Custom JVPs or VJPs with custom_jvp and custom_vjp#
There are two ways to define differentiation rules in JAX:
using
jax.custom_jvpandjax.custom_vjpto define custom differentiation rules for Python functions that are already JAX-transformable; anddefining new
core.Primitiveinstances along with all their transformation rules, for example to call into functions from other systems like solvers, simulators, or general numerical computing systems.
This notebook is about #1. To read instead about #2, see Custom derivative rules with hijax primitives and Defining new JAX types with hijax.
Hijax primitives (see Custom derivative rules with hijax primitives) unify these two approaches, and are the recommended path going forward: a hijax primitive carries a JAX-traceable Python implementation along with custom rules for differentiation and other transformations. The APIs here remain fully supported, and are often the most convenient for simple cases.
For an introduction to JAX’s automatic differentiation API, see The Autodiff Cookbook with JVP and VJP. This notebook assumes some familiarity with jax.jvp and jax.grad, and the mathematical meaning of JVPs and VJPs.
TL;DR: Custom JVPs with jax.custom_jvp#
import jax.numpy as jnp
from jax import custom_jvp
@custom_jvp
def f(x, y):
return jnp.sin(x) * y
@f.defjvp
def f_jvp(primals, tangents):
x, y = primals
x_dot, y_dot = tangents
primal_out = f(x, y)
tangent_out = jnp.cos(x) * x_dot * y + jnp.sin(x) * y_dot
return primal_out, tangent_out
from jax import jvp, grad
print(f(2., 3.))
y, y_dot = jvp(f, (2., 3.), (1., 0.))
print(y)
print(y_dot)
print(grad(f)(2., 3.))
2.7278922
2.7278922
-1.2484405
-1.2484405
# Equivalent alternative using the defjvps convenience wrapper
@custom_jvp
def f(x, y):
return jnp.sin(x) * y
f.defjvps(lambda x_dot, primal_out, x, y: jnp.cos(x) * x_dot * y,
lambda y_dot, primal_out, x, y: jnp.sin(x) * y_dot)
print(f(2., 3.))
y, y_dot = jvp(f, (2., 3.), (1., 0.))
print(y)
print(y_dot)
print(grad(f)(2., 3.))
2.7278922
2.7278922
-1.2484405
-1.2484405
TL;DR: Custom VJPs with jax.custom_vjp#
from jax import custom_vjp
@custom_vjp
def f(x, y):
return jnp.sin(x) * y
def f_fwd(x, y):
# Returns primal output and residuals to be used in backward pass by f_bwd.
return f(x, y), (jnp.cos(x), jnp.sin(x), y)
def f_bwd(res, g):
cos_x, sin_x, y = res # Gets residuals computed in f_fwd
return (cos_x * g * y, sin_x * g)
f.defvjp(f_fwd, f_bwd)
print(grad(f)(2., 3.))
-1.2484405
Example problems#
To get an idea of what problems jax.custom_jvp and jax.custom_vjp are meant to solve, let’s go over one example for each — a custom_jvp for numerical stability, and a custom_vjp for gradient clipping. (More example problems, including enforcing a differentiation convention at a boundary and efficient implicit differentiation of fixed points, are worked in Custom derivative rules with hijax primitives; each translates directly to these APIs.) A more thorough introduction to the jax.custom_jvp and jax.custom_vjp APIs is in the next section.
Example: Numerical stability#
One application of jax.custom_jvp is to improve the numerical stability of differentiation.
Say we want to write a function called log1pexp, which computes \(x \mapsto \log ( 1 + e^x )\). We can write that using jax.numpy:
def log1pexp(x):
return jnp.log(1. + jnp.exp(x))
log1pexp(3.)
Array(3.0485873, dtype=float32, weak_type=True)
Since it’s written in terms of jax.numpy, it’s JAX-transformable:
from jax import jit, grad, vmap
print(jit(log1pexp)(3.))
print(jit(grad(log1pexp))(3.))
print(vmap(jit(grad(log1pexp)))(jnp.arange(3.)))
3.0485873
0.95257413
[0.5 0.7310586 0.8807971]
But there’s a numerical stability problem lurking here:
print(grad(log1pexp)(100.))
nan
That doesn’t seem right: after all, the derivative of \(x \mapsto \log (1 + e^x)\) is \(x \mapsto \frac{e^x}{1 + e^x}\), and so for large values of \(x\) we’d expect the value to be about 1.
