--- jupytext: cell_metadata_filter: -all formats: ipynb,md:myst,py main_language: python notebook_metadata_filter: nosearch text_representation: extension: .md format_name: myst format_version: 0.13 jupytext_version: 1.16.4 nosearch: true --- ```{raw-cell} --- Copyright 2025 The JAX Authors. 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 https://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. --- ``` # `Ref`: mutable arrays for data plumbing and memory control JAX `Array`s are immutable, representing mathematical values. Immutability can make code easier to reason about, and is useful for optimized compilation, parallelization, rematerialization, and transformations like autodiff. But immutability is constraining too: * **expressiveness** --- plumbing out intermediate data or maintaining state, e.g. for normalization statistics or metrics, can feel heavyweight; * **performance** --- it's more difficult to reason about performance, like memory lifetimes and in-place updates. `Ref`s can help! They represent mutable arrays that can be read and written in-place. These array references are compatible with JAX transformations, like `jax.jit` and `jax.grad`: ```{code-cell} import jax import jax.numpy as jnp x_ref = jax.new_ref(jnp.zeros(3)) # new array ref, with initial value [0., 0., 0.] @jax.jit def f(): x_ref[1] += 1. # indexed add-update print(x_ref) # Ref([0., 0., 0.]) f() f() print(x_ref) # Ref([0., 2., 0.]) ``` The indexing syntax follows NumPy's. For a `Ref` called `x_ref`, we can read its entire value into an `Array` by writing `x_ref[...]`, and write its entire value using `x_ref[...] = A` for some `Array`-valued expression `A`: ```{code-cell} def g(x): x_ref = jax.new_ref(0.) x_ref[...] = jnp.sin(x) return x_ref[...] print(jax.grad(g)(1.0)) # 0.54 ``` `Ref` is a distinct type from `Array`, and it comes with some important constraints and limitations. In particular, indexed reading and writing is just about the *only* thing you can do with an `Ref`. References can't be passed where `Array`s are expected: ```{code-cell} x_ref = jax.new_ref(1.0) try: jnp.sin(x_ref) # error! can't do math on refs except Exception as e: print(e) ``` To do math, you need to read the ref's value first, like `jnp.sin(x_ref[...])`. So what _can_ you do with `Ref`? Read on for the details, and some useful recipes. ### API If you've ever used [Pallas](https://docs.jax.dev/en/latest/pallas/quickstart.html), then `Ref` should look familiar. A big difference is that you can create new `Ref`s yourself directly using `jax.new_ref`: ```{code-cell} from jax import Array, Ref def array_ref(init_val: Array) -> Ref: """Introduce a new reference with given initial value.""" ``` `jax.freeze` is its antithesis, invalidating the given ref (so that accessing it afterwards is an error) and producing its final value: ```{code-cell} def freeze(ref: Ref) -> Array: """Invalidate given reference and produce its final value.""" ``` In between creating and destroying them, you can perform indexed reads and writes on refs. You can read and write using the functions `jax.ref.get` and `jax.ref.swap`, but usually you'd just use NumPy-style array indexing syntax: ```{code-cell} import types Index = int | slice | Array | types.EllipsisType Indexer = Index | tuple[Index, ...] def get(ref: Ref, idx: Indexer) -> Array: """Returns `ref[idx]` for NumPy-style indexer `idx`.""" def swap(ref: Ref, idx: Indexer, val: Array) -> Array: """Performs `newval, ref[idx] = ref[idx], val` and returns `newval`.""" ``` Here, `Indexer` can be any NumPy indexing expression: ```{code-cell} x_ref = jax.new_ref(jnp.arange(12.).reshape(3, 4)) # int indexing row = x_ref[0] x_ref[1] = row # tuple indexing val = x_ref[1, 2] x_ref[2, 3] = val # slice indexing col = x_ref[:, 1] x_ref[0, :3] = col # advanced int array indexing vals = x_ref[jnp.array([0, 0, 1]), jnp.array([1, 2, 3])] x_ref[jnp.array([1, 2, 1]), jnp.array([0, 0, 1])] = vals ``` As with `Array`s, indexing mostly follows NumPy behavior, except that an out-of-bounds index raises an error when its value