Neural Networks and Parameters — What It Means to Nudge Hundreds of Billions of Dials

What Does the Spam Folder Look at to Choose?

When you open your inbox in the morning, a few promotional emails are already sitting in the spam folder. Who moved those emails there? One way is for a person to write down rules. Like "'Spam if the word 'free' appears more than three times'." Another way is to show a computer a huge number of spam and normal emails, and let it adjust to the difference between the two by itself.

This article opens the inside of the second method. What comes out when you open it is numbers. It is close to imagining dials lined up in front of audio equipment. Every dial has tick marks, and depending on where you set the ticks, the sound that comes out changes. Here, there are so many dials that a person could never dare count them one by one, and no person adjusts those ticks by hand. There is also one difference from audio equipment. Here, no one pre-assigns what each and every dial will handle, and countless dials move together to produce a result.

What does a single dial do? Who turns it, and how? What changes if you attach many? This article answers those three by following the path of a single email passing through this calculation once.

Calculations That Filter Spam with a Few Dials

For the sake of explanation, let us assume a person extracts only three things from a single email. These are the number of times promotional phrases appear, the number of links, and whether the sender is someone you have previously exchanged messages with. The calculation starts by multiplying each of these three by a number and adding them all up. If the number multiplied by promotional phrases is large, promotional phrases heavily sway the result. If the number multiplied by whether you know the sender is negative, the more familiar the sender is, the more the result drops.

Weights are numbers that determine how strongly and in which direction each input is reflected in the result. The single dial mentioned earlier is a single weight. To the sum obtained by multiplying and adding the weights, a single preset default value is added. The sum is not output as-is, but passed through a predetermined curve once. One example of the curve is a shape that squashes very small values close to 0 and very large values close to 1. Then, you can read the result as a value between 0 and 1, such as "the probability that this email is spam."

There is a name for this single unit. An artificial neuron is a small calculation unit that multiplies several inputs by respective weights, adds them up, and then passes the sum through a designated curve to output it. An artificial network is a computational structure in which a very large number of simple calculation units are connected to one another. Artificial neurons and their curves are the most fundamental concepts that appear when learning this structure (DOI: 10.1002/9781394269297.ch02).

Now, we need a term to call this entire calculation. A model is a computational device that takes inputs and produces outputs, and the numbers inside it can be adjusted to bring it close to a desired answer. In spam filtering, the input is the email and the output is the probability of being spam. Parameters are numbers inside the model that are determined through learning, and these numbers determine how the model calculates inputs to produce answers. Learning is the process of gradually tweaking the numbers inside a model to match example data or experience, and is also called training. How the method of humans writing rules differs from the method of adjusting numbers with examples is covered in another part of this series, "AI Models".

In this small example, the parameters are four in total: a set of weights and one default value. Most of the parameters in a neural network are weights, and other numbers like these default values make up the rest. The four numbers are not decided by humans, but by learning. Dials are the numbers you multiply, and examples adjust the ticks.

What Was Borrowed from the Brain and What Was Not

The names artificial neuron and artificial network came from the brain's nerve cells. One older primer unpacks the motivation behind how this field arose while explaining biological nerve cells and artificial calculation models together (DOI: 10.1109/2.485891). Early artificial neuron models were very simple translations of biological nerve cells, and neuroscience principles have significantly influenced this field multiple times, including through these models (PMID 32477089).

Real nerve cells are drawn as much more complex. Nerve cells have branches that receive and gather inputs, and a long fiber that sends out outputs. Neuroscience separately models the electrical properties of cell membranes, the way signals travel along fibers, the movement of the part that decides whether to fire a signal, and the way connections between cells change (DOI: 10.1142/s0218127492000653). What the artificial neuron carried over from all that is the skeleton of 「receiving multiple inputs with different strengths each, gathering them, and sending out one」.

The way of learning is harder to compare. This is because we still do not know well how the brain learns. The fact that connections between cells can change is included in the basic model from earlier as well (DOI: 10.1142/s0218127492000653), but how such changes accumulate to become learning has not been solved. In particular, one study diagnosed that while respecting the brain's biological constraints, a learning rule that stacks more complex features on top of simple features across multiple stages is not yet known (DOI: 10.48550/arxiv.2010.08262).