We can get a bit more insight into what’s going on by looking at the jaxpr for the gradient computation:
jit(grad(log1pexp)).trace(100.).jaxpr
{ lambda ; a:f32[]. let
b:f32[] = exp a
c:f32[] = add 1.0:f32[] b
_:f32[] = log c
d:f32[] = div 1.0:f32[] c
e:f32[] = mul d b
in (e,) }
Stepping through how the jaxpr would be evaluated, we can see that the last line would involve multiplying values that floating point math will round to 0 and \(\infty\), respectively, which is never a good idea. That is, we’re effectively evaluating lambda x: (1 / (1 + jnp.exp(x))) * jnp.exp(x) for large x, which effectively turns into 0. * jnp.inf.
Instead of generating such large and small values, hoping for a cancellation that floats can’t always provide, we’d rather just express the derivative function as a more numerically stable program. In particular, we can write a program that more closely evaluates the equal mathematical expression \(1 - \frac{1}{1 + e^x}\), with no cancellation in sight.
This problem is interesting because even though our definition of log1pexp could already be JAX-differentiated (and transformed with jit, vmap, …), we’re not happy with the result of applying standard autodiff rules to the primitives comprising log1pexp and composing the result. Instead, we’d like to specify how the whole function log1pexp should be differentiated, as a unit, and thus arrange those exponentials better.
This is one application of custom derivative rules for Python functions that are already JAX transformable: specifying how a composite function should be differentiated, while still using its original Python definition for other transformations (like jit, vmap, …).
Here’s a solution using jax.custom_jvp:
from jax import custom_jvp
@custom_jvp
def log1pexp(x):
return jnp.log(1. + jnp.exp(x))
@log1pexp.defjvp
def log1pexp_jvp(primals, tangents):
x, = primals
x_dot, = tangents
ans = log1pexp(x)
ans_dot = (1 - 1/(1 + jnp.exp(x))) * x_dot
return ans, ans_dot
print(grad(log1pexp)(100.))
1.0
print(jit(log1pexp)(3.))
print(jit(grad(log1pexp))(3.))
print(vmap(jit(grad(log1pexp)))(jnp.arange(3.)))
3.0485873
0.95257413
[0.5 0.7310586 0.8807971]
Here’s a defjvps convenience wrapper to express the same thing:
@custom_jvp
def log1pexp(x):
return jnp.log(1. + jnp.exp(x))
log1pexp.defjvps(lambda t, ans, x: (1 - 1/(1 + jnp.exp(x))) * t)
print(grad(log1pexp)(100.))
print(jit(log1pexp)(3.))
print(jit(grad(log1pexp))(3.))
print(vmap(jit(grad(log1pexp)))(jnp.arange(3.)))
1.0
3.0485873
0.95257413
[0.5 0.7310586 0.8807971]
Example: Gradient clipping#
While in some cases we want to express a mathematical differentiation computation, in other cases we may even want to take a step away from mathematics to adjust the computation autodiff performs. One canonical example is reverse-mode gradient clipping.
For gradient clipping, we can use jnp.clip together with a jax.custom_vjp reverse-mode-only rule:
from functools import partial
from jax import custom_vjp
@custom_vjp
def clip_gradient(lo, hi, x):
return x # identity function
def clip_gradient_fwd(lo, hi, x):
return x, (lo, hi) # save bounds as residuals
def clip_gradient_bwd(res, g):
lo, hi = res
return (None, None, jnp.clip(g, lo, hi)) # use None to indicate zero cotangents for lo and hi
clip_gradient.defvjp(clip_gradient_fwd, clip_gradient_bwd)
import matplotlib.pyplot as plt
from jax import vmap
t = jnp.linspace(0, 10, 1000)
plt.plot(jnp.sin(t))
plt.plot(vmap(grad(jnp.sin))(t))
[<matplotlib.lines.Line2D at 0x76c2f92c50a0>]
def clip_sin(x):
x = clip_gradient(-0.75, 0.75, x)
return jnp.sin(x)
plt.plot(clip_sin(t))
plt.plot(vmap(grad(clip_sin))(t))
[<matplotlib.lines.Line2D at 0x76c2f8f40e00>]
Basic usage of jax.custom_jvp and jax.custom_vjp APIs#
Use jax.custom_jvp to define forward-mode (and, indirectly, reverse-mode) rules#
Here’s a canonical basic example of using jax.custom_jvp, where the comments use
Haskell-like type signatures:
from jax import custom_jvp
import jax.numpy as jnp
# f :: a -> b
@custom_jvp
def f(x):
return jnp.sin(x)
# f_jvp :: (a, T a) -> (b, T b)
def f_jvp(primals, tangents):
x, = primals
t, = tangents
return f(x), jnp.cos(x) * t
f.defjvp(f_jvp)
<function __main__.f_jvp(primals, tangents)>
from jax import jvp
print(f(3.))