is known in advance (unlike with `Array`s, where [out-of-bounds reads clamp and writes are dropped](https://docs.jax.dev/en/latest/notebooks/Common_Gotchas_in_JAX.html#out-of-bounds-indexing)). ### Pure and impure functions A function that takes a ref as an argument (either explicitly or by lexical closure) is considered _impure_. For example: ```{code-cell} # takes ref as an argument => impure @jax.jit def impure1(x_ref, y_ref): x_ref[...] = y_ref[...] # closes over ref => impure y_ref = jax.new_ref(0) @jax.jit def impure2(x): y_ref[...] = x ``` If a function only uses refs internally, it is still considered _pure_. Purity is in the eye of the caller. For example: ```{code-cell} # internal refs => still pure @jax.jit def pure1(x): ref = jax.new_ref(x) ref[...] = ref[...] + ref[...] return ref[...] ``` Pure functions, even those that use refs internally, are familiar: for example, they work with transformations like `jax.grad`, `jax.vmap`, `jax.shard_map`, and others in the usual way. Impure functions are sequenced in Python program order. ### Restrictions `Ref`s are second-class, in the sense that there are restrictions on their use: * **Can't return refs** from `jit`\-decorated functions or the bodies of higher-order primitives like `jax.lax.scan`, `jax.lax.while_loop`, or `jax.lax.cond` * **Can't pass a ref as an argument more than once** to `jit`\-decorated functions or higher-order primitives * **Can only `freeze` in creation scope** * **No higher-order refs** (refs-to-refs) For example, these are errors: ```{code-cell} x_ref = jax.new_ref(0.) # can't return refs @jax.jit def err1(x_ref): x_ref[...] = 5. return x_ref # error! try: err1(x_ref) except Exception as e: print(e) # can't pass a ref as an argument more than once @jax.jit def err2(x_ref, y_ref): ... try: err2(x_ref, x_ref) # error! except Exception as e: print(e) # can't pass and close over the same ref @jax.jit def err3(y_ref): y_ref[...] = x_ref[...] try: err3(x_ref) # error! except Exception as e: print(e) # can only freeze in creation scope @jax.jit def err4(x_ref): jax.freeze(x_ref) try: err4(x_ref) # error! except Exception as e: print(e) ``` These restrictions exist to rule out aliasing, where two refs might refer to the same mutable memory, making programs harder to reason about and transform. Weaker restrictions would also suffice, so some of these restrictions may be lifted as we improve JAX's ability to verify that no aliasing is present. There are also restrictions stemming from undefined semantics, e.g. in the presence of parallelism or rematerialization: * **Can't `vmap` or `shard_map` a function that closes over refs** * **Can't apply `jax.remat`/`jax.checkpoint` to an impure function** For example, here are ways you can and can't use `vmap` with impure functions: ```{code-cell} # vmap over ref args is okay def dist(x, y, out_ref): assert x.ndim == y.ndim == 1 assert out_ref.ndim == 0 out_ref[...] = jnp.sum((x - y) ** 2) vecs = jnp.arange(12.).reshape(3, 4) out_ref = jax.new_ref(jnp.zeros((3, 3))) jax.vmap(jax.vmap(dist, (0, None, 0)), (None, 0, 0))(vecs, vecs, out_ref) # ok! print(out_ref) ``` ```{code-cell} # vmap with a closed-over ref is not x_ref = jax.new_ref(0.) def err5(x): x_ref[...] = x try: jax.vmap(err5)(jnp.arange(3.)) # error! except Exception as e: print(e) ``` The latter is an error because it's not clear which value `x_ref` should be after we run `jax.vmap(err5)`. ### `Ref`s and automatic differentiation Autodiff can be applied to pure functions as before, even if they use array refs internally. For example: ```{code-cell} @jax.jit def pure2(x): ref = jax.new_ref(x) ref[...] = ref[...] + ref[...] return ref[...] print(jax.grad(pure2)(3.0)) # 2.0 ``` Autodiff can also be applied to functions that take array refs as arguments. The simplest case is when those ref arguments are only used for plumbing, and aren't involved in differentiation. For example, `jax.grad` differentiates with respect to its function's first argument by default, so ref arguments in other positions are just along for the