residuals;

2. immediately re-run the forward pass, saving residuals;

3. run the backward pass, consuming residuals from step 2.

That is, in code we’d have something like:

def f_grad_bad(x):
_ = f(x)                  # step 1
_, f_vjp = jax.vjp(f, x)  # step 2
x_bar, = f_vjp(1.0)       # step 3
return x_bar


We also wouldn’t get any memory savings by applying jax.checkpoint to h, the second stage of f. That’s because evaluating jax.grad(lambda x: jax.checkpoint(h)(g(x))) would lead to a computation like:

1. run the forward pass of g, saving residuals;

2. run the forward pass of h, discarding residuals;

3. immediately re-run the forward pass of h, saving residuals;

4. run the backward pass of h, consuming residuals from step 3;

5. run the backward pass of g, consuming residuals from step 1.

That is, in code we’d have something like:

def f_grad_bad2(x):
y, g_vjp = jax.vjp(g, x)  # step 1
z = h(y)                  # step 2
_, h_vjp = jax.vjp(h, y)  # step 3
y_bar, = h_vjp(1.0)       # step 3
x_bar, = g_vjp(y_bar)     # step 5
return x_bar


Slightly more generally, if we had a chain composition of functions, like f = lambda x: f3(f2(f1(x))), and we were interested in evaluating jax.grad(f), we could say that:

• we shouldn’t apply jax.checkpoint to the whole function f, since that wouldn’t save any memory (and will perform wasteful recomputation);

• we shouldn’t apply jax.checkpoint to the last sub-function f3, since that wouldn’t save any memory (and will perform wasteful recomputation);

• we could apply jax.checkpoint to f1, f2, or their composition lambda x: f2(f1(x)), since any of those might save memory and would express different memory/recompute tradeoffs.

### Custom policies for what’s saveable#

As shown so far, using jax.checkpoint switches from one extreme to another:

• without jax.checkpoint, JAX’s autodiff tends to compute everything possible on the forward pass and store it for the backward pass;

• with a jax.checkpoint decorator, we instead compute as little as possible on the forward pass and recompute values as needed on the backward pass.

To operate between these two extremes, saving some things and not others, we can carefully place jax.checkpoint decorators on sub-functions. But that requires editing the function to be differentiated, e.g. model code, which may be inconvenient. It can also be hard to experiment with variations.

So an alternative is to use the policy argument to jax.checkpoint. A policy is a callable (i.e. a function) which takes as input a type-level specification of a first order primitive application and returns a boolean indicating whether the corresponding output value(s) are allowed to be saved as residuals (or instead must be recomputed in the (co)tangent computation as needed). To write robust code, a policy should be selected from the attributes on jax.checkpoint_policies, like jax.checkpoint_policies.dots_with_no_batch_dims_saveable, since the API for writing custom policy callables is considered internal.

For example, consider this function to be differentiated:

def loss(params, x, y):
return jnp.sum((predict(params, x) - y)**2)

def predict(params, x):
*Ws, Wlast = params
for W in Ws:
x = layer(W, x)
x = jnp.dot(Wlast, x)
return x

def layer(W, x):
return jnp.sin(jnp.dot(W, x))

W1 = W2 = W3 = jnp.ones((4, 4))
params = [W1, W2, W3]
x = jnp.ones(4)
y = jnp.ones(4)

print_saved_residuals(loss, params, x, y)

f32[4,4] from the argument 'params'
f32[4,4] from the argument 'params'
f32[4,4] from the argument 'params'
f32 from the argument 'x'
f32 output of sin from <ipython-input-18-3808b5023c3d>:12 (layer)
f32 output of cos from <ipython-input-18-3808b5023c3d>:12 (layer)
f32 output of sin from <ipython-input-18-3808b5023c3d>:12 (layer)
f32 output of cos from <ipython-input-18-3808b5023c3d>:12 (layer)
f32 output of mul from <ipython-input-18-3808b5023c3d>:2 (loss)


