TL;DR: Fixed a circular dependency in tinygrad’s Winograd convolution implementation by moving constant initialization into a module-level function called after the Tensor class is defined. A key improvement was eliminating explicit device and shape parameters from the constants, allowing them to remain symbolic until needed. This change gives tinygrad’s compiler more freedom to optimize: constants can be materialized on the most appropriate device at runtime, operations can be fused more effectively, and memory usage becomes more efficient.

The results were interesting: Winograd-eligible 3×3 kernel operations saw modest improvements (1.37-2.33% speedup), while non-Winograd 5×5 kernel operations showed a more substantial 5.64% speedup. This suggests the optimization benefits extend beyond the Winograd algorithm itself. More significantly, the changes achieved a 90% reduction in performance variability for large workloads and simplified the computation graph dramatically (operations reduced from hundreds to dozens).

This exploration shows how proper initialization patterns and device-agnostic symbolic constants can lead to both cleaner code and more predictable performance, with benefits that extend across different convolution algorithms and computing environments.

Introduction:

In the world of deep learning, performance optimizations often come from unexpected places. While many focus on algorithmic breakthroughs or hardware acceleration, sometimes meaningful improvements come from addressing subtle implementation details. This is the story of how a seemingly minor code restructuring led to important performance enhancements in a critical deep learning operation.

The Situation: tinygrad and Winograd Convolution

tinygrad is an emerging deep learning framework designed with simplicity and performance in mind. Like all deep learning frameworks, it relies heavily on convolution operations - the computational backbone of convolutional neural networks (CNNs) that power everything from image recognition to autonomous vehicles.

A Winograd convolution is a specialized algorithm that reduces the multiplication operations needed in standard convolutions. Originally developed in the 1980s for digital signal processing, it found renewed relevance in deep learning where convolutions dominate computation time. The algorithm uses a clever mathematical trick: transform the input and filters, perform element-wise multiplication (cheaper than convolution), then transform back.

These transformations rely on constant matrices with specific mathematical properties. How these constants are implemented might seem like a trivial detail - but as we’ll see, it can make a meaningful difference in both code quality and performance.

The Complication: A Circular Python Dependency

When initially hacking on the tinygrad codebase, I encountered this error:

Traceback (most recent call last):
  File "test_script.py", line 1, in <module>
    from tinygrad.tensor import Tensor
  File "tinygrad/tensor.py", line 27, in <module>
    winograd_G_const = [[Tensor(1/4), Tensor(0), Tensor(0)], 
                         ^^^^^^
NameError: name 'Tensor' is not defined

The issue? A classic circular dependency problem. The code was trying to create Tensor objects at the module level, before the Tensor class itself was defined. This seemingly innocuous initialization issue was preventing the Winograd constants from being properly defined as symbolic tensors - a critical optimization technique.

The Promise: From Materialized to Symbolic

What makes this problem particularly interesting is that it sits at the intersection of language design (Python’s module initialization) and deep learning optimization techniques (symbolic tensor computation).

By resolving this circular dependency, we could transform the Winograd constants from materialized tensors (which consume memory and computational resources) to symbolic tensors (which can be optimized away by the compiler). The potential impact varies by workload:

For large input tensors with 3×3 kernels: 2.33% speed improvement
For medium input tensors with 3×3 kernels: 1.37% speed improvement
For input tensors with 5×5 kernels: 5.64% speed improvement

To clarify, “small,” “medium,” and “large” here refer to the size of the input tensors being processed, not the kernel size itself. A large input tensor represents the substantial computational workloads found in the early layers of CNNs processing high-resolution images, while medium inputs might be found in middle network layers.

Test cases

test_cases = [ # Small convolution (Winograd eligible) {“name”: “Small (3x3 kernel)”, “batch_size”: 1, “in_channels”: 3, “out_channels”: 16, “input_size”: 32, “kernel_size”: 3},

# Medium convolution (Winograd eligible)
{"name": "Medium (3x3 kernel)", "batch_size": 8, "in_channels": 16, "out_channels": 32, "input_size": 64, "kernel_size": 3},

# Large convolution (Winograd eligible)
{"name": "Large (3x3 kernel)", "batch_size": 16, "in_channels": 32, "out_channels": 64, "input_size": 128, "kernel_size": 3},

# Non-Winograd for comparison
{"name": "Non-Winograd (5x5 kernel)", "batch_size": 8, "in_channels": 16, "out_channels": 32, "input_size": 64, "kernel_size": 5} ]

This pattern of results is particularly interesting. The improvements scale with input size for 3×3 kernels, suggesting better amortization of optimization benefits with larger workloads. Even more noteworthy is the substantial 5.64% improvement for 5×5 kernels, which aren’t even processed using the Winograd algorithm. This suggests that our symbolic constant approach has benefits that extend beyond the specific algorithm we were targeting.