As we will see later, artificial neural networks learn by sending the degree of error backward from the output side to the input side. The rule tailored to the brain proposed as a candidate by the same study does not send backward like this, and instead relies mainly on the activity of the two cells on both sides of the connection. A neural network trained with this rule stacked features of photographs, speech, and video in simulations across multiple stages. This is not a study confirming that the brain actually uses this rule. What is cited here is a preprint version published on arXiv, a version uploaded before undergoing peer review (DOI: 10.48550/arxiv.2010.08262).

Neuroscience uses artificial neural networks as well. Neuroscientists use artificial neural networks as models to reproduce complex behaviors and neural activity, and modify the structure and learning method to be closer to the brain when needed (PMID 32970997). One review asks whether the brain also learns by indiscriminately fitting to massive examples, and views this perspective as a fundamental challenge to various theoretical assumptions in psychology and neuroscience (PMID 32027833). While artificial neural networks got their name and several inspirations from brain research, it has not been confirmed that they operate in the same way as the brain.

Lining Them Up and Stacking the Layers

In filtering spam, humans chose the input set. For the task of identifying cats in photos, what should we choose? Rules written as numbers for ear shape or whiskers do not easily come to mind. The representative neural network that handles photos skips this selection. Even if humans do not extract features in advance, it can take raw photos as they are and handle them, and this point is cited as one of the reasons that significantly boosted photo recognition performance (PMID 34383173).

To do that, multiple artificial neurons are lined up in a row. A layer is a single row of artificial neurons that receive values passed from the front and calculate them side by side. The first layer receives the raw input, and the next layer receives the values sent out by the previous layer. Deep learning is a method of training an artificial neural network with several layers stacked like this. The word "deep" means there are many layers. Humans do not decide what the middle layers will calculate; training decides it by adjusting the weights. In the example with three inputs, it showed which input each dial multiplies, but the dials in the middle layers do not have such name tags.

This is where the curve from the previous section plays its part. Without the curve, what each layer does is just multiplying and adding, so no matter how many layers you stack, it reduces to the same calculation as a single-layer one. A curve must be wedged in to express more complex relationships as you stack layers. The shape of the curve can vary, and a broken line that cuts values smaller than 0 down to 0 is also commonly used in the middle layers. As long as it does not stretch straight in a single line, it can fulfill this role.

If you widen and stack the layers, the dials also swell massively. Assuming that the neurons of one layer receive every single value from the previous layer one by one, one million weights are created between just two layers with a thousand neurons each. At this time, if you increase the number of neurons in each layer, the number of weights grows multiplicatively, and if you stack one more layer, that layer's share is added on top.

As Wrong as It Is, in the Direction of Shrinking

The dials of a freshly made neural network are set at random. How to set the initial values of the weights is also a separate problem handled during training (DOI: 10.1002/9781394269297.ch02). If you feed a spam email into the network in this state, it outputs a nonsensical value. To fix this, you must first measure how wrong it is.

Loss is a single number that measures how far off the model's answer is from the correct answer (DOI: 10.1002/9781394269297.ch02). If the network answers that a spam email has a 0.3 probability of being spam, the loss is large; if it answers 0.95, the loss is small. Training is structured as the task of shrinking this loss.

The next question is which dial to turn in which direction to reduce the loss. The change in loss—in which direction and by how much—when you turn a single dial just a tiny bit is called the gradient of that dial. Gradient descent is a method of looking at the gradient for each parameter and repeating the process of shifting every parameter slightly in the direction that reduces the loss. A human pre-determines the multiplier that decides how large of a step to take at once; if it is too large, the model overshoots, and if it is too small, it takes forever. In practice, various modified versions of gradient descent are used, and each variation has different strengths and weaknesses (DOI: 10.48550/arxiv.1609.04747, preprint version posted on arXiv).

The remaining problem is how to calculate all those gradients when there are billions of dials. Backpropagation is a method that starts from the loss measured at the output, works backward, and calculates how much influence each individual parameter has on the loss (the gradient). Backpropagation only calculates the gradients, and it is gradient descent that takes those gradients and actually turns the dials. Because earlier layers inherit and use the results calculated by later layers, this yields all the gradients at once with far fewer calculations than trying out each dial individually. Calculating forward to measure the loss, tracing backward to find the gradients, and turning the dials slightly based on those gradients to shrink the loss—these three steps form the skeleton of a single training step (DOI: 10.1002/9781394269297.ch02).

It is generally believed that backpropagation is unlikely to happen as-is in the brain. Therefore, some researchers view the search for alternatives that could also work in the brain as a fundamental challenge in artificial neural network research (DOI: 10.1609/aaai.v33i01.33014651). Whether a drop in loss truly means the model has learned well is yet another question. The difference between memorizing and learning is covered in another part of this series, "AI Models".

