- Aesara applies many kinds of graph optimizations, with different objectives:
The optimizations are listed in roughly chronological order. The table below gives a quick summary of the optimizations included in the default modes. The descriptions are brief and point to further reading.
If you would like to add an additional optimization, refer to Graph optimization in the guide to extending Aesara.
When compiling, we can make a tradeoff between compile-time and run-time. Faster compile times will result in fewer optimizations being applied, hence generally slower run-times. For making this tradeoff when compiling, we provide a set of 4 optimization modes, ‘o1’ to ‘o4’, where ‘o1’ leads to fastest compile-time and ‘o4’ leads to fastest run-time in general. For an even faster run-time, we could disable assertions (which could be time consuming) for valid user inputs, using the optimization mode ‘unsafe’, but this is, as the name suggests, unsafe. (Also see note at Unsafe optimization.)
This list is partial.
The print_summary method allows several OpDBs and optimizers to list the executed optimizations. This makes it possible to have an up-to-date list.
python -c "import aesara; aesara.compile.optdb.query(aesara.compile.predefined_optimizers['<OPT_ID>']).print_summary()"
|Optimization||o4 *||o3||o2||o1 †||Stabilization||unsafe|
*) o4 is equivalent to fast_run
†) o1 is equivalent to fast_compile
A simple optimization in which redundant Apply nodes are combined. For example, in
function([x,y], [(x+y)*2, (x+y)*3])the merge optimization will ensure that
yare only added once.
This optimization is very useful because it frees users to write highly redundant mathematical code. Aesara will make sure to compute just what is necessary.
- constant folding¶
When all the inputs to an expression are constant, then the expression can be pre-computed at compile-time.
- shape promotion¶
Aesara often knows how to infer the shape of an output from the shape of its inputs. Without this optimization, it would otherwise have to compute things (e.g.
log(x)) just to find out the shape of it!
- fill cut¶
Fill(a,b)means to make a tensor of the shape of
afull of the value
b. Often when fills are used with elementwise operations (e.g. f) they are un-necessary: *
f(fill(a,b), c) -> f(b, c)*
f(fill(a, b), fill(c, d), e) -> fill(a, fill(c, f(b, d, e)))
- inc_subtensor serialization¶
Incrementing a small subregion of a large tensor can be done quickly using an inplace operation, but if two increments are being done on the same large tensor, then only one of them can be done inplace. This optimization reorders such graphs so that all increments can be done inplace.
inc_subtensor(a,b,idx) + inc_subtensor(a,c,idx) -> inc_subtensor(inc_subtensor(a,b,idx),c,idx)
This optimizes graphs like
reshape(reshape(x, shape1), shape2)->
- constant elimination¶
Many constants indicate special cases, such as
pow(x,1) -> x. Aesara recognizes many of these special cases.
- add canonicalization¶
Rearrange expressions of additions and subtractions to a canonical form:
- mul canonicalization¶
Rearrange expressions of multiplication and division to a canonical form:
This simple optimization replaces dot(matrix, matrix) with a special
dot22op that only works for matrix multiplication. This op is implemented with a call to GEMM, and sometimes replaced entirely by the gemm optimization.
Aesara has a sparse matrix multiplication algorithm that is faster in many cases than scipy’s (for dense matrix output). This optimization swaps scipy’s algorithm for ours.
This optimizes graphs like
sum(scalar * tensor)->
scalar * sum(tensor)
Composition of two negatives can be cancelled out.
Matching negatives in both the numerator and denominator can both be removed.
- add specialization¶
This optimization simplifies expressions involving the addition of zero.
- mul specialization¶
Several special cases of mul() exist, and this optimization tries to recognize them. Some examples include: *
- pow specialization¶
Several special cases of pow() exist, and this optimization tries to recognize them. Some examples include: *
In order to be a pure Op, setsubtensor must copy its entire input, and modify just the subtensor in question (possibly a single element). It is much more efficient to modify that element inplace.
Numerical libraries such as MKL and ATLAS implement the BLAS-level-3 interface, and provide a function
GEMMthat implements , for matrices
Z, and scalars .
This optimization tries to rearrange a variety of linear algebra expressions into one or more instances of this motif, and replace them each with a single
When one of the inputs to an elementwise expression has the same type and shape as the output, and is no longer needed for computation after the elemwise expression is evaluated, then we can reuse the storage of the input to store the output.
Typically when a graph uses random numbers, the RandomState is stored in a shared variable, used once per call and, updated after each function call. In this common case, it makes sense to update the random number generator in-place.
- elemwise fusion¶
This optimization compresses subgraphs of computationally cheap elementwise operations into a single Op that does the whole job in a single pass over the inputs (like loop fusion). This is a win when transfer from main memory to the CPU (or from graphics memory to the GPU) is a bottleneck.
- GPU transfer¶
The current strategy for choosing which expressions to evaluate on the CPU and which to evaluate on the GPU is a greedy one. There are a number of Ops *TODO* with GPU implementations and whenever we find a graph copying data from GPU to CPU in order to evaluate an expression that could have been evaluated on the GPU, we substitute the GPU version of that Op for the CPU version. Likewise if we are copying the output of a Op with a GPU implementation to the GPU, then we substitute the GPU version for the CPU version. In this way, if all goes well, this procedure will result in a graph with the following form:
- copy non-shared inputs to GPU
- carry out most/all computations on the GPU
- copy output back to CPU
When using a GPU,
shared()will default to GPU storage for ‘float32’ ndarray arguments, and these shared variables act as seeds for the greedy algorithm.
- This is a stabilization optimization. It can happen due to rounding errors that the softmax probability of one value gets to 0. Taking the log of 0 would generate -inf that will probably generate NaN later. We return a closer answer.
This is an unsafe optimization. For the fastest possible Aesara, this optimization can be enabled by setting
optimizer_including=local_remove_all_assertwhich will remove all assertions in the graph for checking user inputs are valid. Use this optimization if you are sure everything is valid in your graph.