y, y_dot = jvp(f, (3.,), (1.,))
print(y)
print(y_dot)
0.14112
0.14112
-0.9899925
In words, we start with a primal function f that takes inputs of type a and produces outputs of type b. We associate with it a JVP rule function f_jvp that takes a pair of inputs representing the primal inputs of type a and the corresponding tangent inputs of type T a, and produces a pair of outputs representing the primal outputs of type b and tangent outputs of type T b. The tangent outputs should be a linear function of the tangent inputs.
You can also use f.defjvp as a decorator, as in
@custom_jvp
def f(x):
...
@f.defjvp
def f_jvp(primals, tangents):
...
Even though we defined only a JVP rule and no VJP rule, we can use both forward- and reverse-mode differentiation on f. JAX will automatically transpose the linear computation on tangent values from our custom JVP rule, computing the VJP as efficiently as if we had written the rule by hand:
from jax import grad
print(grad(f)(3.))
print(grad(grad(f))(3.))
-0.9899925
-0.14112
For automatic transposition to work, the JVP rule’s output tangents must be linear as a function of the input tangents. Otherwise a transposition error is raised.
Multiple arguments work like this:
@custom_jvp
def f(x, y):
return x ** 2 * y
@f.defjvp
def f_jvp(primals, tangents):
x, y = primals
x_dot, y_dot = tangents
primal_out = f(x, y)
tangent_out = 2 * x * y * x_dot + x ** 2 * y_dot
return primal_out, tangent_out
print(grad(f)(2., 3.))
12.0
The defjvps convenience wrapper lets us define a JVP for each argument separately, and the results are computed separately then summed:
@custom_jvp
def f(x):
return jnp.sin(x)
f.defjvps(lambda t, ans, x: jnp.cos(x) * t)
print(grad(f)(3.))
-0.9899925
Here’s a defjvps example with multiple arguments:
@custom_jvp
def f(x, y):
return x ** 2 * y
f.defjvps(lambda x_dot, primal_out, x, y: 2 * x * y * x_dot,
lambda y_dot, primal_out, x, y: x ** 2 * y_dot)
print(grad(f)(2., 3.))
print(grad(f, 0)(2., 3.)) # same as above
print(grad(f, 1)(2., 3.))
12.0
12.0
4.0
As a shorthand, with defjvps you can pass a None value to indicate that the JVP for a particular argument is zero:
@custom_jvp
def f(x, y):
return x ** 2 * y
f.defjvps(lambda x_dot, primal_out, x, y: 2 * x * y * x_dot,
None)
print(grad(f)(2., 3.))
print(grad(f, 0)(2., 3.)) # same as above
print(grad(f, 1)(2., 3.))
12.0
12.0
0.0
Calling a jax.custom_jvp function with keyword arguments, or writing a jax.custom_jvp function definition with default arguments, are both allowed so long as they can be unambiguously mapped to positional arguments based on the function signature retrieved by the standard library inspect.signature mechanism.
When you’re not performing differentiation, the function f is called just as if it weren’t decorated by jax.custom_jvp:
@custom_jvp
def f(x):
print('called f!') # a harmless side-effect
return jnp.sin(x)
@f.defjvp
def f_jvp(primals, tangents):
print('called f_jvp!') # a harmless side-effect
x, = primals
t, = tangents
return f(x), jnp.cos(x) * t
from jax import vmap, jit
print(f(3.))
called f!
0.14112
print(vmap(f)(jnp.arange(3.)))
print(jit(f)(3.))
called f!