The Journey of a Single Email

Now let's follow a single spam email. When the mail enters the first layer, each neuron multiplies the inputs by weights, adds them up, passes the result through a curve, and hands it over to the next layer. This forward calculation continues all the way to the final layer, and out comes "0.3 probability of being spam" at the end.

The correct answer is that it is spam. The distance between 0.3 and the correct answer is calculated as the loss. Now, backpropagation traces backward from the final layer to the first layer, writing down the slope for each dial. Gradient descent takes those slopes and turns every dial just a tiny bit in the direction that reduces the loss. In practice, it is common to put multiple emails into a batch, look at the loss for the entire batch, and turn them all at once.

The amount that a single dial moves in one step is very tiny. When the next batch comes in, the same thing happens again, and this single step is repeated countless times. Adjusting billions of numbers means that a single step of turning every dial just a little bit at once is repeated countless times for each batch of examples. This method of optimizing millions of weights to match millions of observations has produced models that hold up even in real-world complex situations (PMID 32027833). However, what this process reduces is the loss on the examples shown, and whether it will make correct predictions on emails it sees for the first time must be checked separately.

Making Them Bigger Makes Them Better

The neural network Chinchilla, which writes continuing text, had 70 billion parameters. Chinchilla was published in 2022, and the models compared against it were Gopher with 280 billion and GPT-3 with 175 billion (arXiv:2203.15556, preprint version published on arXiv). Behind this size competition lies the observation that "making them bigger makes them better."

The scaling law is a tracked trend of how much the loss decreases when you increase the model size, the amount of training data, and the compute used for training. Looking at results compiled in 2020, the loss of text-continuing neural networks decreased in a consistent pattern as all three were increased, and some of the trends continued across a range where the scale varied by more than 10 million times (arXiv:2001.08361, preprint version published on arXiv).

This pattern is one where every time you increase the scale by a certain multiple, the loss decreases by almost the same proportion. To reduce the loss by that same proportion again, you have to increase the scale by that same multiple once more. Improving here means that the loss decreases, and whether useful capabilities increased by that much must be measured separately.

The scaling law is a trend obtained from observation, and the answers to what and how much should be scaled differed across studies. Here, compute acts like a budget. The compute required for training increases the larger the model and the more data there is. Therefore, if the compute is fixed, spending more on growing the model reduces the share available for data. The 2020 results viewed it as efficient to concentrate this budget on the model side and train a very large model with relatively little data (arXiv:2001.08361).

The answer from the 2022 study was different. This study trained over 400 models ranging from 70 million to over 16 billion parameters. And it concluded that the large models of the time had not been trained sufficiently relative to their data. To use a given amount of compute most sparingly, when you double the model, you must also double the data. Chinchilla, built according to this prescription, used the same amount of compute as Gopher. Instead, its parameters were reduced to 1/4 of Gopher's, and its data was increased 4 times. As a result, it consistently outperformed Gopher and GPT-3 across multiple tasks (arXiv:2203.15556). This means that even with the same budget, depending on how it is split between the model and the data, a smaller model could outperform a larger model.

There is also the criticism that the trend is tied to conditions. It is a theory that if machine-generated text increasingly mixes into the training data, this trend can break, and even the phenomenon of forgetting skills already learned can occur. The work cited here is also a preprint version published on arXiv, and the theory was verified through arithmetic tasks and writing experiments on a single publicly available large model (DOI: 10.48550/arxiv.2402.07043). That making things bigger generally made them better is an observation, and a guarantee that they will unconditionally get better is not contained within that observation.

Does Capability Suddenly Emerge?

There is an even more dramatic claim about the relationship between size and capability. Emergent capabilities are abilities that do not exist in small models but appear in large models. A 2022 study outlining this concept argued that such abilities cannot be predicted by extrapolating the performance of small models, meaning that scaling them up further could widen the scope of their capabilities (arXiv:2206.07682, a preprint version published on arXiv).