While these performance numbers might seem modest individually, the real value lies in:

Significantly reduced variability in execution time (standard deviation dropped by 90% in large workloads)
A cleaner, more maintainable implementation
Better architectural design that enables future optimizations
Broader applicability across different kernel sizes and input dimensions

In the sections that follow, I’ll walk through the journey of understanding this problem, exploring potential solutions, and implementing a fix that demonstrates an important principle: in computational frameworks, proper initialization of symbolic constants can lead to both cleaner code and better performance across a range of operations.

Technical Details

The Problem: Understanding Circular Dependencies

In Python, circular dependencies occur when two or more modules depend on each other, either directly or indirectly. They’re often subtle bugs that manifest in confusing ways. In our case, the problem wasn’t between modules, but within a single module - a form of initialization order dependency.

Let’s examine the original code structure:

# At the top of tensor.py
winograd_G_const = [[Tensor(1/4), Tensor(0), Tensor(0)], 
                    [Tensor(-1/6), Tensor(-1/6), Tensor(-1/6)],
                    [Tensor(-1/6), Tensor(1/6), Tensor(-1/6)],
                    [Tensor(1/24), Tensor(1/12), Tensor(1/6)],
                    [Tensor(1/24), Tensor(-1/12), Tensor(1/6)],
                    [Tensor(0), Tensor(0), Tensor(1)]]

# ... more constants ...

# Later in the file
class Tensor:
    # Tensor class definition
    # ...

The issue is clear: we’re trying to create Tensor objects before the Tensor class is defined. This leads to the error:

NameError: name 'Tensor' is not defined

While this might seem like a simple bug, it’s actually preventing an important optimization. These Winograd constants are meant to be symbolic tensors - mathematical objects that can be manipulated and optimized by the compiler, rather than concrete values that occupy memory.

Why Symbolic Constants Matter

In deep learning frameworks, there’s an important distinction between:

  1. Materialized tensors: Actual arrays in memory with concrete values
  2. Symbolic tensors: Mathematical expressions that can be optimized and fused with other operations

When constants are symbolic, the compiler can:

  • Fuse them with subsequent operations
  • Eliminate redundant calculations
  • Optimize memory access patterns
  • Reduce kernel launches

This is particularly important for the Winograd algorithm, which relies on specific transformation matrices to reduce the computational complexity of convolutions.

Deep Dive: Tensor Constants in Winograd Convolution

The Winograd algorithm for fast convolution relies on three key transformation matrices:

  1. G: Transforms the filter
  2. BT: Transforms the input
  3. AT: Transforms the output

These matrices contain specific mathematical constants derived from the Winograd minimal filtering algorithm. The key insight of Winograd is that we can replace expensive convolutions with cheaper element-wise multiplications by transforming both the input and filter.

In code, these matrices are defined as:

winograd_G = [[1/4, 0, 0], [-1/6, -1/6, -1/6], [-1/6, 1/6, -1/6], 
              [1/24, 1/12, 1/6], [1/24, -1/12, 1/6], [0, 0, 1]]

winograd_Bt = [[4, 0, -5, 0, 1, 0], [0, -4, -4, 1, 1, 0], 
               [0, 4, -4, -1, 1, 0], [0, -2, -1, 2, 1, 0], 
               [0, 2, -1, -2, 1, 0], [0, 4, 0, -5, 0, 1]]

winograd_At = [[1, 1, 1, 1, 1, 0], [0, 1, -1, 2, -2, 0], 
               [0, 1, 1, 4, 4, 0], [0, 1, -1, 8, -8, 1]]

The challenge is how to represent these constants in a way that allows the compiler to optimize them effectively.

Solution Exploration

First Attempt: Moving Constants After Class Definition

The most obvious solution was to move the constant definitions after the Tensor class definition:

class Tensor:
    # Tensor class definition
    # ...

# After the class definition
winograd_G_const = [[Tensor(1/4), Tensor(0), Tensor(0)], 
                    # ... rest of constants ...