[0. 0.84147096 0.9092974 ]
called f!
0.14112
The custom JVP rule is invoked during differentiation, whether forward or reverse:
y, y_dot = jvp(f, (3.,), (1.,))
print(y_dot)
called f_jvp!
called f!
-0.9899925
print(grad(f)(3.))
called f_jvp!
called f!
-0.9899925
Notice that f_jvp calls f to compute the primal outputs. In the context of higher-order differentiation, each application of a differentiation transform will use the custom JVP rule if and only if the rule calls the original f to compute the primal outputs. (This represents a kind of fundamental tradeoff, where we can’t make use of intermediate values from the evaluation of f in our rule and also have the rule apply in all orders of higher-order differentiation.)
grad(grad(f))(3.)
called f_jvp!
called f_jvp!
called f!
Array(-0.14112, dtype=float32, weak_type=True)
You can use Python control flow with jax.custom_jvp:
@custom_jvp
def f(x):
if x > 0:
return jnp.sin(x)
else:
return jnp.cos(x)
@f.defjvp
def f_jvp(primals, tangents):
x, = primals
x_dot, = tangents
ans = f(x)
if x > 0:
return ans, 2 * x_dot
else:
return ans, 3 * x_dot
print(grad(f)(1.))
print(grad(f)(-1.))
2.0
3.0
Use jax.custom_vjp to define custom reverse-mode-only rules#
While jax.custom_jvp suffices for controlling both forward- and, via JAX’s automatic transposition, reverse-mode differentiation behavior, in some cases we may want to directly control a VJP rule, for example in the latter two example problems presented above. We can do that with jax.custom_vjp:
from jax import custom_vjp
import jax.numpy as jnp
# f :: a -> b
@custom_vjp
def f(x):
return jnp.sin(x)
# f_fwd :: a -> (b, c)
def f_fwd(x):
return f(x), jnp.cos(x)
# f_bwd :: (c, CT b) -> CT a
def f_bwd(cos_x, y_bar):
return (cos_x * y_bar,)
f.defvjp(f_fwd, f_bwd)
from jax import grad
print(f(3.))
print(grad(f)(3.))
0.14112
-0.9899925
In words, we again start with a primal function f that takes inputs of type a and produces outputs of type b. We associate with it two functions, f_fwd and f_bwd, which describe how to perform the forward- and backward-passes of reverse-mode autodiff, respectively.
The function f_fwd describes the forward pass, not only the primal computation but also what values to save for use on the backward pass. Its input signature is just like that of the primal function f, in that it takes a primal input of type a. But as output it produces a pair, where the first element is the primal output b and the second element is any “residual” data of type c to be stored for use by the backward pass. (This second output is analogous to PyTorch’s save_for_backward mechanism.)
The function f_bwd describes the backward pass. It takes two inputs, where the first is the residual data of type c produced by f_fwd and the second is the output cotangents of type CT b corresponding to the output of the primal function. It produces an output of type CT a representing the cotangents corresponding to the input of the primal function. In particular, the output of f_bwd must be a sequence (e.g. a tuple) of length equal to the number of arguments to the primal function.
So multiple arguments work like this:
from jax import custom_vjp
@custom_vjp
def f(x, y):
return jnp.sin(x) * y
def f_fwd(x, y):
return f(x, y), (jnp.cos(x), jnp.sin(x), y)
def f_bwd(res, g):
cos_x, sin_x, y = res
return (cos_x * g * y, sin_x * g)
f.defvjp(f_fwd, f_bwd)
print(grad(f)(2., 3.))
-1.2484405
Calling a jax.custom_vjp function with keyword arguments, or writing a jax.custom_vjp function definition with default arguments, are both allowed so long as they can be unambiguously mapped to positional arguments based on the function signature retrieved by the standard library inspect.signature mechanism.
As with jax.custom_jvp, the custom VJP rule comprised by f_fwd and f_bwd is not invoked if differentiation is not applied. If function is evaluated, or transformed with jit, vmap, or other non-differentiation transformations, then only f is called.
@custom_vjp
def f(x):
print("called f!")
return jnp.sin(x)
def f_fwd(x):
print("called f_fwd!")
return f(x), jnp.cos(x)
def f_bwd(cos_x, y_bar):
print("called f_bwd!")
return (cos_x * y_bar,)
f.defvjp(f_fwd, f_bwd)
print(f(3.))
called f!