A counterargument emerged in 2023. The study presenting this counterargument offered an explanation that phenomena looking like emergence may arise because of the scoring metrics chosen by researchers. When looking at already published answers from a specific task and model family, measuring them with a metric that drops off abruptly or curves steeply makes capabilities look as though they appeared suddenly, whereas measuring them with a continuously varying metric shows the same model's performance changing smoothly and predictably. For example, a scoring method that awards points only when an entire answer is correct marks answers that were getting incrementally closer as near-zero up to a certain size. This study reanalyzed results from the GPT-3 family of models and a benchmark suite gathering multiple tasks, reporting that the claimed emergent capabilities disappear when the metric is changed or statistics are computed better, and even demonstrated creating nonexistent emergence at will by selecting specific metrics in several neural networks dealing with images. The conclusion is that emergence may not be a fundamental property of scaling up models (arXiv:2304.15004, a preprint version published on arXiv).

The conclusion of the counterargument stops at "may not be a fundamental property," and does not assert that emergence is absent everywhere. Emergence is still under debate, and there are grounds for both sides on whether capabilities suddenly appear according to size or merely look that way because of the metrics.

The Numbers Are All Visible, But the Meaning Is Unread

Since the parameters of such models are all just numbers, the person who owns the model can pull them out one by one. The problem is that individual numbers do not have their own separate meanings attached to them. What the model does lies in the calculations made collectively by countless numbers. One review suggests that such models learn neither simple human-recognizable rules nor world representations in a human-recognizable form. Instead, they operate by interpolating between examples in a parameter space of very many dimensions (PMID 32027833). Picturing parameters as drawers where knowledge is placed one compartment at a time does not fit this picture.

That is why deep neural networks are often called opaque boxes. Deep neural networks pay the price of making their internal representations hard to inspect in exchange for gaining high discriminatory power (DOI: 10.1631/fitee.1700808). Interpretability is the degree to which a model can explain why it gave a certain answer in terms that humans can understand. This property connects to whether humans can trust the system, and also to ethical issues such as algorithm-driven discrimination (DOI: 10.1109/tetci.2021.3100641).

Research looking inside the "black box" is ongoing. When a neural network that recognizes photos answers "dog," there is a method that colors and shows which part of the photo played a large role in that answer. This picture showed that even seemingly absurd incorrect answers had their own reasons, and it was also used to find biases in training data. In tests with human subjects, non-experts who saw this explanation were able to pick the better one between two neural networks that gave the same answer (DOI: 10.1109/iccv.2017.74).

A 2018 overview noted that these explanations were neither standardized nor systematically evaluated (DOI: 10.1109/dsaa.2018.00018). A 2022 overview still counts figuring out representations inside neural networks as human-readable concepts as a major unresolved challenge (DOI: 10.1214/21-ss133). Humans cannot yet read all together what these many parameters are doing, and while tools to peer inside are being built, they are not yet finished.

Where to Look Further

• How a massively scaled neural network becomes a conversational program is covered in greater depth by "How Guessing the Next Word Becomes a Conversation — The Minimum Foundation for Reading Language Models". Reading "How Generative AI Answers", another piece in this series, first makes it easier.
• When backpropagation emerged and what it made possible—the history of deep learning—can be found in "From Backpropagation to AlphaFold: What Deep Learning Stood On".
• The question of whether artificial neural networks resemble the brain leads into "Does AI Really 'Think'?".
• The issue of how evaluation metrics change conclusions is covered by "The Score Went Up. But What Went Up?".
• How much electricity the computation required to scale models consumes can be seen in "How Much Electricity Does AI Use?".
• The ways in which training goes wrong continue in "How AI Gets Things Wrong", another piece in this series.

Source Papers

Items marked with an arXiv number and a 10.48550 DOI cite preprint versions hosted on arXiv—meaning they were uploaded before undergoing peer review. Some of these may have later been published in conferences or journals.