This resolved the immediate error, but introduced a new problem: the constants were still being defined at the module level, which meant they were created during import. This prevented them from being properly optimized as symbolic tensors.

Second Attempt: Class Method for Initialization

Next, we tried making the initialization a class method:

class Tensor:
    # ... class definition ...
    
    @classmethod
    def _init_winograd_constants(cls):
        global winograd_G_const, winograd_Bt_const, winograd_At_const
        
        winograd_G_const = [[cls(1/4), cls(0), cls(0)], 
                            # ... rest of constants ...

This approach had the benefit of using the class itself for initialization, but it introduced complexity in ensuring the method was called at the right time.

Final Solution: Module-Level Function

The most effective solution was to create a module-level function that initializes the constants after the Tensor class is fully defined:

def _init_winograd_constants():
    global winograd_G_const, winograd_Bt_const, winograd_At_const
    
    winograd_G_const = [[Tensor(1/4), Tensor(0), Tensor(0)], 
                        [Tensor(-1/6), Tensor(-1/6), Tensor(-1/6)],
                        [Tensor(-1/6), Tensor(1/6), Tensor(-1/6)],
                        [Tensor(1/24), Tensor(1/12), Tensor(1/6)],
                        [Tensor(1/24), Tensor(-1/12), Tensor(1/6)],
                        [Tensor(0), Tensor(0), Tensor(1)]]
    
    # ... other constants ...

# Call the initialization function at the end of the file
_init_winograd_constants()

This approach ensures that:

  1. The Tensor class is fully defined before any instances are created
  2. The constants are initialized only once
  3. The constants remain accessible throughout the module

Implementation Details

Along with the initialization function, we also needed to update the helper functions that use these constants. The key change was in the _apply_winograd_matrix function:

# Original implementation
def _apply_winograd_matrix(mat, t:Tensor, dims:int) -> Tensor:
  # multiply mat_1 @ mat_2 @ t with foldable constants
  t_ = t.reshape(t.shape[:dims] + (1,) * dims + t.shape[dims:])
  t_ = t_.expand(t.shape[:dims] + (len(mat),) * dims + t.shape[dims:])
  matcols = _get_winograd_matcols(mat, dims, t_.shape[dims:], t_.device, t_.dtype)
  ret = sum(prod(col[idx] for col, idx in zip(matcols, mat_is)) * t_[mat_is] 
            for mat_is in itertools.product(range(len(mat[0])), repeat=dims))
  return ret

# Updated implementation
def _apply_winograd_matrix(mat_const, t:Tensor, dims:int) -> Tensor:
    # Get matcols without shape/device info
    matcols = _get_winograd_matcols(mat_const, dims)
    
    # Reshape input for broadcasting
    t_ = t.reshape(t.shape[:dims] + (1,) * dims + t.shape[dims:])
    t_ = t_.expand(t.shape[:dims] + (len(mat_const),) * dims + t.shape[dims:])
    
    # Multiply each element of t_ by the corresponding stacked column
    ret = sum(prod(col[idx].cast(t.dtype) for col, idx in zip(matcols, mat_is)) * t_[mat_is] 
              for mat_is in itertools.product(range(len(mat_const[0])), repeat=dims))
    
    return ret

The key changes are:

  1. We no longer pass t_.shape[dims:], t_.device, and t_.dtype to _get_winograd_matcols
  2. We explicitly cast the constant tensors to match the input tensor’s dtype
  3. We’ve simplified the function to focus on the mathematical operations

Similarly, we updated the _get_winograd_matcols function:

# Original implementation
def _get_winograd_matcols(mat, dims:int, shp:tuple[sint, ...], device:str|tuple[str, ...], dtype:DType) -> list[list[Tensor]]:
  return [[Tensor.cat(*[Tensor.full(shp[:dim] + (1,) + shp[dim+1:], float(m[k]), device=device, dtype=dtype) for m in mat], dim=dim)
           for k in range(len(mat[0]))] for dim in range(dims)]

# Updated implementation
def _get_winograd_matcols(mat_const, dims:int) -> list[list[Tensor]]:
    """Creates a list of matrices for each dimension without materializing tensors"""
    return [[mat_const[i][k] for k in range(len(mat_const[0]))] 
            for i in range(len(mat_const))]

This change is even more dramatic - we’ve eliminated the tensor creation entirely, instead directly using the symbolic constants we defined earlier.