Instead of saving so many values on the forward pass, perhaps we only want to save the results of matrix multiplications with no batch dimension (since they may be FLOP- rather than memory-bound). We can do that using the policy jax.checkpoint_policies.dots_with_no_batch_dims_saveable:

loss_checkpoint = jax.checkpoint(loss, policy=jax.checkpoint_policies.dots_with_no_batch_dims_saveable)
print_saved_residuals(loss_checkpoint, params, x, y)

f32[4,4] from the argument 'params'
f32[4,4] from the argument 'params'
f32[4,4] from the argument 'params'
f32 from the argument 'x'
f32 from the argument 'y'
f32 output of dot_general from <ipython-input-18-3808b5023c3d>:12 (layer)
f32 output of dot_general from <ipython-input-18-3808b5023c3d>:12 (layer)
f32 output of dot_general from <ipython-input-18-3808b5023c3d>:8 (predict)


Notice also that by providing a policy, we didn’t need to edit the code defining loss, predict, or layer. That is particularly convenient if we want to experiment with policies in calling code (e.g. a training script) without changing library code (e.g. the neural network library).

Some policies can refer to values named with jax.ad_checkpoint.checkpoint_name:

from jax.ad_checkpoint import checkpoint_name

def predict(params, x):
*Ws, Wlast = params
for i, W in enumerate(Ws):
x = layer(W, x)
x = checkpoint_name(x, name=f'layer{i}_output')
x = jnp.dot(Wlast, x)
return x


By itself, checkpoint_name is just an identity function. But because some policy functions know to look for them, we can use the names to control whether certain values output by checkpoint_name are considered saveable:

print_saved_residuals(loss, params, x, y)

f32[4,4] from the argument 'params'
f32[4,4] from the argument 'params'
f32[4,4] from the argument 'params'
f32 from the argument 'x'
f32 output of cos from <ipython-input-18-3808b5023c3d>:12 (layer)
f32 named 'layer0_output' from <ipython-input-22-e48aedf368ad>:7 (predict)
f32 output of cos from <ipython-input-18-3808b5023c3d>:12 (layer)
f32 named 'layer1_output' from <ipython-input-22-e48aedf368ad>:7 (predict)
f32 output of mul from <ipython-input-18-3808b5023c3d>:2 (loss)

loss_checkpoint2 = jax.checkpoint(loss, policy=jax.checkpoint_policies.save_any_names_but_these('layer1_output'))
print_saved_residuals(loss_checkpoint2, params, x, y)

f32[4,4] from the argument 'params'
f32[4,4] from the argument 'params'
f32[4,4] from the argument 'params'
f32 from the argument 'x'
f32 from the argument 'y'


Another policy which refers to names is jax.checkpoint_policies.save_only_these_names.

Some of the policies are:

• everything_saveable (the default strategy, as if jax.checkpoint were not being used at all)

• nothing_saveable (i.e. rematerialize everything, as if a custom policy were not being used at all)

• dots_saveable or its alias checkpoint_dots

• dots_with_no_batch_dims_saveable or its alias checkpoint_dots_with_no_batch_dims

• save_anything_but_these_names (save any values except for the output of checkpoint_name with any of the names given)

• save_any_names_but_these (save only named values, i.e. any outputs of checkpoint_name, except for those with the names given)

• save_only_these_names (save only named values, and only among the names given)

Policies only indicate what is saveable; a value is only saved if it’s actually needed by the backward pass.

### Advanced: recursive jax.checkpoint#

By applying jax.checkpoint in the right way, there are many tradeoffs between memory usage and (re)computation that can be expressed. One surprising example is recursive checkpointing, where we apply jax.checkpoint to a function which itself calls jax.checkpoint-decorated functions in a way so that memory usage from the chain composition of $$D$$ functions scales like $$\mathcal{O}(\log_2 D)$$ rather than $$\mathcal{O}(D)$$.

As a toy example, consider the chain composition of multiple jnp.sin functions:

def chain_compose(funs):
def f(x):
for fun in funs:
x = fun(x)
return x
return f

f = chain_compose([jnp.sin] * 8)
print_saved_residuals(f, 3.)

f32[] output of cos from <ipython-input-25-46b5594773cb>:4 (f)
f32[] output of cos from <ipython-input-25-46b5594773cb>:4 (f)
f32[] output of cos from <ipython-input-25-46b5594773cb>:4 (f)
f32[] output of cos from <ipython-input-25-46b5594773cb>:4 (f)
f32[] output of cos from <ipython-input-25-46b5594773cb>:4 (f)
f32[] output of cos from <ipython-input-25-46b5594773cb>:4 (f)
f32[] output of cos from <ipython-input-25-46b5594773cb>:4 (f)
f32[] output of cos from <ipython-input-25-46b5594773cb>:4 (f)