• "Fundamentals of Deep Learning" (2025, book chapter) — DOI: 10.1002/9781394269297.ch02 — Basic concepts such as neurons, layers, and activation functions, forward and backward computation, loss functions that reduce errors, gradient descent, weight initialization, and learning rates.
• "Artificial neural networks: a tutorial" (1996) — DOI: 10.1109/2.485891 — Artificial neural networks are parallel systems linking massive numbers of simple processors. Introduces biological neurons and artificial computational models side by side.
• "Crossing the Cleft: Communication Challenges Between Neuroscience and Artificial Intelligence" (2020) — PMID 32477089 — The perceptron, a simple model of a biological neuron, influenced artificial neural networks. The priorities of the two fields have diverged.
• "TUTORIAL ON NEUROBIOLOGY: FROM SINGLE NEURONS TO BRAIN CHAOS" (1992) — DOI: 10.1142/s0218127492000653 — Nerve cells have dendrites and axons, with membranes, fibers, trigger zones, and synapses described by individual models.
• "Local plasticity rules can learn deep representations using self-supervised contrastive predictions" (2020, preprint version hosted on arXiv) — DOI: 10.48550/arxiv.2010.08262 — Brain learning is not well understood, and rules for learning deep representations while respecting biological constraints are still unknown. Proposes local rules without error backpropagation between layers, allowing neural networks trained with these rules to build deep hierarchical representations in images, audio, and video.
• "Artificial Neural Networks for Neuroscientists: A Primer" (2020) — PMID 32970997 — Neuroscience uses artificial neural networks as models of behavior and neural activity, tweaking their structure and learning to align more closely with neurobiology.
• "Direct Fit to Nature: An Evolutionary Perspective on Biological and Artificial Neural Networks" (2020) — PMID 32027833 — The power of optimization in fitting millions of weights to millions of observations. Such models interpolate in a high-dimensional parameter space rather than learning human-readable simple rules. Raises the question of whether the brain does something similar.
• "A primer on deep learning and convolutional neural networks for clinicians" (2021) — PMID 34383173 — Convolutional neural networks handle raw data directly without manual feature extraction, greatly enhancing the learning capacity of computer vision.
• "An overview of gradient descent optimization algorithms" (2016, preprint version hosted on arXiv) — DOI: 10.48550/arxiv.1609.04747 — Various variants and challenges of gradient descent, along with the strengths and weaknesses of each algorithm.
• "Biologically Motivated Algorithms for Propagating Local Target Representations" (2019) — DOI: 10.1609/aaai.v33i01.33014651 — Finding biologically plausible alternatives to backpropagation is a fundamentally important challenge in artificial neural network research.
• "Training Compute-Optimal Large Language Models" (2022, preprint version hosted on arXiv) — arXiv:2203.15556 — About 400 models ranging from 70 million to 16 billion parameters. Large models of the time were under-trained; model size and data should scale in equal proportion. Chinchilla (70 billion parameters, 4 times the data) outperformed Gopher (280 billion) and GPT-3 (175 billion) across a wide range of tasks.
• "Scaling Laws for Neural Language Models" (2020, preprint version hosted on arXiv) — arXiv:2001.08361 — Loss follows a power law relative to model size, data, and compute, with some trends continuing past 7 orders of magnitude. Width and depth have a minor impact. When compute is fixed, a very large model should be trained with relatively less data.
• "A Tale of Tails: Model Collapse as a Change of Scaling Laws" (2024, preprint version hosted on arXiv) — DOI: 10.48550/arxiv.2402.07043 — The phenomenon where mixing synthetic data into the training corpus breaks scaling or causes the model to forget skills. Theory plus arithmetic tasks and Llama 2 experiments.
• "Emergent Abilities of Large Language Models" (2022, preprint version hosted on arXiv) — arXiv:2206.07682 — Defines emergence as abilities not present in small models but appearing in large ones. Cannot be predicted by extrapolating from small models; scaling up further can broaden the range of abilities.
• "Are Emergent Abilities of Large Language Models a Mirage?" (2023, preprint version hosted on arXiv) — arXiv:2304.15004 — Emergence can appear due to the choice of metrics. Nonlinear and discontinuous metrics look like emergence, whereas continuous metrics show smooth changes. Emergence may not be a fundamental property.
• "Visual interpretability for deep learning: a survey" (2018) — DOI: 10.1631/fitee.1700808 — Deep neural networks trade high discriminative power for the low interpretability of black-box representations.
• "A Survey on Neural Network Interpretability" (2021) — DOI: 10.1109/tetci.2021.3100641 — Black-box properties affect ethical issues such as trust and algorithmic bias. Definitions and classifications of interpretability.
• "Grad-CAM: Visual Explanations from Deep Networks via Gradient-Based Localization" (2017) — DOI: 10.1109/iccv.2017.74 — Maps highlighting image regions important for predictions. Helps discover reasons for wrong answers and data biases, allowing non-experts to screen for better neural networks.
• "Explaining Explanations: An Overview of Interpretability of Machine Learning" (2018) — DOI: 10.1109/dsaa.2018.00018 — Explanations lack standardization and systematic evaluation, and current approaches for deep neural networks are insufficient.
• "Interpretable machine learning: Fundamental principles and 10 grand challenges" (2022) — DOI: 10.1214/21-ss133 — Ten challenges of interpretable machine learning include the supervised and unsupervised disentanglement of neural network representations.