Overcoming the Limitations of Training Models in AI with GPUs

I have been discussing with the techbros around here for days different alternatives to the ludicrously brute-force approaches used to train mainly ANNs, just by realizing how economical a human brain's consumption is in comparison to an equivalent network of GPUs.

We use logarithmic units to take a exact idea of the comparison of energy efficiency between the human brain and the GPU network:

1. Energy Consumption of the Human Brain per Day: 20W/hr 480 W/d.
2. Energy Consumption of the GPU Network per Day (100): 7200000 W/d.

The  REG ( Ratio Energy Consumption) =  GPU Network EC / Human Brain EC
The  REG in logunits = 4.18  or roughly 15K times, more energy-consuming than the human brain for comparable computational task.          

So it seems pretty clear that the AI technology we have today is doing something very wrong and, above all, is ludicrously inefficient by several orders of magnitude compared to what nature itself has achieved without the need for any engineering intelligence.

Thus, for the techbro group, it becomes increasingly evident that traditional training models are terribly inefficient and expensive, beyond any justified measure, regarding issues of scalability and efficiency, and in adapting to diverse network architectures. In the midst of these challenges, our latest article, "Overcoming the Limitations of Training Models in AI," tries to show a trace of hope by using an alternative approach to handle this complexity in a more effective manner.

The novel approach synthesizes the robustness of functional programming with the raw power of parallel computing. It integrates the elegance of Abstract Algebra (exemplified in profunctor abstractions) with a C++ fiber-based framework (fibers are closer to the equivalent biological brain synapses both in detailed communication capabilities and in economical energetic functioning than traditional threads). With this fusion of concepts, we not only start addressing the existing limitations in a scientific way but also pave the way for a more versatile, efficient, and scalable AI training model. In this series of articles, we will showcase the power of this approach with realistic and highly profitable engineering examples. Let's start with the training series.

I.- INTEGRATION with FIBERS instead of THREADS.

Our scenario is to train a machine learning model on a multi-processor system with several different architectures, using modern C++ and Boost.Fiber. We aim to train the model on a large dataset with the least consumption of energy compared to threads, and using a more effective architecture of fiber switch and intercommunication, like NUMA. Let's take a brief closer look at those architectures to optimize access, memory, and distribution of tasks along our fibers/threads/processes (NUMA, Work-Stealing, and Shared-Work Scheduling).

ARCHITECTURES WE WILL USE WITH OUR FIBERS

NUMA

• Purpose: NUMA optimizations are more about how memory is accessed in relation to processors, while work-stealing/shared-work is about efficiently distributing computational tasks among processors. So NUMA architecture is designed to optimize the memory access patterns in multi-processor systems. In NUMA, processors access their local memory faster than non-local memory (memory local to other processors).
• Advantage Memory access speed. When a processor accesses memory that is local to it, the operation is faster, leading to better overall performance, especially in memory-intensive applications.
• Use Cases: Multi-processor systems where memory access speed is a bottleneck.

WORK-STEALING AND SHARED-WORK SCHEDULING

- Purpose: These are algorithms are used to balance CPUs utilization with distributed parallel tasks. Work-stealing, for example, allows idle processors to 'steal' tasks from busy processors.

- Advantage: the CPU utilization is more efficient. By dynamically redistributing tasks, all processors are kept busy, minimizing idle time.

- Use Case: Scenarios where tasks are unevenly distributed, causing some threads or processors to be idle while others are overloaded.

COMPLEMENTARY USE OF ARCHITECTURES

In many high-performance systems, NUMA optimizations are implemented alongside intelligent task scheduling algorithms like work-stealing. They can complement each other – NUMA ensures efficient memory access, and work-stealing ensures efficient CPU usage.

In summary, Our use of these architectures will be depended on the specific performance bottlenecks and architectural needs of the system in question. In an ideal setup, both concepts can be leveraged together for optimal performance.

Both NUMA (Non-Uniform Memory Access) and work-stealing/shared-work architectures, individually or in combination, are crucial in several engineering domains. In these engineering/business domains, the choice between NUMA, work-stealing, shared-work, or a combination depends on specific requirements such as data size, processor architecture, and performance goals. Properly leveraging these architectures can lead to brutal improvements in efficiency, speed, and overall system performance.

High-Performance Computing (HPC):

- Application: Used in scientific research, weather forecasting, climate research, oil and gas exploration, and more.

