Add set_gradients method for JAX backend. (#278)

Co-authored-by: Rémi Flamary <remi.flamary@gmail.com>

Author: AdrienCorenflos

ot/backend.py

@@ -287,16 +287,16 @@ def from_numpy(self, a, type_as=None):
             return jnp.array(a).astype(type_as.dtype)
 
     def set_gradients(self, val, inputs, grads):
-        # no gradients for jax because it is functional
+        from jax.flatten_util import ravel_pytree
+        val, = jax.lax.stop_gradient((val,))
 
-        # does not work
-        # from jax import custom_jvp
-        # @custom_jvp
-        # def f(*inputs):
-        #     return val
-        # f.defjvps(*grads)
-        # return f(*inputs)
+        ravelled_inputs, _ = ravel_pytree(inputs)
+        ravelled_grads, _ = ravel_pytree(grads)
 
+        aux = jnp.sum(ravelled_inputs * ravelled_grads) / 2
+        aux = aux - jax.lax.stop_gradient(aux)
+
+        val, = jax.tree_map(lambda z: z + aux, (val,))
         return val
 
     def zeros(self, shape, type_as=None):

test/test_backend.py

@@ -345,7 +345,8 @@ def test_gradients_backends():
 
     rnd = np.random.RandomState(0)
     v = rnd.randn(10)
-    c = rnd.randn(1)
+    c = rnd.randn()
+    e = rnd.randn()
 
     if torch:
 
@@ -362,3 +363,15 @@ def test_gradients_backends():
 
         assert torch.equal(v2.grad, v2)
         assert torch.equal(c2.grad, c2)
+
+    if jax:
+        nx = ot.backend.JaxBackend()
+        with jax.checking_leaks():
+            def fun(a, b, d):
+                val = b * nx.sum(a ** 4) + d
+                return nx.set_gradients(val, (a, b, d), (a, b, 2 * d))
+            grad_val = jax.grad(fun, argnums=(0, 1, 2))(v, c, e)
+
+        np.testing.assert_almost_equal(fun(v, c, e), c * np.sum(v ** 4) + e, decimal=4)
+        np.testing.assert_allclose(grad_val[0], v, atol=1e-4)
+        np.testing.assert_allclose(grad_val[2], 2 * e, atol=1e-4)