0.14112
print(grad(f)(3.))
called f_fwd!
called f!
called f_bwd!
-0.9899925
from jax import vjp
y, f_vjp = vjp(f, 3.)
print(y)
called f_fwd!
called f!
0.14112
print(f_vjp(1.))
called f_bwd!
(Array(-0.9899925, dtype=float32, weak_type=True),)
Forward-mode autodiff cannot be used on the jax.custom_vjp function and will raise an error:
from jax import jvp
try:
jvp(f, (3.,), (1.,))
except TypeError as e:
print('ERROR! {}'.format(e))
called f_fwd!
called f!
ERROR! can't apply forward-mode autodiff (jvp) to a custom_vjp function.
If you want to use both forward- and reverse-mode, use jax.custom_jvp instead.
We can use jax.custom_vjp together with pdb to insert a debugger trace in the backward pass:
import pdb
@custom_vjp
def debug(x):
return x # acts like identity
def debug_fwd(x):
return x, x
def debug_bwd(x, g):
pdb.set_trace()
return g
debug.defvjp(debug_fwd, debug_bwd)
def foo(x):
y = x ** 2
y = debug(y) # insert pdb in corresponding backward pass step
return jnp.sin(y)
jax.grad(foo)(3.)
> <ipython-input-113-b19a2dc1abf7>(12)debug_bwd()
-> return g
(Pdb) p x
Array(9., dtype=float32)
(Pdb) p g
Array(-0.91113025, dtype=float32)
(Pdb) q
custom_vjp functions can’t be forward-differentiated#
One important limitation: defining a custom_vjp rule makes the function
reverse-mode only. JAX has no way to derive a JVP rule from your VJP rules,
so applying forward-mode autodiff — jax.jvp or jax.jacfwd — to a
function that contains a custom_vjp function raises an error:
from jax import jvp
jvp(lambda x: clip_gradient(-1., 1., x), (3.,), (1.,))
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
Cell In[50], line 3
1 from jax import jvp
2
----> 3 jvp(lambda x: clip_gradient(-1., 1., x), (3.,), (1.,))
[... skipping hidden 4 frame]
Cell In[50], line 3, in <lambda>(x)
----> 3 jvp(lambda x: clip_gradient(-1., 1., x), (3.,), (1.,))
[... skipping hidden 8 frame]
File ~/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.12/site-packages/jax/_src/interpreters/ad.py:1117, in raise_custom_vjp_error_on_jvp(*_, **__)
1116 def raise_custom_vjp_error_on_jvp(*_, **__):
-> 1117 raise TypeError("can't apply forward-mode autodiff (jvp) to a custom_vjp "
1118 "function.")
TypeError: can't apply forward-mode autodiff (jvp) to a custom_vjp function.
Note that reverse mode composes fine: jax.grad-of-grad,
jax.hessian, and forward-over-reverse Hessian-vector products all work on
functions containing custom_vjp, because after one reverse-mode pass the
remaining computation is built from your fwd and bwd rules, which are
themselves differentiable. It’s only forward mode applied to the
custom_vjp function directly that fails — as in jax.jacfwd of your
model, or jax.jvp-based sensitivity analysis. If you need both modes,
define a custom_jvp instead (JAX derives reverse mode from it
automatically), or use a hijax primitive (Custom derivative rules with hijax primitives), which
can carry rules for both modes at once.
More features and details#
Working with list / tuple / dict containers (and other pytrees)#
You should expect standard Python containers like lists, tuples, namedtuples, and dicts to just work, along with nested versions of those. In general, any pytrees are permissible, so long as their structures are consistent according to the type constraints.