- Why NUMA and Work-Stealing/Shared-Work?: HPC tasks involve complex calculations and massive data sets. NUMA helps in efficient memory access across multiple processors, and work-stealing ensures efficient CPU utilization, crucial for parallel processing tasks.

Data Centers and Cloud Computing:

- Application: Running large-scale web services, cloud storage, and processing big data.

- Why NUMA and Work-Stealing/Shared-Work?: Data centers house a vast number of multi-processor systems. Optimizing memory access and CPU workload distribution is key to improving response times and handling multiple simultaneous requests efficiently.

3. Real-Time Systems and Simulations:

- Application: Used in flight simulations, real-time analytics, gaming, and virtual reality.

- Goals: These systems require rapid processing and memory access to deliver real-time responses. Efficient task scheduling and memory management are critical to maintaining performance and responsiveness.

4. Financial Trading Systems:

- Application: High-frequency trading platforms and risk management systems.

- Goals: extremely fast processing for transactions and data analysis. The low-latency memory access of NUMA architectures and efficient task distribution can provide the needed speed.

5. Telecommunications:

- Application: Network infrastructure, signal processing, and data routing.

- Goals: Telecom systems require the management of large data flows and most of the time rely on multi-processor systems. Efficient memory and processor utilization are key to maintaining high data throughput and low latency.

6. Bioinformatics and Genomics:

- Application: DNA sequencing, protein structure prediction, and genomic analysis.

- Goals: These fields deal with extremely large datasets. Efficient parallel processing and memory usage are essential for analyzing and processing genomic data within reasonable timeframes.

7. Artificial intelligence and Machine learning

- Application: Neural networks, deep learning, and large-scale machine learning algorithms.

- Goals: AI and ML workloads involve complex computations distributed across many processors. Optimizing both memory access and CPU usage is crucial for training models efficiently.

II.- IMPLEMENTATION WITH FIBERS

1. NUMA-Aware Allocation (for Memory):

• As mentioned crucial to optimize memory access patterns. Utilize NUMA-aware memory allocators we can assure that data segments are allocated in the memory local to the CPU that processes them.
• We can use libraries like libnuma for NUMA-specific memory management.

2. Advanced Scheduling for Fibers (For tasks):

• Implement a custom scheduler for fibers to efficiently manage CPU resources. This scheduler should be capable of work-stealing, where idle CPUs as said can take over tasks from busy ones.
• Consider dynamic load balancing based on runtime metrics to optimize resource utilization.

3. Data Segment Management:

• The splitting of the dataset into segments accounts for the uneven distribution of data complexity. Some segments may require more processing time than others.
• We can Implement a mechanism to dynamically adjust segment sizes or redistribute segments based on processing time predictions or real-time metrics.

4. Efficient Synchronization:

• We can go beyond barriers for synchronization, and use a lock-free data structures and atomic operations to minimize synchronization overhead.
• Implementantion of fine-grained synchronization to reduce idle times and improve overall throughput is further much easier with fibers than with threads.

5. Error Handling and Fault Tolerance:

• Design the system to handle failures gracefully. This includes handling exceptions within fibers and ensuring data integrity in case of partial failures (We will replace this with Monads to avoid the gaps in the type system associated with exceptions and errors. Monads offer a more structured and reliable way of handling these issues, yielding a smoother and more robust system operation.)
• Consider checkpointing mechanisms to save the state of the model at intervals, enabling recovery from a known good state.

6. Profiling and Optimization:

• We can further profit of profiling tools to identify bottlenecks in the system. This includes CPU profiling, memory access patterns, and fiber scheduling efficiency.
• Continuously optimize based on the profiling data. This might include tuning the size of data segments, adjusting the fiber-to-CPU bindings, or modifying the scheduling algorithm.

7. Integration with Machine Learning Libraries:

• Easy integration with parallelization capabilities and optimizations provided by our machine learning libraries.
• Otherwise we can adapt those ones to our NUMA architectures and integrate with our fiber-based design.

8. Scalability and Flexibility:

• Design the system to be scalable, both in terms of dataset size and the number of CPUs becomes much easier-
• The system remains flexible to adapt to different machine learning models and varying computational requirements.

GRAL CODE FOR FIBERS: example of a fiber pool implementation using Boost.Fiber in C++.

