Algebraic Simplification

Mathematical equivalence allows compilers to rewrite computational graphs into more efficient forms without altering the final result. Algebraic simplification is an optimization technique that focuses on reducing the arithmetic complexity of the model itself. This process utilizes axioms of elementary algebra and linear algebra to identify redundant operations, replace expensive instructions with cheaper alternatives, and rearrange computations for better performance.

Frontend frameworks often generate sub-optimal graphs due to the usage of high-level abstractions. A user might write code that logically makes sense but introduces unnecessary transposes or effectively multiplies tensors by identity matrices. The goal of the algebraic simplification pass is to canonicalize the graph and prune these inefficiencies before the heavy lifting of tensor scheduling begins.

Strength Reduction

Strength reduction is the practice of replacing a computationally expensive operation with a functionally equivalent but hardware-cheaper instruction. In the context of deep learning accelerators, the cost difference between specific arithmetic logic units (ALUs) can be significant. Transcendental functions (like pow, exp, log) and division operations typically consume more clock cycles and die area than simple multiplication or addition.

Consider the power operation used frequently in variance calculations or Euclidean distance metrics. A generic power kernel handles arbitrary floating-point exponents, requiring complex logic. However, if the exponent is a small integer, the compiler can rewrite the node:

$y = x^2 \rightarrow y = x \cdot x$

$y = x^3 \rightarrow y = x \cdot x \cdot x$

This transformation replaces a call to a special function unit with simple multiplication instructions, which are often vectorizable and have higher throughput on GPUs. Similarly, division by a constant $C$ can be transformed into multiplication by the reciprocal $1/C$. Since multiplication is generally faster than division on most instruction set architectures (ISAs), this yields a distinct speedup.

$y = \frac{x}{C} \rightarrow y = x \cdot (C^{-1})$

The compiler pre-calculates $C^{-1}$ during the compilation phase (constant folding), so the runtime only sees a multiplication.

Identity and Redundancy Elimination

Graphs often contain operations that serve no computational purpose, appearing as artifacts of automatic differentiation or modular model design. Eliminating these nodes reduces kernel launch overhead and memory bandwidth usage.

Common identity transformations include:

• Additive Identity: $x + 0 \rightarrow x$
• Multiplicative Identity: $x \cdot 1 \rightarrow x$
• Idempotence: $\text{max}(x, x) \rightarrow x$

In tensor algebra, transpose cancellations are a frequent target for optimization. A transpose operation essentially swaps the strides of a tensor. Applying a transpose twice reverses the stride swap, returning the tensor to its original layout. If a graph contains two consecutive transpose operations that are inverses of each other, both nodes can be removed entirely.

$(A^T)^T \rightarrow A$

This logic extends to slicing and concatenation. If a tensor is sliced and then immediately concatenated back in the same order, the sequence is a no-op. The diagram below illustrates the structural simplification of a graph containing redundant linear algebra operations.

Optimization of a redundant subgraph. The transpose pair cancels out, and adding zero is an identity operation, allowing the Input to flow directly to Output without computation.

Associative Reordering in GEMM

One of the most impactful algebraic simplifications involves reordering matrix multiplications (GEMM). Matrix multiplication is associative, meaning $(AB)C = A(BC)$. However, the computational cost of these two evaluation orders is rarely identical. The cost depends heavily on the dimensions of the matrices involved.

Let us define the computational complexity of multiplying a matrix of size $M \times K$ by a matrix of size $K \times N$ as roughly $O(M \cdot N \cdot K)$.

Consider a chain of three matrices with dimensions:

• $A: 10 \times 100$
• $B: 100 \times 10$
• $C: 10 \times 100$

Order 1: $(A \times B) \times C$

1. Compute $T = A \times B$. Result $T$ is $10 \times 10$. Cost: $10 \cdot 10 \cdot 100 = 10,000$ FLOPs.
2. Compute $R = T \times C$. Result $R$ is $10 \times 100$. Cost: $10 \cdot 100 \cdot 10 = 10,000$ FLOPs.
3. Total: 20,000 FLOPs.

Order 2: $A \times (B \times C)$

1. Compute $T = B \times C$. Result $T$ is $100 \times 100$. Cost: $100 \cdot 100 \cdot 10 = 100,000$ FLOPs.
2. Compute $R = A \times T$. Result $R$ is $10 \times 100$. Cost: $10 \cdot 100 \cdot 100 = 100,000$ FLOPs.
3. Total: 200,000 FLOPs.

In this specific scenario, the first order is an order of magnitude faster than the second. A compiler can analyze the static shapes of tensors in the IR and inject a reordering pass to minimize total floating-point operations (FLOPs).

Comparison of floating-point operations required for different associative groupings of matrix multiplication. The optimized order significantly reduces computational load.

Broadcast and Layout Simplification

Modern deep learning frameworks rely heavily on broadcasting semantics (e.g., NumPy-style broadcasting). While convenient for the developer, broadcasting can obscure the underlying linear algebra. Simplification passes analyze broadcast patterns to identify opportunities for reduction.

For instance, a common pattern in normalization layers is the reduction of a tensor followed by a broadcast back to the original shape for division. If subsequent operations effectively mask or slice this tensor, the compiler may prove that the full broadcast is unnecessary.

Furthermore, algebraic properties allow for the propagation of layout transformations. Operations like ReLU or Tanh are element-wise and commutative with respect to data layout permutations. This allows the compiler to push transpose operations past activation functions to group them with other layout-sensitive operations (like Convolution), potentially enabling the fusion or elimination of the transposes discussed earlier.

$\text{ReLU}(\text{Transpose}(A)) \Leftrightarrow \text{Transpose}(\text{ReLU}(A))$

Floating Point Precision and Safety

It is important to note that while algebraic simplification is mathematically sound in the domain of real numbers $\mathbb{R}$, it is not always precise in the domain of floating-point arithmetic. Floating-point addition is not strictly associative due to rounding errors.

$(a + b) + c \neq a + (b + c)$

This discrepancy becomes significant when summing numbers of different magnitudes. Deep learning compilers often provide "fast math" flags. When enabled, the compiler treats floating-point numbers as real numbers, permitting aggressive reordering and simplification. When precision is critical (e.g., in scientific computing or gradient accumulation for high-precision training), these algebraic passes must be constrained to value-safe transformations only.

In the context of inference, where models are often resilient to minor noise, compilers generally default to aggressive algebraic simplification to maximize throughput. Implementers of compiler passes must explicitly categorize rewrite rules based on whether they preserve bitwise accuracy or introduce numerical divergence.