ride. Only non-differentiated values can be written into such plumbing refs: ```{code-cell} # error def err6(x, some_plumbing_ref): y = x + x some_plumbing_ref[...] += y return y # fine def foo(x, some_plumbing_ref): y = x + x some_plumbing_ref[...] += jax.lax.stop_gradient(y) return y ``` Differentiating _with respect to_ a ref-typed argument is another matter: pointing `jax.grad` at one (e.g. via `argnums`) is an error, since the gradient for a ref must itself live in a ref. Instead, use `jax.vjp` and `with_refs`, described below. You can combine plumbing refs with `custom_vjp` to plumb data out of the backward pass of a differentiated function: ```{code-cell} # First, define the helper `stash_grads`: @jax.custom_vjp def stash_grads(grads_ref, x): return x def stash_grads_fwd(grads_ref, x): return x, grads_ref def stash_grads_bwd(grads_ref, g): grads_ref[...] = g return None, g stash_grads.defvjp(stash_grads_fwd, stash_grads_bwd) ``` ```{code-cell} # Now, use `stash_grads` to stash intermediate gradients: def f(x, grads_ref): x = stash_grads(grads_ref, x) x = jnp.sin(x) return x grads_ref = jax.new_ref(0.) jax.grad(f)(1., grads_ref) print(grads_ref) # Ref(0.54), the gradient at the stash point: cos(1.) ``` Notice `stash_grads_fwd` is returning a `Ref` here. That's a special allowance for `custom_vjp` fwd rules: it's really syntax for indicating which ref arguments should be shared by both the fwd and bwd rules. So any refs returned by a fwd rule must be arguments to that fwd rule. #### Differentiating with respect to `Ref` arguments The plumbing refs above are just passengers: they carry data out of the computation, but no gradients flow through them. We can also differentiate _with respect to_ a ref argument. Since the gradient for a `Ref`-typed input is itself `Ref`-typed, `jax.grad` doesn't apply here. Instead we use `jax.vjp`, and bind a gradient ref to the VJP function using its `with_refs` method: ```{code-cell} def f(x_ref): return x_ref[...] ** 2 x_ref = jax.new_ref(2.) y, f_vjp = jax.vjp(f, x_ref) x_grad_ref = jax.new_ref(0.) f_vjp.with_refs(x_grad_ref)(1.0) # bind the gradient ref, then apply the VJP print(x_grad_ref) # Ref(4.) ``` Here `with_refs` takes one entry per argument of the differentiated function and returns a new VJP function with those gradient refs bound. When differentiating with respect to a ref argument, using `with_refs` is mandatory; the gradient needs a ref to be accumulated into, so calling the VJP function without binding one is an error: ```{code-cell} _, f_vjp = jax.vjp(f, jax.new_ref(2.)) try: f_vjp(1.0) # error! no ref bound for the ref-typed argument's gradient except Exception as e: print(e) # ... gradient must be accumulated into a ref ... `with_refs` ... ``` The gradient is accumulated into the bound ref via `+=`, added to whatever the ref already contains rather than overwriting it: ```{code-cell} x_grad_ref = jax.new_ref(100.) _, f_vjp = jax.vjp(f, jax.new_ref(2.)) f_vjp.with_refs(x_grad_ref)(1.0) print(x_grad_ref) # Ref(104.), i.e. 100. + 4.: accumulated, not set ``` Accumulating rather than setting might seem like an odd choice, but it means one gradient ref can collect contributions from several backward passes with no extra memory traffic, as in the examples below. In-place updates to the ref argument inside the differentiated function are differentiated too. The result is the gradient with respect to the ref's initial value: ```{code-cell} def g(x_ref): x_ref[...] = jnp.sin(x_ref[...]) return x_ref[...] ** 2 x_ref = jax.new_ref(2.) _, g_vjp = jax.vjp(g, x_ref) # runs g, so x_ref is updated in-place here g_grad_ref = jax.new_ref(0.) g_vjp.with_refs(g_grad_ref)(1.0) print(g_grad_ref) # Ref(-0.757), i.e. 2*sin(2)*cos(2) ``` #### Gradient refs for value arguments When differentiating with respect to an ordinary `Array` argument, `with_refs` is optional: we can call the VJP function directly and get the gradient back as a value in the usual way, or we can bind a ref and have the gradient accumulated into it in-place: ```{code-cell} _, sin_vjp = jax.vjp(jnp.sin, 