// File: fiber_fun_pool.cpp
#include <boost/fiber/all.hpp>
#include "super_semaphore.cpp"

thread minionCreator(boost::fibers::barrier& giggleGate, super_semaphore& partyOver, int giggleCount) {
    return move(thread([&]() {
        boost::fibers::use_scheduling_algorithm<boost::fibers::algo::shared_laughter>();
        giggleGate.wait();
        while (partyOver.try_acquire() == false)
            boost::this_fiber::chillax();
    }));
}

void chuckleFest() {
    for (int chuckle = 0; chuckle < 10; chuckle++) {
        boost::this_fiber::sleep_for(chrono::duration<int, milli>(rand() % 100));
        cout << " Chuckle #" << chuckle << " from fiber " << boost::this_fiber::get_id()
             << " rocking on thread " << this_thread::get_id() << endl;
        boost::this_fiber::chillax();
    }
}

int main() {
    int laughTracks = thread::hardware_concurrency();
    boost::fibers::barrier giggleGate(static_cast<size_t>(laughTracks));
    super_semaphore partyOver(0);
    unique_ptr<thread> minions[laughTracks - 1];

    boost::fibers::use_scheduling_algorithm<boost::fibers::algo::shared_laughter>();

    for (int i = 0; i < laughTracks - 1; i++)
        minions[i] = make_unique<thread>(minionCreator(giggleGate, partyOver, laughTracks));

    giggleGate.wait();

    boost::fibers::fiber funFibers[100];
    for (int i = 0; i < 100; i++) {
        funFibers[i] = boost::fibers::fiber(chuckleFest);
        // funFibers[i].detach();
    }

    for (int i = 0; i < 100; i++)
        funFibers[i].join();

    partyOver.release(laughTracks - 1);

    for (int i = 0; i < laughTracks - 1; i++)
        minions[i]->join();

    return 0;
}
        

KEY POINTS of the code:

• Barrier Implementation- Boost.Fiber Barrier
• The barrier (boost::fibers::barrier) is used to synchronize threads. It ensures that all worker threads set up their fiber scheduler before any fibers are created. This is crucial to avoid race conditions where a fiber might be scheduled before all threads are ready.
• Thread Pool SetupThread Creation The workerSetup function creates worker threads. Each thread runs a lambda function that first sets a scheduling algorithm for fibers then waits at the barrier . It then enters a loop , yielding control if the done semaphore is not acquired, effectively waiting for work to be finished.
• In main (Main Thread):(a) It determines the number of logical cores (minions) and sets up an equal number of threads(b) Each thread, including the main thread, sets up a fiber scheduler and synchronizes at the barrier. (c) After passing the barrier, the main thread creates fibers to execute

Reminder on the Key Concepts

1. Thread vs. Fiber: Threads are kernel-level entities managed by the OS, while fibers are user-level entities managed by the application. Fibers are lightweight and provide a means for cooperative multitasking within a thread.

2. Shared Work Scheduler: The scheduler used here (boost::fibers::algo::shared_work) allows fibers to be executed by any thread in the pool, providing load balancing and efficient use of system resources.

3. Concurrency Control: The use of barriers and semaphores is crucial for coordinating the start and end of threaded work and ensuring that no thread or fiber runs ahead of the setup sequence.

This implementation showcases our blend of threading and fiber technologies to achieve efficient task parallelism and concurrency control, leveraging the capabilities of modern GPUS too.

SAME CODE BUT NOW USING NUMA INSTEAD...

// File: fiber_fun_pool_numa.cpp
#include <boost/fiber/all.hpp>
#include "super_semaphore.cpp"
#include <numa.h> // Include NUMA library

thread minionCreator(boost::fibers::barrier& giggleGate, super_semaphore& partyOver, int giggleCount, int cpuId) {
    return move(thread([&]() {
        // NUMA: Bind thread to a specific CPU
        // Example: Set thread affinity to cpuId

        boost::fibers::use_scheduling_algorithm<boost::fibers::algo::shared_laughter>();
        giggleGate.wait();
        while (partyOver.try_acquire() == false)
            boost::this_fiber::chillax();
    }));
}

void chuckleFest() {
    // NUMA: Ensure that memory used here is local to the thread's CPU
    for (int chuckle = 0; chuckle < 10; chuckle++) {
        boost::this_fiber::sleep_for(chrono::duration<int, milli>(rand() % 100));
        cout << " Chuckle #" << chuckle << " from fiber " << boost::this_fiber::get_id()
             << " rocking on thread " << this_thread::get_id() << endl;
        boost::this_fiber::chillax();
    }
}

int main() {
    int laughTracks = thread::hardware_concurrency();
    boost::fibers::barrier giggleGate(static_cast<size_t>(laughTracks));
    super_semaphore partyOver(0);
    unique_ptr<thread> minions[laughTracks - 1];