In general, the number of stored residuals scales linearly with the length of the chain:

f = chain_compose([jnp.sin] * 16)
print_saved_residuals(f, 3.)

f32[] output of cos from <ipython-input-25-46b5594773cb>:4 (f)
f32[] output of cos from <ipython-input-25-46b5594773cb>:4 (f)
f32[] output of cos from <ipython-input-25-46b5594773cb>:4 (f)
f32[] output of cos from <ipython-input-25-46b5594773cb>:4 (f)
f32[] output of cos from <ipython-input-25-46b5594773cb>:4 (f)
f32[] output of cos from <ipython-input-25-46b5594773cb>:4 (f)
f32[] output of cos from <ipython-input-25-46b5594773cb>:4 (f)
f32[] output of cos from <ipython-input-25-46b5594773cb>:4 (f)
f32[] output of cos from <ipython-input-25-46b5594773cb>:4 (f)
f32[] output of cos from <ipython-input-25-46b5594773cb>:4 (f)
f32[] output of cos from <ipython-input-25-46b5594773cb>:4 (f)
f32[] output of cos from <ipython-input-25-46b5594773cb>:4 (f)
f32[] output of cos from <ipython-input-25-46b5594773cb>:4 (f)
f32[] output of cos from <ipython-input-25-46b5594773cb>:4 (f)
f32[] output of cos from <ipython-input-25-46b5594773cb>:4 (f)
f32[] output of cos from <ipython-input-25-46b5594773cb>:4 (f)


But we can apply jax.checkpoint recursively to improve the scaling:

def recursive_checkpoint(funs):
if len(funs) == 1:
return funs
elif len(funs) == 2:
f1, f2 = funs
return lambda x: f1(f2(x))
else:
f1 = recursive_checkpoint(funs[:len(funs)//2])
f2 = recursive_checkpoint(funs[len(funs)//2:])
return lambda x: f1(jax.checkpoint(f2)(x))

f = recursive_checkpoint([jnp.sin] * 8)
print_saved_residuals(f, 3.)

f32[] from the argument 'x'
f32[] output of sin from <ipython-input-27-86f83c871e81>:6 (<lambda>)
f32[] output of cos from <ipython-input-27-86f83c871e81>:6 (<lambda>)
f32[] output of cos from <ipython-input-27-86f83c871e81>:6 (<lambda>)

f = recursive_checkpoint([jnp.sin] * 16)
print_saved_residuals(f, 3.)

f32[] from the argument 'x'
f32[] output of sin from <ipython-input-27-86f83c871e81>:6 (<lambda>)
f32[] output of sin from <ipython-input-27-86f83c871e81>:6 (<lambda>)
f32[] output of cos from <ipython-input-27-86f83c871e81>:6 (<lambda>)
f32[] output of cos from <ipython-input-27-86f83c871e81>:6 (<lambda>)


The cost here, as usual, is recomputation: in particular, we end up performing $$\mathcal{O}(\log_2 D)$$ times as many FLOPs:

f = chain_compose([jnp.sin] * 8)
print_fwd_bwd(f, 3.)


forward computation:                  backward computation:

{ lambda ; a:f32[]. let               { lambda ; a:f32[] b:f32[] c:f32[] d:f32[] e:f32[] f:f32[] g:f32[] h:f32[] i:f32[]. let
b:f32[] = sin a                       j:f32[] = mul i a
c:f32[] = cos a                       k:f32[] = mul j b
d:f32[] = sin b                       l:f32[] = mul k c
e:f32[] = cos b                       m:f32[] = mul l d
f:f32[] = sin d                       n:f32[] = mul m e
g:f32[] = cos d                       o:f32[] = mul n f
h:f32[] = sin f                       p:f32[] = mul o g
i:f32[] = cos f                       q:f32[] = mul p h
j:f32[] = sin h                     in (q,) }
k:f32[] = cos h
l:f32[] = sin j
m:f32[] = cos j
n:f32[] = sin l
o:f32[] = cos l
p:f32[] = sin n
q:f32[] = cos n
in (p, q, o, m, k, i, g, e, c) }

f = recursive_checkpoint([jnp.sin] * 8)
print_fwd_bwd(f, 3.)