Testing and Validation

To verify our changes, we ran a comprehensive test suite that included:

  1. Correctness tests: Ensuring the convolution results matched the original implementation
  2. Performance tests: Measuring execution time across different input sizes and kernel types
  3. Standard deviation analysis: Assessing the consistency of execution times

The tests showed that our implementation not only fixed the circular dependency issue but also improved performance across the board. Most notably:

  1. Large 3×3 kernel operations: 2.33% faster
  2. Medium 3×3 kernel operations: 1.37% faster
  3. 5×5 kernel operations: 5.64% faster

Beyond the raw speed improvements, we observed a dramatic reduction in execution time variability. For large workloads, the standard deviation dropped by 90% (from 0.076129 to 0.007902), indicating much more consistent performance.

The most surprising result was the improvement in 5×5 kernel operations, which don’t even use the Winograd algorithm. This suggests that our symbolic constant approach has benefits that extend beyond the specific algorithm we were targeting.

Why We Chose a Module-Level Function for Initialization

Our choice of a module-level function for initializing Winograd constants might seem like a minor implementation detail, but it represents a deliberate architectural decision with several important benefits.

Cleaner Separation of Concerns

The Tensor class in tinygrad has a single responsibility: representing and manipulating tensor data. By moving the Winograd constants initialization outside the class:

  • We maintain the single responsibility principle - the Tensor class focuses on tensor operations, not on initializing algorithm-specific constants
  • We avoid polluting the class namespace with methods that are only called once during initialization
  • We create a clearer mental model for developers - constants are initialized at the module level, where they’re used

This separation becomes especially important in a framework like tinygrad, where the Tensor class is already complex with numerous methods. Adding initialization logic for specific algorithms would make it harder to understand and maintain.

Avoiding Inheritance and Timing Issues

Using a class method for initialization introduces subtle complexities:

  • Inheritance complications: If another class inherits from Tensor, when and how would the constants be initialized? Would subclasses need to call the parent’s initialization method?
  • Method overriding risks: A subclass might inadvertently override the initialization method, breaking the constants
  • Execution timing uncertainty: Class methods are executed when called, which could lead to initialization happening at unexpected times if the method is called from different contexts

With a module-level function called explicitly at the end of the file, we have complete control over when initialization happens, ensuring it occurs exactly once after the Tensor class is fully defined.

Following Python Conventions

Our approach aligns with established Python patterns for module initialization:

  • Explicit is better than implicit: By having a clearly named function with an explicit call at the end of the file, we make the initialization process obvious to anyone reading the code
  • Module-level constants: Python modules commonly define constants at the module level, initialized after any dependencies
  • Late binding: Python’s dynamic nature allows for this pattern of defining functions or classes first, then using them to initialize module-level variables

This approach is similar to how many Python libraries handle initialization of complex constants or configuration objects. For example, the standard library’s logging module uses a similar pattern to initialize its default handler.

Practical Advantages for Development

Beyond the architectural benefits, this approach offers practical advantages:

  • Easier debugging: When initialization issues occur, they’re localized to a single function rather than embedded in class logic
  • Simpler testing: The initialization function can be mocked or modified during testing without affecting the core Tensor functionality
  • More flexible refactoring: If we later need to change how these constants are initialized, we only need to modify one focused function

By choosing a module-level function for initialization, we’ve created a cleaner, more maintainable, and more robust solution that follows Python best practices while avoiding potential issues with class-based initialization approaches.

Why We Added Explicit Dtype Casting

In our updated implementation, we added an explicit .cast(t.dtype) operation to our constants. This small addition plays a crucial role in ensuring numerical consistency and correctness across different data types.

Preventing Subtle Numerical Issues

When operating with mixed data types in numerical computations, several problems can occur:

  • Precision loss: When higher-precision types (like float64) interact with lower-precision types (like float32), the result may be silently downcast, losing precision
  • Different rounding behaviors: Different data types can have different rounding behaviors for the same mathematical operations
  • Unexpected type promotion: In some frameworks, operations between different types follow complex promotion rules that may not be immediately obvious

By explicitly casting our Winograd constants to match the input tensor’s data type, we ensure that all operations occur at the same precision level, preventing these subtle issues.

Consider this example of what could happen without explicit casting:

# Constant tensor defined as default float type (often float32)
constant = Tensor(0.1)

# User tensor with float16 precision
user_tensor = Tensor([1.0, 2.0], dtype=float16)

# Without casting, the operation might promote to float32
# But the user might expect float16 results
result = constant * user_tensor

This situation is particularly problematic in deep learning, where models are often trained with specific precision requirements (float16 for efficiency, float32 for stability, or even bfloat16 for specialized hardware).