    // NUMA: Initialize NUMA library and configure memory allocation strategies

    boost::fibers::use_scheduling_algorithm<boost::fibers::algo::shared_laughter>();

    // NUMA: Evenly distribute thread creation across CPUs
    for (int i = 0; i < laughTracks - 1; i++) {
        int cpuId = i % numa_num_configured_cpus(); // Determine CPU ID for binding
        minions[i] = make_unique<thread>(minionCreator(giggleGate, partyOver, laughTracks, cpuId));
    }

    giggleGate.wait();

    boost::fibers::fiber funFibers[100];
    for (int i = 0; i < 100; i++) {
        funFibers[i] = boost::fibers::fiber(chuckleFest);
        // funFibers[i].detach();
    }

    for (int i = 0; i < 100; i++)
        funFibers[i].join();

    partyOver.release(laughTracks - 1);

    for (int i = 0; i < laughTracks - 1; i++)
        minions[i]->join();

    // NUMA: Clean up, e.g., free NUMA resources

    return 0;
}
        

III.- INTEGRATION with ABSTRACT ALGEBRA (PROFUNCTORS).

To integrate our profunctor-based model with the provided C++ fiber-based implementation for training Artificial Neural Networks (ANNs), we need to adjust the design to incorporate the profunctor abstractions for both the forward and backward passes of the neural network layers. Our integration aims to enhance the modularity and flexibility of the neural network training process, particularly in managing different layer types and their gradient computations.

1. Defining Profunctor-Based Layer Operations:Each neural network layer (e.g., convolutional, fully connected, recurrent) is abstracted as a profunctor. This profunctor will manage the transformation of inputs and outputs for both forward and backward propagation.Implement LayerOp and GradientOp as profunctors, with dimap for adapting input and output transformations.
2. Incorporating Profunctors into the Fiber-Based System -Modify the trainModelSegment function to use these profunctor-based layer operations. Each segment of the dataset processed by a fiber corresponds to a segment of the neural network model.Ensure that each fiber handles the computation for its assigned segment using the profunctor abstractions.
3. Parallel Execution of Layer Operations. Each fiber is responsible for executing a series of profunctor-transformed operations, corresponding to the neural network layers assigned to it.The forward and backward passes through these layers are managed using the LayerOp and GradientOp profunctors.
4. NUMA and Work-Stealing Considerations. Maintain the NUMA-aware design for optimized memory access.If implementing a work-stealing/shared-work scheduler, ensure that it accounts for the dependencies and sequence of layer operations.
5. Future considerations. The steps in our abstract multiprocessor machine can be further integrated with another profunctor into a Moore automaton (state machine). This integration enhances the system's capability to manage states and transitions even more effectively.

NOW IT IS THE TURN FOR HASKELL...

To develop the Haskell functions for our profunctor abstraction applied to AI training, we will focus on the trainLayerHaskell function . This function will utilize the LayerOp and GradientOp profunctors ( using dimap ) to handle the forward and backward passes of a neural network layer. Here is an outline of how these Haskell functions can be structured:

Refined Haskell Implementation with Profunctor Abstraction:

1.- Profunctor Definitions

First, let's reiterate the profunctor definitions:

-- Layer operation as a profunctor

data LayerOp i o = LayerOp (i -> o)

instance Profunctor LayerOp where

    dimap inputTransform outputTransform (LayerOp f) =

        LayerOp (outputTransform . f . inputTransform)

-- Gradient operation as a profunctor

data GradientOp i o = GradientOp (i -> o)

instance Profunctor GradientOp where

    dimap errorTransform gradientTransform (GradientOp g) =

        GradientOp (gradientTransform . g . errorTransform)        

2.- Profunctor-Based Layer Training:

Now, let's use these profunctors in the trainLayerHaskell function. This function will utilize dimap to adapt the input and output transformations for both the forward and backward passes:

-- Function to train a single layer using profunctor abstraction

trainLayerHaskell :: (LayerData -> LayerData) -- input transformation for the forward pass

                  -> (LayerData -> LayerData) -- output transformation for the forward pass

                  -> (LayerData -> LayerData) -- error transformation for the backward pass

                  -> (LayerData -> LayerData) -- gradient transformation for the backward pass

                  -> LayerOp LayerData LayerData -- Layer operation

                  -> GradientOp LayerData LayerData -- Gradient operation

                  -> LayerData -- Input data

                  -> (LayerData, LayerData) -- Output data and gradients

trainLayerHaskell inputTransform outputTransform errorTransform gradientTransform (LayerOp forward) (GradientOp backward) inputData =

    let forwardOp = dimap inputTransform outputTransform (LayerOp forward)

        backwardOp = dimap errorTransform gradientTransform (GradientOp backward)

        outputData = forwardOp inputData

        gradients = backwardOp outputData

    in (outputData, gradients)        

