1. Abstract

In this report, we delve into a range of significant challenges and their corresponding solutions, which are pertinent to tensors, training accelerators, and quality assurance. Our intended audience comprises decision-makers who are tasked with implementing such systems in professional applications.

The combination of AI’s dual capabilities, offering both quantitative estimations and narrative depth, positions it as the preferred choice for enhancing efficiency in critical sectors such as finance, medical, pharmaceuticals, manufacturing, and transportation. This imminent progress will also give rise to some unexpected shifts within the AI industry:

(a) All investments in AI infrastructure will exhibit a shorter technological life-cycle and higher costs than initially anticipated

(b) Emerging and innovative players will exert a disruptive influence on the AI infrastructure and algorithmic market

(c) The ecosystem of established AI producers may prove to be of lesser value than what current market expectations suggest

This report provides an in-depth exploration of the challenges, solutions, and underlying reasons driving these developments.

2. Exploring AI Challenges: Unveiling New Horizons

The advent of AI services, leveraging technologies like LLM and GAN, has marked a remarkable surge in the realm of natural language processing and visual generative applications. However, these advancements have been largely empirically driven, often evolving from the crucible of challenging machine-learning competitions such as Kaggle. It’s a common trajectory in AI development: empirical experimentation precedes theoretical integration, giving rise to new layers of abstraction that, in turn, open doors to fresh applications. In essence, it’s a natural process of labor division, creating new jobs and skills that didn’t exist just a few years ago. Therefore, it’s reasonable to anticipate that AI is poised at the threshold of an exciting new phase, both theoretically and technologically.

In the landscape of AI, there are numerous critical challenges steering the industry’s evolution, and we’ll discuss three prominent ones here:

1. The Model Selection and Training Challenge: Within the AI community, it’s widely acknowledged that training neural networks is an intensive process that involves a multitude of experiments. While AI may seem to reduce the workforce required for certain tasks, this is primarily true for AI products themselves. The training phase demands extensive work. Tasks like selecting the network architecture, determining the number of hidden nodes, choosing activation functions, and defining the number of training epochs involve experimenting with a plethora of parameters on diverse datasets. This process doesn’t have a clear endpoint. An encouraging observation is that deep-learning models exhibit a high degree of portability – a model trained for image classification, for instance, can be adapted for significantly different tasks with minor modifications. However, this is an empirical finding, and extensive training trials remain an essential part of the process. The engineering modeling field has a practice called Model Governance, which outlines the safeguards and measures necessary for correct model design and usage. AI Model Governance is still in its early stages.
2. The Model Output Challenge: Often, having a correct output forecast from a model isn’t sufficient. Users may also require knowledge or estimations regarding how hidden or unmeasurable parameters influence the output. For instance, an econometrician might need both a forecast of the next quarter’s unemployment and a model-estimated inverse elasticity of labor supply. Artificial Neural Networks (ANNs) do not inherently provide access to these latent parameters, which presents a challenge. In mathematical terms, this problem is akin to solving the inverse problem (IP). As detailed in this report, neural networks can be harnessed to tackle the model output challenge.
3. The Model Consistency Challenge: AI predominantly relies on probability theory, Monte Carlo methods, and various search heuristics. Consequently, AI-based estimations often lack the consistency found in analytical methods. For example, a forecast might yield varying estimations with each evaluation. While this variability may be acceptable in certain applications, it’s a drawback in domains like finance and diagnostics. AI has developed strategies to mitigate this issue, such as employing larger datasets, model averaging, hyperparameter tuning, reinforcement learning, Bayesian methods, and more. Nevertheless, as this report further elucidates, an approach grounded in external references may offer a preferable solution.

 

3. Training Accelerators

For a significant period, the AI landscape has revolved around the development of training and inference accelerators, such as Graphical Processing Units (GPUs) and Tensor Processor Units (TPUs). To fully comprehend the future of AI, encompassing algorithms, applications, and technology, it is imperative to delve into the realm of GPUs and TPUs. Unfortunately, many business and financial circles have held somewhat immature and overhyped perspectives on this matter.

An accelerator, like a GPU or TPU, is an electronic chip mounted on a circuit board, linked to a central computer via a bus or high-speed serial communication. These chips are designed for mathematical computations. While a Central Processing Unit (CPU) performs sequences of operations over time, a GPU operates in parallel across space. To illustrate, consider a task involving 100 multiplications. If a CPU can perform one multiplication in 1 nanosecond (ns), then performing 100 multiplications sequentially would take 100 ns (1 ns x 100). In contrast, an accelerator features 100 multiplicative units permanently etched into the chip at the transistor level, all working simultaneously. Consequently, the same task, with each multiplication taking 1 ns, would still complete in 1 ns. The speed advantage of the accelerator is evident.