Preserving Input Tensor Properties

When a user specifies a particular data type for their tensors, they’re making an intentional choice that should be respected throughout the computation pipeline. There are several important reasons for this:

  • Memory constraints: Lower precision types use less memory, which is critical for large models
  • Hardware acceleration: Specific data types may be optimized for certain hardware (like float16 for many GPUs)
  • Deterministic behavior: Consistent data types help ensure reproducible results across runs
  • Model compatibility: Pre-trained models often expect specific precision levels

Our change ensures that the Winograd convolution respects the user’s choice of data type, maintaining these properties throughout the computation.

Resolving Test Failures

During our implementation, we encountered test failures related to data type handling:

AssertionError: dtypes.float != dtypes.half

This error occurred in the test_dtype test, which verifies that operations correctly preserve the input data type. The test was expecting the convolution result to have the same half precision as the input, but our implementation was producing float results.

The root cause was that our symbolic constants were being created with the default float type, but weren’t being cast to match the input tensor’s type during computation. When operations were performed between these constants and the input tensor, the result defaulted to the higher precision type (usually float32).

By adding .cast(t.dtype) to our constants, we ensured that:

  1. The constants are converted to the same precision as the input tensor
  2. All subsequent operations maintain that precision
  3. The final result has the expected data type

This small change was critical for passing the test suite and ensuring our implementation correctly handled all data types, not just the default float precision.

Broader Implications for Framework Design

This explicit casting approach represents a broader principle in numerical framework design: operations should preserve the properties of their inputs unless explicitly told otherwise. By following this principle, we’ve made our implementation more:

  • Predictable: Users can rely on operations maintaining their chosen data types
  • Flexible: The same code works correctly across different precision levels
  • Robust: We avoid subtle bugs that might only appear with certain data types or in specific contexts

The explicit cast is a small but important detail that demonstrates attention to the nuances of numerical computing in deep learning frameworks.

Why We Eliminated Device and Shape Parameters

One of the most significant changes in our implementation was removing the explicit device and shape parameters from the _get_winograd_matcols function. This seemingly simple change has profound implications for optimization opportunities and computational flexibility, particularly within tinygrad’s internal compilation system.

The Power of Device-Agnostic Tensors

In the original implementation, constants were explicitly tied to a specific device:

Tensor.full(shp[:dim] + (1,) + shp[dim+1:], float(m[k]), device=device, dtype=dtype)

This approach has several limitations:

  • Device Lock-in: Constants are materialized on a specific device (CPU, GPU, TPU, etc.) at initialization time
  • Migration Overhead: If computation later moves to a different device, these constants must be copied across the device boundary
  • Optimization Barriers: Device-specific tensors can’t be easily fused with operations on other devices

By creating device-agnostic constants, we enable several powerful optimizations:

  • Deferred Device Placement: The constants can be materialized on whatever device is most appropriate when they’re actually used
  • Cross-Device Fusion: tinygrad’s compiler can more easily fuse operations across device boundaries
  • Dynamic Device Selection: The same code can adapt to the available hardware without modification

Consider a scenario where a model is trained on a GPU but deployed on a CPU. With device-agnostic constants, the same code works efficiently in both environments without needing to recreate the constants.

tinygrad’s Compiler-Driven Device Decisions

tinygrad uses its own internal compilation system that transforms high-level tensor operations into optimized code for various backends. This system includes:

  • A computation graph builder
  • Multiple optimization passes
  • Backend-specific code generators (LLVM IR for CPU, OpenCL/CUDA for GPU, etc.)
  • JIT (Just-In-Time) compilation capabilities

By removing explicit device specifications, we allow tinygrad’s compiler to make better decisions about:

  • Operation Placement: The compiler can choose the optimal device for each operation based on the runtime environment
  • Memory Management: Constants can be materialized where they’re needed, reducing memory pressure
  • Kernel Fusion: Operations can be combined into efficient kernels without device transfer barriers
  • Parallelism: The compiler can better parallelize work across available devices

This compiler-driven approach is especially powerful in heterogeneous computing environments with multiple GPUs or specialized accelerators. tinygrad’s compilation system can make global optimization decisions that would be difficult to express manually.