In this implementation:

• The function trainLayerHaskell takes transformations for both the forward and backward passes, along with the LayerOp and GradientOp profunctors.
• It uses dimap to adapt the input and output transformations for the forward pass and the error and gradient transformations for the backward pass.
• The function then applies these adapted operations to the input data to obtain the output data and the gradients.

3.- Integration with C++

• The Haskell function trainLayerHaskell needs to be exposed to C++ via an FFI (Foreign Function Interface).
• Marshalling data between C++ and Haskell will be key, especially for complex structures like LayerData.

4.- Considerations:

• Type Safety and Abstraction: This implementation leverages Haskell's strong type system and the abstraction capabilities of profunctors, ensuring a flexible and safe design.
• Performance: It's important to consider the performance implications of this abstraction, especially in the context of a high-performance computing task like neural network training.

By correctly applying the profunctor abstraction, this implementation provides a more flexible and composable way of handling neural network layer operations, enhancing the modularity and scalability of the system.

5.- Concrete Implementation for Training of ANNs

To illustrate the use of the profunctorial abstraction in training different types of Artificial Neural Networks (ANNs) in Haskell, let's create a main function that demonstrates how this approach can be applied to three different training models under various contexts. We'll consider three types of neural network layers: a fully connected layer, a convolutional layer and a recurrent layer. Each of these layers will require different forward and backward transformations.

Step 1: Define Layer Types and Data Structures

For simplicity, let's define basic types and data structures to represent different layers and the data they process:

type FullyConnectedLayer = -- (define structure)

type ConvolutionalLayer = -- (define structure)

type RecurrentLayer = -- (define structure)

type LayerData = -- (define data structure, e.g., matrices or tensors)        

Step 2: Define Specific Layer Operations

We define specific operations for each layer type. These are placeholders and should be replaced with actual implementations:

fullyConnectedForward :: FullyConnectedLayer -> LayerData -> LayerData

fullyConnectedForward = -- (define forward pass for fully connected layer)

fullyConnectedBackward :: FullyConnectedLayer -> LayerData -> LayerData

fullyConnectedBackward = -- (define backward pass for fully connected layer)
-- Similar functions for convolutional and recurrent layers...        

Step 3: Integrate with Profunctor Abstraction

Now, let's use the profunctor abstraction with these specific layer operations:

trainFullyConnectedLayer :: FullyConnectedLayer -> LayerData -> (LayerData, LayerData)

trainFullyConnectedLayer layer inputData =

    trainLayerHaskell -- input transformation

                      -- output transformation

                      -- error transformation

                      -- gradient transformation

                      (LayerOp $ fullyConnectedForward layer)

                      (GradientOp $ fullyConnectedBackward layer)

                      inputData

-- AGAIN imilar functions for training convolutional and recurrent layers...        

Step 4: Main Function to Demonstrate Different Training Contexts

Finally, we create a main function to demonstrate training with different layer types:

-- LETS GO WITH MONADS AFTER PROFUNCT
main :: IO ()

main = do

    let fullyConnectedLayer = -- (initialize a fully connected layer)

        convolutionalLayer = -- (initialize a convolutional layer)

        recurrentLayer = -- (initialize a recurrent layer)

        inputData = -- (initialize input data)

    -- Train each type of layer

    let (fcOutput, fcGradients) = trainFullyConnectedLayer fullyConnectedLayer inputData

        -- (train convolutional and recurrent layers similarly)

    -- Output results or further processing...

    print fcOutput

    -- (print or process outputs for other layers)        

And we have done this:

• Each specific layer type (fully connected, convolutional, recurrent) has its own forward and backward operations.
• The trainLayerHaskell function is used to apply these operations within the profunctorial framework, allowing for different transformations depending on the layer type.
• The main function demonstrates how to train different types of layers using this framework, handling different contexts (data and layer types).

Considerations:

- Actual Implementations: The placeholders for layer operations (`fullyConnectedForward`, etc.) should be replaced with actual implementations for each layer type.

- Data Structures: The LayerData type and specific layer structures should be appropriately defined to represent the actual data and layer configurations used in ANNs.