However, accelerators come with three main disadvantages:

1. The mathematical operations on an accelerator are typically fixed with Application-Specific Integrated Circuit (ASIC) chips. As the field of AI rapidly advances in mathematical complexity, accelerators quickly become technologically outdated, necessitating hardware replacement.
2. Substantial amounts of data need to be transferred between the accelerator and the central processor unit. The speed advantage is maintained only if the accelerator receives data at or near its full capacity.
3. Data must be converted into a format suitable for the accelerator and results must be sent back to the central processing unit. This process, known as data orchestration, is managed by the CPU using various programs such as TensorFlow, cuDNN, Keras, cuBlas, etc. These programs need to be compatible with the accelerator chip, which, as mentioned, undergoes frequent changes. A version mismatch can be a significant inconvenience for many AI practitioners. Moreover, the communication flow and data adjustments consume substantial CPU resources.

From the description above, it is clear that technological advancements in AI involve integrating an accelerator onto the same chip as the CPU to reduce communication costs. Furthermore, as much orchestration as possible should be hardware-based. The key challenge lies in developing algorithms that remain stable over extended periods. This challenge encompasses both algorithmic improvements and the manufacturing of large ASIC chips. The existing ecosystem of a specific GPU/TPU manufacturer holds less significance in this context.

 

4. Tensors

To grasp the concept of tensors, imagine an Excel spreadsheet. In this context, a single cell containing a number represents a scalar. A column with numbers can be understood as a vector, and an entire sheet filled with numbers under multiple columns and rows is, mathematically speaking, a matrix. In mathematics, a tensor is an extension of this idea, encompassing Excel files containing sheets with scalars, vectors, matrices, or even tensors of higher dimensions.

The concept of tensors was initially introduced in mathematics by Bernhard Riemann and later by Gregorio Ricii, around 1888. In mathematics, vectors, matrices, and tensors are always associated with a set of possible operations. For example, the length of a vector (vector norm) is computed using the Pythagorean theorem, which can be generalized to calculate the norm (or “length”) of a matrix and a tensor. By defining tensor addition and subtraction as element-wise operations, you can compute the distance between two tensors as the norm of their difference. These concepts are further extended to compute tensor gradients, among other operations. It’s important to note that the operations associated with tensors in physics and differential geometry differ from those in AI.

It’s crucial to understand that the primary reason for using tensors in AI is not merely fascination with high-dimensional mathematics but the practical need to facilitate parallel computations on hardware accelerators. While future developments may lead to AI tensors having mathematical properties akin to Hilbert or Banach spaces used in engineering and physics, we haven’t reached that stage yet. Such developments, if they occur, could result in secure, high-performance accelerators.

Common operations performed with tensors in AI include norms, distances, activation functions, convolutions, and gradient computations.

 

5. Network Orchestration and Training

Step 1: Data Curation

The initial stage of network training, data curation, is crucial due to common issues like outliers, missing samples, mixed formats, or non-representative samples in input data. Automated curation tools play a vital role in machine learning by ensuring data quality.

Step 2: Tensorization

Data undergoes a transformation into tensors of suitable dimensions. The primary objective of tensorization is technical in nature, aimed at facilitating high-speed data flow into training accelerators. During this step, the resulting tensor is partitioned into training, validation, and test sets. This division serves two purposes: it helps control overfitting and allows for the assessment of the model’s ability to generalize to new data.

Step 3: Orchestration

This phase involves using a high-level programming language like Python or Matlab to set up the neural network’s architecture. It may involve employing a design language, including graphical interfaces. The process includes compiling the network and programming the actual flow of tensors into the accelerator for training. This flow of tensors is managed by one or more layers of programs, such as TensorFlow. Training continues until validation tests indicate the onset of overfitting.

Step 4: Generalization Testing

Following training, the resulting neural network undergoes testing to evaluate its generalization strength. This involves assessing the model’s predictive capabilities using data that was not part of the training process.

 

6. Inference and Transfer Learning

The culmination of the training process yields a fully prepared network, ready for inference. The inference stage, like training, begins with data curation and tensorization. The resulting tensor is then passed through the trained network. The output of the inference stage is often a tensor, which may need to be de-tensorized for practical use. For instance, if the network is a generative algorithm for images, a function is employed to convert the output tensor into a matrix and eventually into a viewable image.

As a network is utilized over time, there often arises a need for fine-tuning. This fine-tuning can serve either to enhance the network’s inference capabilities or to adapt it to a domain-specific dataset. Consider the example of a bank wishing to employ a general-purpose LLM as an advisory service while incorporating bank-specific financial products into the system. The process of training and evolving a network unfolds over time instances, denoted as t=1, 2, and so on.

At the initial instance (t=0), the network starts with random weights generated by the network builder, as discussed in the Orchestration section. The training data includes a known query Q(1) and a known response R(1). Training with Q(1) and R(1) results in a network with adjusted weights, denoted as N(1). Subsequent training at t=2 with query Q(2) and response R(2) produces an improved set of network weights, N(2). This process continues until the network starts exhibiting overfitting, at some time t = k. The result is a trained network suitable for inferences.

As further enhancements are made through fine-tuning and transfer learning, additional training sets T(n+1), T(n+2), produce corresponding network weights N(n+1), N(n+2). It’s important to note that these network weights can always be represented as tensors. For example, if a network consists of multiple layers, each layer corresponds to a dimension in a tensor. These network tensors are inherently associated with the operations performed by the network.