1.0) x_bar, = sin_vjp(1.0) # the usual way: gradient returned as a value print(x_bar) # 0.54 grad_ref = jax.new_ref(0.) _, sin_vjp = jax.vjp(jnp.sin, 1.0) sin_vjp.with_refs(grad_ref)(1.0) # gradient accumulated into grad_ref print(grad_ref) # Ref(0.54) ``` We can mix and match. Each entry of `with_refs` can be: * a `Ref`, meaning accumulate this argument's gradient into the ref in-place (the VJP call then returns a `jax.ad.GradRef()` placeholder in that position); * `jax.ad.GradValue()`, meaning return this argument's gradient as a value in the usual way (the default); or * `jax.ad.DontWant()`, meaning don't compute this argument's gradient at all (the VJP call returns a `jax.ad.DidntWant()` placeholder in that position — more on this below). One reason to use a gradient ref here is to exploit sparsity. Consider differentiating a function that slices its input: ```{code-cell} @jax.jit def take(x, i): return x[i] x = jnp.arange(10.) _, take_vjp = jax.vjp(take, x, 3) x_bar, _ = take_vjp(1.0) print(x_bar) # [0., 0., 0., 1., 0., 0., 0., 0., 0., 0.] ``` The gradient with respect to `x` is one-hot: the backward pass materializes a dense array of zeros and sets a single element of it. If we compute many such gradients and sum them, say over a loop of sparse accesses, we pay for a dense array's worth of memory traffic on each one, even though each contribution only touches one element. If we instead bind a gradient ref with `with_refs`, each backward pass performs a sparse in-place add-update, writing only where it needs to: ```{code-cell} grad_ref = jax.new_ref(jnp.zeros(10)) for i in [3, 5, 3]: _, take_vjp = jax.vjp(take, x, i) take_vjp.with_refs(grad_ref, jax.ad.GradValue())(1.0) # no gradient ref for i print(grad_ref) # Ref([0., 0., 0., 2., 0., 1., 0., 0., 0., 0.]) ``` We can check that the update is sparse by inspecting the jaxpr of a single VJP application: ```{code-cell} @jax.make_jaxpr def take_vjp_jaxpr(): _, take_vjp = jax.vjp(take, x, 3) take_vjp.with_refs(grad_ref, jax.ad.GradValue())(1.0) print(take_vjp_jaxpr()) ``` The backward pass boils down to `b[c:c+1] += j`, an in-place add-update of one element of the gradient ref, with no dense one-hot array in sight. #### `DontWant`: skipping unneeded gradients The VJP function returned by `jax.vjp` computes gradients for all the arguments of the differentiated function. But sometimes we don't need all of them, like when differentiating with respect to parameters but not data. Passing `jax.ad.DontWant()` for an argument tells the backward pass not to compute its gradient at all, playing the same role for `jax.vjp` that `argnums` plays for `jax.grad`: ```{code-cell} def predict(W, x): return W @ x W = jnp.ones((4, 4)) x = jnp.ones(4) _, f_vjp = jax.vjp(predict, W, x) W_bar, x_bar = f_vjp.with_refs(jax.ad.GradValue(), jax.ad.DontWant())(jnp.ones(4)) print(W_bar[0]) # [1., 1., 1., 1.] print(x_bar) # DidntWant() ``` This is more than a convenience: it can save real work in the backward pass, like an eager form of dead code elimination. Transpose rules can check for `DontWant` and skip computing the corresponding cotangents. For example, the transpose of matrix multiplication usually computes two dot products, one for each operand's gradient, but with `DontWant` it computes only one. We can see that by counting the `dot_general` operations in the jaxpr of each VJP application: ```{code-cell} _, f_vjp = jax.vjp(predict, W, x) both = jax.make_jaxpr(lambda: f_vjp(jnp.ones(4)))() only_W = jax.make_jaxpr( lambda: f_vjp.with_refs(jax.ad.GradValue(), jax.ad.DontWant())(jnp.ones(4)))() print(str(both).count('dot_general')) # 2, one dot for each gradient print(str(only_W).count('dot_general')) # 1, the dot for x_bar is skipped ``` #### Example: gradient accumulation over microbatches Here's a more realistic recipe that puts these pieces together. In pipelined training, we often split a batch into microbatches, run a forward and backward pass one microbatch at a time, and accumulate weight gradients as we go. Using `with_refs`, each microbatch's backward