Does tensorflow use automatic or symbolic gradients?

Question

I haven\'t been able to find a clear statement of whether tensorflow uses automatic or symbolic differentiation.

I skimmed the tensorflow paper and they mention auto

Answer 1

By "automatic differentiation" you may be thinking of "differentiation by finite differences" where you approximate derivative of f(x) as [f(x+e)-f(x-e)]/(2e). However, automatic differentiation is different and the finite difference method is an example of "numerical differentiation".

TensorFlow uses reverse mode automatic differentiation for it's gradients operation and finite difference method for tests that check validity of gradient operation like here.

Finite difference method is not practical for high dimensional problems, whereas reverse mode automatic differentiation gives you derivative of a "many->1" function at roughly the same cost as computing original function

You can see the core of the implementation here.

Implementation of differentiation method is symbolic in a sense that gradients operation takes computational graph and produces computational graph that can be fed back into "gradients" operation to get higher level derivatives.

Here's an example

tf.reset_default_graph()
x = tf.Variable(0.)
y = tf.square(x)
z = tf.gradients([y], [x])

Here's the graph you get

There are some extra operators since same graph code will work for higher dimensional x, but the point is that you see both x^2 and 2*x in the same graph.

Answer 2

Afaik symbolic differentiation means working with a mathematical, symbolic equation (i.e., symbolic math equation in, derivative of the equation out). Automatic differentiation computes derivatives based on computational functions (which in turn are broken down into basic operations such as addition/subtraction and multipliation/division).

Since TensorFlow does differentiation based on a computation graph of operations, I'd intuitively say that it's automatic differentiation (I don't know of any other technique that would be appropriate here; I think the possibility that TensorFlow is converting the computation graph into a mathematical equation that is then parsed to compute the derivative of that equation is prob. out of question). The authors say "symbolic differentiation" in the TensorFlow whitepaper though -- however, I think this may be a misnomer similar to "Tensor" instead of "(multi-dimensional) data array" if you'd ask a mathematician.

Answer 3

TF uses automatic differentiation and more specifically reverse-mode auto differentiation.

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There are 3 popular methods to calculate the derivative:

1. Numerical differentiation
2. Symbolic differentiation
3. Automatic differentiation

Numerical differentiation relies on the definition of the derivative: , where you put a very small h and evaluate function in two places. This is the most basic formula and on practice people use other formulas which give smaller estimation error. This way of calculating a derivative is suitable mostly if you do not know your function and can only sample it. Also it requires a lot of computation for a high-dim function.

Symbolic differentiation manipulates mathematical expressions. If you ever used matlab or mathematica, then you saw something like this

Here for every math expression they know the derivative and use various rules (product rule, chain rule) to calculate the resulting derivative. Then they simplify the end expression to obtain the resulting expression.

Automatic differentiation manipulates blocks of computer programs. A differentiator has the rules for taking the derivative of each element of a program (when you define any op in core TF, you need to register a gradient for this op). It also uses chain rule to break complex expressions into simpler ones. Here is a good example how it works in real TF programs with some explanation.

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You might think that Automatic differentiation is the same as Symbolic differentiation (in one place they operate on math expression, in another on computer programs). And yes, they are sometimes very similar. But for control flow statements (`if, while, loops) the results can be very different:

symbolic differentiation leads to inefficient code (unless carefully done) and faces the difficulty of converting a computer program into a single expression