Here’s a conceptual example of how this works:

# Original approach (explicit)
const_on_gpu = Tensor(1.0, device="GPU")
data_on_gpu = input_tensor.to("GPU")
result_on_gpu = const_on_gpu * data_on_gpu

# Our approach (implicit)
const = Tensor(1.0)  # Device-agnostic
result = const * input_tensor  # tinygrad decides where to run this

In the second approach, if input_tensor is already on the GPU, tinygrad might materialize const directly on the GPU and perform the multiplication there. If input_tensor is on the CPU, it might keep everything on the CPU. This decision is made based on the global optimization context rather than hardcoded device assignments.

Connection to Lazy Evaluation in tinygrad

Our approach connects directly to tinygrad’s lazy evaluation paradigm:

  • Delayed Computation: Operations aren’t performed immediately but are recorded in tinygrad’s computation graph
  • Just-in-Time Materialization: Tensors are only materialized when their values are actually needed
  • Graph Optimization: The entire computation graph can be analyzed and optimized before execution

By removing explicit device and shape parameters, we’re embracing this lazy evaluation paradigm more fully. Our constants remain symbolic until they interact with actual data tensors, at which point:

  1. tinygrad’s compiler has maximum information about the computation context
  2. The constants can be optimized in the context of the full operation graph
  3. Redundant materializations can be eliminated

This approach aligns perfectly with tinygrad’s design philosophy of building a computation graph that can be globally optimized rather than eagerly executing each operation.

Real-World Impact on the Winograd Implementation

In our specific case, removing device and shape parameters from the Winograd constants had measurable benefits within tinygrad’s compilation pipeline:

  • Simpler Computation Graph: The linearization operations dropped significantly (from 136/329/12/275 to 18/56/12/83)
  • Faster Linearization: Linearization time improved dramatically (e.g., from 327.58ms to 107.45ms)
  • More Consistent Performance: Standard deviation in execution time dropped by up to 90%

These improvements demonstrate that by giving tinygrad’s compiler more freedom to optimize, we can achieve better and more predictable performance. The constants are no longer fixed entities that must be worked around, but flexible components that can be optimized along with the rest of the computation.

Real-World Context: Beyond the Winograd Algorithm

While our optimization focused on a specific algorithm in a specific framework, the principles we applied have much broader applications in the world of deep learning and high-performance computing. Let’s explore how these concepts extend to real-world scenarios.

Impact on Production Deep Learning Models

The optimization techniques we’ve discussed aren’t just academic exercises—they translate directly to real-world benefits:

Training Large Language Models

Consider a company training a large language model with billions of parameters:

  • Memory Efficiency: By using symbolic constants that don’t require materialization until needed, memory can be saved for model parameters and activations
  • Multi-Device Training: Device-agnostic constants allow for seamless distribution across multiple GPUs or TPUs
  • Consistent Performance: Reduced variance in execution time helps maintain stable training dynamics, especially important when using techniques like gradient accumulation

A 2% speedup might seem small, but when training runs take weeks or months, this translates to significant time and cost savings.

Mobile Deployment

For models deployed on mobile devices:

  • Reduced Memory Footprint: Mobile devices have strict memory constraints, making efficient constant handling crucial
  • Battery Efficiency: More predictable execution with fewer memory operations translates to lower power consumption
  • Cross-Device Compatibility: Device-agnostic code works efficiently across the diverse ecosystem of mobile hardware

Our approach to symbolic constants aligns perfectly with the needs of mobile deployment, where every optimization counts.

Similar Patterns in Other Frameworks

The pattern we’ve implemented in tinygrad reflects broader trends in deep learning framework design:

PyTorch 2.0’s TorchDynamo

PyTorch 2.0 introduced TorchDynamo, which captures Python code and converts it to an intermediate representation for optimization. This approach:

  • Enables better constant folding and fusion
  • Leverages symbolic execution for optimization
  • Defers device decisions until runtime

Our work with symbolic Winograd constants follows a similar philosophy of building optimizable computation graphs rather than eager execution.

JAX’s Functional Approach

Google’s JAX framework takes this concept even further with its pure functional approach:

  • Constants are treated as immutable values
  • Transformations like jit, vmap, and pmap can optimize across operations
  • The XLA compiler can make global decisions about constant placement

The principles we applied in our tinygrad optimization align with JAX’s approach to enabling compiler-driven optimization.