- Performance and Practicality: While this example provides a high-level view of how to use profunctors in training different types of neural network layers, practical implementation would require careful consideration of performance and data handling in a real-world context.

AND NOW THE TURN FOR C++

So, now we can use our abstract algebra monadic and profunctorial creatures in the upper layer and combine it with the lower-FiberBundle Layer implementation using fibers and the optimal architecture, all in C++. This demonstrates how our category theory-based abstraction can be applied to a practical neural network scenario to bring together in real time how many new different architectures of training we want without starting from scratch an spend several magnitudes greater the same amount of energy.

#include <boost/fiber/all.hpp>
// Include other necessary libraries and headers

// Define Profunctors for Layers and Gradient Operations
template <typename Input, typename Output>
struct LayerOp {
    std::function<Output(const Input&)> forward;
    std::function<Input(const Output&)> backward;
};

template <typename Input, typename Output>
struct GradientOp {
    std::function<Output(const Input&)> gradient;
};

// Define Profunctor Instances
template <typename Input, typename Middle, typename Output>
LayerOp<Input, Output> compose(const LayerOp<Input, Middle>& layer1, const LayerOp<Middle, Output>& layer2) {
    return {
        [layer1, layer2](const Input& input) { return layer2.forward(layer1.forward(input)); },
        [layer1, layer2](const Output& output) { return layer1.backward(layer2.backward(output)); }
    };
}

template <typename Input, typename Output>
GradientOp<Input, Output> compose(const LayerOp<Input, Output>& layer) {
    return {
        [layer](const Input& input) { return layer.forward(input); },
        [layer](const Output& output) { return layer.backward(output); }
    };
}

// Function to train a segment of the model using Profunctors
template <typename Input, typename Output>
void trainModelSegment(const LayerOp<Input, Output>& layer, const Input& input) {
    // Forward pass
    Output output = layer.forward(input);
    
    // Backpropagation
    Input gradient = layer.backward(output);
    
    // Update model parameters using the gradient
    // ...
}

// Fiber function to manage training
template <typename Input, typename Output>
void modelTrainer(boost::fibers::barrier& syncBarrier, const LayerOp<Input, Output>& layer, const Input& segment, int cpuId) {
    // NUMA: Bind fiber to a specific CPU for localized memory access
    bindFiberToCpu(cpuId);

    syncBarrier.wait(); // Synchronize before starting
    trainModelSegment(layer, segment); // Train on a data segment
}

int main() {
    int numCpus = detectNumCpus(); // Detect number of CPUs
    boost::fibers::barrier syncBarrier(numCpus);

    // Split the dataset into segments, one for each CPU
    std::vector<InputSegment> inputSegments = splitDataset(numCpus);

    // Create Profunctors for layers
    LayerOp<Input, Middle> layer1 = createLayer1Profunctor();
    LayerOp<Middle, Output> layer2 = createLayer2Profunctor();

    // Compose layers using Profunctor composition
    LayerOp<Input, Output> composedLayer = compose(layer1, layer2);

    // Create fibers for each CPU
    std::vector<boost::fibers::fiber> fibers;
    for (int i = 0; i < numCpus; ++i) {
        fibers.emplace_back(modelTrainer, std::ref(syncBarrier), composedLayer, inputSegments[i], i);
    }

    // Join all fibers
    for (auto& fiber : fibers) {
        fiber.join();
    }

    // Post-processing, such as aggregating model updates
    aggregateModelUpdates();

    return 0;
}
        

Key Points:

• Profunctor-Based Abstraction: The LayerOp and GradientOp structs represent layers and gradient operations, respectively, using your Profunctor-based approach.
• Composition: Profunctor composition is used to combine different layers and gradient operations in a modular and composable manner.
• Training with Profunctors: The trainModelSegment function trains a segment of the model using the Profunctor representing the composed layer.
• Composability: The code showcases how your abstraction allows for the easy composition of layers and gradient operations, enhancing modularity and flexibility in designing neural networks.

So, now we can use our abstract algebra monadic and profunctorial creatures in the upper layer and combine them with the lower-FiberBundle Layer implementation using fibers and the optimal architecture, all in C++. This demonstrates how our category theory-based abstraction can be applied to a practical neural network scenario, bringing together in real time as many new different architectures of training as we want, without starting from scratch and spending several magnitudes greater the same amount of energy (QED)

#ArtificialIntelligence #MachineLearning #CategoryTheory #NeuralNetworks #CPlusPlusProgramming