pass accumulates directly into a single fixed gradient buffer: ```{code-cell} NUM_LAYERS = 3 NUM_MUBATCHES = 5 MUBATCH_SIZE = 7 def mubatch_loss(Ws, xs): # inner loop over layers act, _ = jax.lax.scan(lambda x, W: (jnp.dot(x, W), None), xs, Ws) return jnp.mean(act) def process_batch(Ws, xs_batch): grad_acc = jax.new_ref(jnp.zeros_like(Ws)) def process_mubatch(_, xs): loss, f_vjp = jax.vjp(lambda Ws: mubatch_loss(Ws, xs), Ws) f_vjp.with_refs(grad_acc)(jnp.ones_like(loss)) # accumulate in-place return (), loss xs_mubatches = xs_batch.reshape(NUM_MUBATCHES, MUBATCH_SIZE, -1) # outer loop over microbatches (), losses = jax.lax.scan(process_mubatch, (), xs_mubatches) return jax.freeze(grad_acc), losses Ws = jnp.ones((NUM_LAYERS, 4, 4)) xs_batch = jnp.ones((NUM_MUBATCHES * MUBATCH_SIZE, 4)) grads, losses = process_batch(Ws, xs_batch) ``` Each iteration of the outer scan runs a forward and backward pass for one microbatch, and `with_refs(grad_acc)` makes the backward pass add that microbatch's gradient contribution directly into `grad_acc`. Note that the scan body closes over `grad_acc`, which is fine for `scan` (though it wouldn't be for `vmap` or `shard_map`, as discussed above). Once all the microbatches are processed, we `freeze` the accumulator to get the total batch gradient as an immutable `Array`. The result matches what we'd get by differentiating the summed loss directly: ```{code-cell} xs_mubatches = xs_batch.reshape(NUM_MUBATCHES, MUBATCH_SIZE, -1) grads_expected = jax.grad( lambda Ws: sum(mubatch_loss(Ws, xs) for xs in xs_mubatches))(Ws) print(jnp.allclose(grads, grads_expected, atol=1e-3, rtol=1e-3)) # True ``` But unlike that version, the ref-based version never materializes per-microbatch gradients as separate arrays: there's one gradient buffer, allocated once, no matter how many microbatches we process. ### `Ref`s and performance At the top level, when calling `jit`\-decorated functions, `Ref`s obviate the need for donation, since they are effectively always donated: ```{code-cell} @jax.jit def sin_inplace(x_ref): x_ref[...] = jnp.sin(x_ref[...]) x_ref = jax.new_ref(jnp.arange(3.)) print(x_ref.unsafe_buffer_pointer(), x_ref) sin_inplace(x_ref) print(x_ref.unsafe_buffer_pointer(), x_ref) ``` Here `sin_inplace` operates in-place, updating the buffer backing `x_ref` so that its address stays the same. Under a `jit`, you should expect array references to point to fixed buffer addresses, and for indexed updates to be performed in-place. **Temporary caveat:** dispatch from Python to impure `jit`\-compiled functions that take `Ref` inputs is currently slower than dispatch to pure `jit`\-compiled functions, since it takes a less optimized path. ### `foreach`, a new way to write `scan` As you may know, `jax.lax.scan` is a loop construct with a built-in fixed access pattern for scanned-over inputs and outputs. The access pattern is built in for autodiff reasons: if we were instead to slice into immutable inputs directly, reverse-mode autodiff would end up creating one-hot gradients and summing them up, which can be asymptotically inefficient. See [Sec 5.3.3 of the Dex paper](https://arxiv.org/pdf/2104.05372). But reading slices of `Ref`s doesn't have this efficiency problem: when we apply reverse-mode autodiff, we always generate in-place accumulation operations. As a result, we no longer need to be constrained by `scan`'s fixed access pattern. We can write more flexible loops, e.g. with non-sequential access. Moreover, having mutation available allows for some syntax tricks, like in this recipe for a `foreach` decorator: ```{code-cell} import jax import jax.numpy as jnp from jax.lax import scan def foreach(*args): def decorator(body): return scan(lambda _, elts: (None, body(*elts)), None, args)[1] return decorator ``` ```{code-cell} r = jax.new_ref(0) xs = jnp.arange(10) @foreach(xs) def ys(x): r[...] += x return x * 2 print(r) # Ref(45, dtype=int32) print(ys) # [ 0 2 4 6 8 10 12 14 16 18] ``` Here, the loop runs immediately, updating `r` in-place and binding `ys` to be the mapped result.