In essence, regarding the mathematical practice of associating tensors with the operations they can execute, at a specific time t=k, the training and transfer learning phases can be succinctly described by a pair <T(1:k), N(k)>, where T(1:k) represents all the training tensors up to time k, and N(k) signifies the weight tensor at time k.

7. Inverse Problems: Bridging Mathematics and Machine Learning

Inverse problems in mathematics revolve around the challenging task of deducing the causal parameters when the outputs of a system are known. The objective is to reverse this process through mathematical modeling, aiming to estimate the original inputs or parameters. This problem frequently arises in diverse fields, including economics, finance, medical diagnostics, geophysics, and engineering, where it’s employed for error detection and identification. Solving these problems is pivotal for constructing narratives that explain how policy measures influence economic changes. Inverse modeling, by nature, falls into the category of ill-posed problems, where the number of parameters to identify exceeds the available data, akin to the challenges encountered in one-shot or few-shot learning within the domain of machine learning.

This process can be likened to identifying a neural network (as depicted in the figure). We begin with known data and utilize an inverse problem solver to pinpoint the root cause as a variation in certain parameters. Subsequently, we gain insights into all model parameters, typically represented as a tensor T_P. With this information, including the model M, we can evaluate the accuracy of our diagnostic efforts by posing pertinent queries (Q) to obtain a response (R_M, with the “M” indicating Model). This response is then compared with the input data, helping to elucidate the causal impact of the parameter change.

In the context of machine learning, the approach involves working with the input data to train an artificial neural network (ANN). This training leads to the acquisition of a parameter tensor Ta specific to the ANN. However, due to the universal adaptability of ANNs, they do not provide the same clarity in identifying the cause of a problem within the parameter space, as we achieved with explicitly identified model factors.

To address this challenge, we introduce the concept of an invertible mapping—a bridge—between the tensors T_A and T_P, denoted as B, allowing for inverse determinations:

T_P = B(T_A)

T_A = inv(T_P)

This tensor bridge serves various crucial purposes, including:

• Narrative: Identifying parameter changes (Δ(T_P) = T_P – B(T_A)).
• Governance: Verifying the correctness of ANN identification by assessing the estimation error
  (Δ(T_A) = T_A – inv(B(T_P)).
• Robust Inference: Leveraging the ensemble (R_MA) generated by both the ANN and the model, where
  R_MA = ENS(B(T_A), T_P), for robust responses to queries (Q).

To solve inverse problems, a range of techniques can be applied, encompassing traditional mathematical solutions associated with analytical models, as well as machine learning solutions. These include:

1. Inverse Reinforcement Learning: Employed to identify reward functions used by optimizing agents.
2. Deconvolution in Deep Learning: Useful for unravelling complex data transformations.
3. Inference Probabilistic Models: Including Bayesian statistics, which frame parameter estimation as a form of solving inverse problems.

This discussion provides a foundational overview of the principle behind these approaches.

8. Conclusion: Navigating the Intersection of Mathematics and Machine Learning

In this article, we have explored the intriguing intersection of mathematics and machine learning, with a particular focus on addressing complex challenges encountered in the fields of AI, neural networks, and data-driven problem-solving. The narrative unfolds through four key stages:

1. Accelerators: Our journey begins with a deep dive into accelerators like GPUs and TPUs. These electronic chips are instrumental in AI, yet they present trade-offs in terms of flexibility and data transfer speed. The emphasis is placed on the importance of integrating accelerators onto the same chip as the CPU, a step that can significantly streamline the data flow and computation processes.
2. Tensors: Next, we unravel the concept of tensors, emphasizing that they are not merely mathematical abstractions but practical tools for handling high-dimensional data, crucial for parallel processing. Tensors are the linchpin connecting AI’s insatiable hunger for data with hardware accelerators.
3. Network Orchestration and Training: The orchestration of neural networks entails meticulous steps, from data curation to tensorization, programming, and fine-tuning. As we navigate this journey, it becomes evident that orchestrating neural networks involves both art and science, with each step finely calibrated to maximize the model’s performance.
4. Inference and Transfer Learning: The culmination of training leads to inference, where the network’s capabilities are put to the test. The role of transfer learning in adapting networks to domain-specific datasets is unveiled. As we traverse this phase, we witness the need for bridging the gap between machine learning models and explicit factors, allowing for a more interpretable and robust AI system.

This confluence of mathematics and machine learning is underscored by the relevance of inverse problems, an intriguing challenge that arises in various domains, including economics, finance, and medical diagnostics. It’s through the lens of inverse problems that we find common ground between mathematical principles and machine learning’s quest for optimal solutions.

In closing, our exploration has unearthed both the complexities and the opportunities in this dynamic intersection. By navigating these challenges and harnessing the potential of mathematical principles, we are poised to unlock new frontiers in AI, enhancing its effectiveness and robustness across a range of real-world applications.

With these insights, we set our sights on a future where the collaborative power of mathematics and machine learning continues to shape the evolving landscape of artificial intelligence, one innovation at a time.