Update limitations.md

docs/src/limitations.md

@@ -20,7 +20,6 @@ Let's explore this with a more concrete example. Here we define a simple mutatin
 ```julia
 function f!(x)
   x .= 2 .* x
-
   return x
 end
 ```
@@ -42,43 +41,36 @@ Stacktrace:
   ...
 ```
 We got an error message and a long stacktrace. The error informs us that our code performs array mutation by calling `copyto!` (we might not have directly called this function, but it is being invoked somewhere in the call stack). We see that our code includes `x .= ...` which is given as an example of array mutation. Other examples of mutating operations include:
-- setting values (`x .= ...`)
-- appending/popping values (`push!(x, v)` / `pop!(x)`)
-- calling mutating functions (`mul!(C, A, B)`)
+- setting values (`x[i] = val` or `x .= values`)
+- appending/popping values (`push!(x, v)` or `pop!(x)`)
+- calling mutating functions (such as `LinearAlgebra.mul!(C, A, B)`)
 
 !!! warning
 
     Non-mutating functions might also use mutation under the hood. This can be done for performance reasons or code re-use.
 
 ```julia
-function g!(x, y)
-  x .= 2 .* y
-
+function g_inner!(x, y)
+  for i in eachindex(x, y)
+    x[i] = 2 * y[i]
+  end
   return x
 end
-g(y) = g!(similar(y), y)
-```
-Here `g` is a "non-mutating function," and it indeed does not mutate `y`, its only argument. But it still allocates a new array and calls `g!` on this array which will result in a mutating operation. You may encounter such functions when working with another package.
-
-Specifically for array mutation, we can use [`Zygote.Buffer`](@ref) to re-write our function. For example, let's fix the function `g!` above.
-```julia
-function g!(x, y)
-  x .= 2 .* y
 
-  return x
+function g_outer(y)
+  z = similar(y)
+  g_inner!(z, y)
+  return z
 end
+```
+Here `g_outer` does not mutate `y`, its only argument. But it still allocates a new array `z` and calls `g_inner!` on this array, which will result in a mutating operation. You may encounter such functions when working with another package.
 
-function g(y)
-  x = Zygote.Buffer(y) # Buffer supports syntax like similar
-  g!(x, y)
-  return copy(x) # this step makes the Buffer immutable (w/o actually copying)
-end
+How can you solve this problem?
+* Re-write the code not to use mutation. Here we can obviously write `g_better(y) = 2 .* y` using broadcasting. Many other cases may be solved by writing comprehensions `[f(x, y) for x in xs, y in ys]` or using `map(f, xs, ys)`, instead of explicitly allocating an output array and then writing into it.
+* Write a custom rule, defining `rrule(::typeof(g), y)` using what you know about `g` to derive the right expression.
+* Use another AD package instead of Zygote for part of the calculation. Replacing `g(y)` with `Zygote.forwarddiff(g, y)` will compute the same value, but when it is time to find the gradient, this job is outsourced to [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl). ForwardDiff has its own limitations but mutation isn't one of them.
 
-julia> gradient(rand(3)) do y
-         sum(g(y))
-       end
-([2.0, 2.0, 2.0],)
-```
+Finally, there is also [`Zygote.Buffer`](@ref) which aims to handle the pattern of allocating space and then mutating it. But it has many bugs and is not really recommended.
 
 ## Try-catch statements
 
@@ -136,7 +128,8 @@ For all of the errors above, the suggested solutions are similar. You have the f
 2. define a [custom `ChainRulesCore.rrule`](https://juliadiff.org/ChainRulesCore.jl/stable/rule_author/example.html)
 3. open an [issue on Zygote](https://github.com/FluxML/Zygote.jl/issues)
 
-Avoiding the operation is simple, just don't do it! If you are using a mutating function, try to use a non-mutating variant. If you are using `try`/`catch` statements, try to use more graceful error handling such as returning `nothing` or another sentinel value. Recall that array mutation can also be avoided by using [`Zygote.Buffer`](@ref) as discussed above.
+Avoiding the operation is simple, just don't do it! If you are using a mutating function, try to use a non-mutating variant. Instead of allocating an array and writing into it, try to make the output directly using broadcasting, `map`, or a comprehension.
+If you are using `try`/`catch` statements, try to use more graceful error handling such as returning `nothing` or another sentinel value.
 
 Sometimes, we cannot avoid expressions that Zygote cannot differentiate, but we may be able to manually derive a gradient. In these cases, you can write [a custom `rrule`](https://juliadiff.org/ChainRulesCore.jl/stable/rule_author/example.html) using ChainRules.jl. Please refer to the linked ChainRules documentation for how to do this. _This solution is the only solution available for foreign call expressions._ Below, we provide a custom `rrule` for `jclock`.
 ```julia