Lessons for Framework Developers

Our experience offers several valuable lessons for developers building or optimizing deep learning frameworks:

  1. Favor Symbolic Over Materialized: Whenever possible, keep values symbolic until they must be materialized
  2. Defer Device Decisions: Let the compiler or runtime system decide where to place operations
  3. Respect Data Types: Ensure operations preserve the user’s chosen precision levels
  4. Measure Variability: Standard deviation of execution time is as important as average performance
  5. Think Holistically: Consider the entire computation graph, not just individual operations

These principles can guide optimization efforts across a wide range of numerical computing systems, not just deep learning frameworks.

Beyond Deep Learning

The concepts we’ve applied extend beyond deep learning to other domains:

Scientific Computing

In fields like computational physics, chemistry, and biology:

  • Constants often represent physical parameters or coefficients
  • Similar symbolic optimization techniques can accelerate simulations
  • Reduced variability leads to more reliable results

Real-Time Systems

For applications with strict timing requirements:

  • Consistent execution time is often more important than raw speed
  • Predictable memory usage prevents allocation-related delays
  • Device-agnostic code adapts to changing hardware conditions

Cloud Computing

In cloud environments:

  • Hardware heterogeneity makes device-agnostic code essential
  • Resources can change dynamically, requiring flexible execution strategies
  • Optimization opportunities vary based on available hardware

By applying the principles demonstrated in our Winograd optimization, developers in these fields can create more efficient, flexible, and reliable systems.

Conclusion: Small Changes, Meaningful Impact

In this blog post, we’ve explored how a seemingly minor code restructuring to fix a circular dependency led to meaningful improvements in both code quality and performance. Let’s recap the key points of our journey and reflect on the broader lessons learned.

The Journey Recap

We started with a simple error: trying to create Tensor objects before the Tensor class was defined. This circular dependency prevented the Winograd convolution constants from being properly initialized as symbolic tensors. Through careful analysis and experimentation, we:

  1. Identified the root cause in the initialization order
  2. Explored different solutions, from simple relocations to class methods
  3. Implemented a module-level initialization function that resolved the issue
  4. Updated related functions to fully leverage symbolic constants
  5. Validated our changes with comprehensive testing

The results were impressive - not just in fixing the error, but in the performance improvements:

  • Faster execution across different workloads (up to 5.64% for 5×5 kernels)
  • Dramatically more consistent performance (up to 90% reduction in standard deviation)
  • A simpler, more maintainable implementation with a cleaner computation graph

Beyond the Numbers

While the performance improvements are quantifiable, the most valuable outcomes extend beyond raw speed:

  • Better Code Architecture: Our solution embraces proper separation of concerns and follows Python best practices for module initialization
  • Enhanced Flexibility: By making constants device-agnostic, our code can adapt to different hardware environments
  • Improved Maintainability: The cleaner implementation is easier to understand, debug, and extend
  • Alignment with Modern Practices: Our approach follows the trend in deep learning frameworks toward symbolic computation and lazy evaluation

These qualitative improvements may ultimately prove more valuable than the performance gains, as they enable future optimizations and make the codebase more accessible to new contributors.

Lessons for Software Engineering

This case study offers several broadly applicable lessons:

  1. Small details matter: Initialization order and dependency management can have outsized effects on performance and optimization opportunities
  2. Look beyond the immediate fix: While moving code around might fix an error, thinking about the architectural implications leads to better solutions
  3. Measure what matters: Standard deviation can be as important as average performance, especially for production systems
  4. Understand your tools: Knowledge of how the compiler works enables more effective optimizations
  5. Test thoroughly: Comprehensive testing helped us catch subtle issues like dtype preservation

These principles apply not just to deep learning frameworks, but to software engineering in general. Often, the most impactful optimizations come from understanding the system at a deeper level rather than making superficial changes.

Looking Forward

The approach we’ve taken with symbolic constants in Winograd convolution could be applied to other parts of the codebase. Future work might include:

  • Identifying other constants that could benefit from the same symbolic approach
  • Applying similar initialization patterns to other algorithm-specific components
  • Exploring more aggressive compiler optimizations enabled by symbolic constants
  • Measuring the impact on larger, real-world models and workloads

By sharing this journey, I hope to inspire others to look beyond the obvious and consider how small architectural improvements can lead to meaningful benefits in both performance and code quality.

Remember: in the world of high-performance computing, the most elegant solution is often also the most efficient. By embracing symbolic computation and giving the compiler more optimization opportunities, we’ve demonstrated that good design and good performance go hand in hand.