Here’s a contrived example with jax.custom_jvp:
from collections import namedtuple
Point = namedtuple("Point", ["x", "y"])
@custom_jvp
def f(pt):
x, y = pt.x, pt.y
return {'a': x ** 2,
'b': (jnp.sin(x), jnp.cos(y))}
@f.defjvp
def f_jvp(primals, tangents):
pt, = primals
pt_dot, = tangents
ans = f(pt)
ans_dot = {'a': 2 * pt.x * pt_dot.x,
'b': (jnp.cos(pt.x) * pt_dot.x, -jnp.sin(pt.y) * pt_dot.y)}
return ans, ans_dot
def fun(pt):
dct = f(pt)
return dct['a'] + dct['b'][0]
pt = Point(1., 2.)
print(f(pt))
{'a': 1.0, 'b': (Array(0.84147096, dtype=float32, weak_type=True), Array(-0.41614684, dtype=float32, weak_type=True))}
print(grad(fun)(pt))
Point(x=Array(2.5403023, dtype=float32, weak_type=True), y=Array(0., dtype=float32, weak_type=True))
And an analogous contrived example with jax.custom_vjp:
@custom_vjp
def f(pt):
x, y = pt.x, pt.y
return {'a': x ** 2,
'b': (jnp.sin(x), jnp.cos(y))}
def f_fwd(pt):
return f(pt), pt
def f_bwd(pt, g):
a_bar, (b0_bar, b1_bar) = g['a'], g['b']
x_bar = 2 * pt.x * a_bar + jnp.cos(pt.x) * b0_bar
y_bar = -jnp.sin(pt.y) * b1_bar
return (Point(x_bar, y_bar),)
f.defvjp(f_fwd, f_bwd)
def fun(pt):
dct = f(pt)
return dct['a'] + dct['b'][0]
pt = Point(1., 2.)
print(f(pt))
{'a': 1.0, 'b': (Array(0.84147096, dtype=float32, weak_type=True), Array(-0.41614684, dtype=float32, weak_type=True))}
print(grad(fun)(pt))
Point(x=Array(2.5403023, dtype=float32, weak_type=True), y=Array(-0., dtype=float32, weak_type=True))
Handling non-differentiable arguments#
Some use cases, like the final example problem, call for non-differentiable arguments like function-valued arguments to be passed to functions with custom differentiation rules, and for those arguments to also be passed to the rules themselves. In the case of fixed_point, the function argument f was such a non-differentiable argument. A similar situation arises with jax.experimental.odeint.
jax.custom_jvp with nondiff_argnums#
Use the optional nondiff_argnums parameter to jax.custom_jvp to indicate arguments like these. Here’s an example with jax.custom_jvp:
from functools import partial
@partial(custom_jvp, nondiff_argnums=(0,))
def app(f, x):
return f(x)
@app.defjvp
def app_jvp(f, primals, tangents):
x, = primals
x_dot, = tangents
return f(x), 2. * x_dot
print(app(lambda x: x ** 3, 3.))
27.0
print(grad(app, 1)(lambda x: x ** 3, 3.))
2.0
One gotcha: no matter where in the argument list these parameters appear, they’re placed at the start of the signature of the corresponding JVP rule. Here’s another example:
@partial(custom_jvp, nondiff_argnums=(0, 2))
def app2(f, x, g):
return f(g((x)))
@app2.defjvp
def app2_jvp(f, g, primals, tangents):
x, = primals
x_dot, = tangents
return f(g(x)), 3. * x_dot
print(app2(lambda x: x ** 3, 3., lambda y: 5 * y))
3375.0
print(grad(app2, 1)(lambda x: x ** 3, 3., lambda y: 5 * y))
3.0
jax.custom_vjp with nondiff_argnums#
A similar option exists for jax.custom_vjp, and, similarly, the convention is that the non-differentiable arguments are passed as the first arguments to the _bwd rule, no matter where they appear in the signature of the original function. The signature of the _fwd rule remains unchanged - it is the same as the signature of the primal function. Here’s an example:
@partial(custom_vjp, nondiff_argnums=(0,))
def app(f, x):
return f(x)
def app_fwd(f, x):
return f(x), x
def app_bwd(f, x, g):
return (5 * g,)
app.defvjp(app_fwd, app_bwd)
print(app(lambda x: x ** 2, 4.))
16.0
print(grad(app, 1)(lambda x: x ** 2, 4.))
5.0
See fixed_point above for another usage example.
You don’t need to use nondiff_argnums with array-valued arguments, for example ones with integer dtype. Instead, nondiff_argnums should only be used for argument values that don’t correspond to JAX types (essentially don’t correspond to array types), like Python callables or strings. If JAX detects that an argument indicated by nondiff_argnums contains a JAX Tracer, then an error is raised. The clip_gradient function above is a good example of not using nondiff_argnums for integer-dtype array arguments.