forward computation:                                                              backward computation:

{ lambda ; a:f32[]. let                                                           { lambda ; a:f32[] b:f32[] c:f32[] d:f32[]. let
b:f32[] = remat2[                                                                 e:f32[] = mul d a
differentiated=False                                                            f:f32[] = mul e b
jaxpr={ lambda ; c:f32[]. let d:f32[] = sin c; e:f32[] = sin d in (e,) }        g:f32[] = remat2[
policy=None                                                                       differentiated=True
prevent_cse=True                                                                  jaxpr={ lambda ; h:f32[] i:f32[]. let
] a                                                                                     j:f32[] = sin h
f:f32[] = sin b                                                                         k:f32[] = cos h
g:f32[] = sin f                                                                         l:f32[] = cos j
h:f32[] = sin g                                                                         m:f32[] = mul i l
i:f32[] = sin h                                                                         n:f32[] = mul m k
j:f32[] = sin i                                                                       in (n,) }
k:f32[] = cos i                                                                     policy=None
l:f32[] = sin j                                                                     prevent_cse=True
m:f32[] = cos j                                                                   ] c f
in (l, m, k, g, a) }                                                                o:f32[] = remat2[
differentiated=True
jaxpr={ lambda ; p:f32[] q:f32[]. let
r:f32[] = sin p
s:f32[] = sin r
t:f32[] = sin s
u:f32[] = cos s
v:f32[] = cos t
w:f32[] = mul q v
x:f32[] = mul w u
y:f32[] = remat2[
differentiated=True
jaxpr={ lambda ; z:f32[] ba:f32[]. let
bb:f32[] = sin z
bc:f32[] = cos z
bd:f32[] = cos bb
be:f32[] = mul ba bd
bf:f32[] = mul be bc
in (bf,) }
policy=None
prevent_cse=True
] p x
in (y,) }
policy=None
prevent_cse=True
] 3.0 g
in (o,) }


## Practical notes#

When differentiated functions are staged out to XLA for compilation, for example by applying jax.jit to a function which contains a jax.grad call, XLA will automatically optimize the computation, including decisions about when to compute or rematerialize values. As a result, jax.checkpoint often isn’t needed for differentiated functions under a jax.jit. XLA will optimize things for you.

One exception is when using staged-out control flow, like jax.lax.scan. Automatic compiler optimizations across multiple control flow primitives, e.g. across a forward-pass scan and the corresponding backward-pass scan, typically aren’t aren’t as thorough. As a result, it’s often a good idea to use jax.checkpoint on the body function passed to jax.lax.scan.

For example, one common pattern in large Transformer models is to express the architecture as a jax.lax.scan over layers so as to reduce compilation times. That is, using a simple fully-connected network as an analogy, instead of writing something like this:

LayerParam = tuple[jnp.ndarray, jnp.ndarray]  # weights, bias pair for a layer
ParamsList = list[LayerParam]

def net(params: ParamsList, x: jnp.ndarray):
for W, b in params:
x = jnp.maximum(jnp.dot(x, W) + b, 0.)
return x


We would instead iterate over the layer application with jax.lax.scan:

StackedWeights = jnp.ndarray  # all weight matrices stacked together
StackedBiases = jnp.ndarray   # all bias vectors stacked together

all_weights = jnp.stack([W for W, _ in params])
all_biases = jnp.stack([b for _, b in params])

def layer(x, W_b_pair):
W, b = W_b_pair
out = jnp.maximum(jnp.dot(x, W) + b, 0.)
return out, None

def net(all_weights, all_biases, x):
x, _ = jax.lax.scan(layer, x, (all_weights, all_biases))
return x


This scan-over-layers version reduces compile times, but by foiling some compiler optimizations it can lead to inefficient computation of gradients. To mitigate the issue, we would use jax.checkpoint on the scanned function:

from functools import partial

@partial(jax.checkpoint,
policy=jax.checkpoint_policies.dots_with_no_batch_dims_saveable)
def layer(x, W_b_pair):
W, b = W_b_pair
out = jnp.maximum(jnp.dot(x, W) + b, 0.)
return out, None


By using jax.checkpoint this way, we’re manually controlling which values JAX’s autodiff saves between the forward and backward passes, and hence not relying on XLA optimizations to choose for us.