Complexity from Simplicity

Our world is intricate, yet it can be distilled into a surprisingly limited number of basic interactions. For instance, the formation of intricate snowflakes and bubble films can be traced back to simple attractive forces between molecules.

This begs the question: how do we account for the emergence of complex structures? In physics, complexity arises as we transition from the microscopic to the macroscopic scale.

To draw a parallel to language, English begins with a small set of basic elements—26 letters—that can combine to create approximately 100,000 words, each with a unique meaning. These words can then form countless sentences, paragraphs, books, and entire libraries.

This hierarchical structure is akin to those found in physics. The Standard Model of particle physics is based on a limited array of elementary particles like quarks and electrons, which interact via force carriers such as photons. These particles and forces coalesce to form atoms, each with specific chemical properties, leading to an enormous variety of molecules, structures, and living organisms.

In nature, complex systems, despite their varied origins, often exhibit universal traits. For instance, numerous liquids, though chemically distinct, share three primary states: liquid, solid, and gas. In an even broader sense, the physics governing certain materials, like Type-I superconductors, can be applied to describe fundamental phenomena, such as the Higgs Mechanism.

While it is crucial to recognize the divergence between language and physics—where physical laws are dictated by nature and languages are human constructs—there is no inherent reason that linguistic complexity must mirror physical complexity.

However, as we will discuss, models like ChatGPT exhibit structures akin to those found in particle physics. Understanding these structures may illuminate the successes of LLMs and suggest that the complexities of language may share similarities with physical complexities.

The Physics of Language Models

While physical laws are encapsulated in equations, what about LLMs like ChatGPT? To bridge the gap between LLMs and physics, we must examine their underlying mathematical frameworks. In physics, the motion of particles—or more broadly, fields or states—can be expressed schematically as follows:

A schematic form of physical equations

This indicates that particles move due to forces derived from the gradients of an abstract entity known as potential. This is analogous to water flowing downstream, where the potential is determined by gravitational forces and fluid dynamics.

Interestingly, the architecture of LLMs mirrors this: they decompose sentences into fundamental units called tokens, which are adjusted layer by layer in a similar fashion:

A schematic equation describing the essence of LLMs

This analogy suggests that transformer-based language models treat words as particles, which interact and form intriguing patterns.

Just as water molecules create beautiful snowflakes or liquid soap forms intricate bubble designs, the remarkable outputs from ChatGPT may be linked to its physics-like characteristics.

In the next section, we will delve deeper into this analogy and explore how it can enhance our understanding of LLMs.

Technical Detour

In this section, we will elaborate on how to conceptualize LLMs through the lens of physics.

At the microscopic level, each particle is influenced by every other particle in the system. For instance, consider a hypothetical scenario with only three particles; this yields a total of 3 × 3 = 9 potential interactions. We can represent this schematically as follows:

A sketch for the equation of motion for 3 particles

To relate this to LLMs, we start with some foundational points: To input data into LLMs, text is segmented into tokens. These tokens represent the smallest indivisible elements within an LLM.

LLMs operate through multiple layers, with each layer modifying all tokens via self-attention modules. The ultimate output layer consolidates these tokens to generate predictions, which can be used for classifications or for generating text/images.

For a three-token example (let's say from the phrase "I enjoy physics"), how would the equations appear?

BERT Models

In BERT-like models, often utilized for classification, each layer modifies the tokens as follows:

The involvement of layer₁ results from the residual connections.

Considering the layers akin to a temporal dimension, the structure of the equation resembles the equations governing the movements of three particles, although in LLMs, the layers are discrete rather than continuous.

To finalize the analogy, we must translate the attention mechanism into a kind of potential. Let's delve deeper mathematically. For a specific token, tᵢ, at each layer, it is modified according to the self-attention mechanism (neglecting multiple attention heads):

Where Q, K, and V represent the Query, Key, and Value matrices typically observed in an attention module. For now, we will overlook normalization layers. The key point is that the exponential form can be reframed as the derivative of a potential term!

This perspective allows us to interpret the passage of tokens through layers in LLMs as akin to particles interacting under certain pairwise interactions. It’s similar to gas molecules colliding and forming weather patterns.

In this view, we can consider the normalization and matrix multiplication M as a form of projection, ensuring that the token-particles remain constrained within the system—much like a roller coaster confined to its tracks.

GPT

For (Chat)GPT models, the discussion shifts slightly. The attention mechanism incorporates a causal structure—tokens can only be influenced by preceding tokens. This means certain terms are absent from the equations.

In our analogy, this implies that particles enter one at a time, each becoming fixed after traversing all interaction layers—similar to how a crystal grows one atom at a time.

It's important to remember that our physics analogy isn't wholly precise, as fundamental principles like symmetries and conservation laws prevalent in physics do not necessarily apply to LLMs.

Emergence in Language Models

With our physics analogies in place, how do they enhance our understanding of LLMs? The hope is that, like complex physical systems, we can draw parallels from well-studied systems to gain insights into LLMs. However, readers should approach this with caution, as confirming these ideas would necessitate detailed experimental investigations on LLMs.

(* If I had more resources, I would envision these concepts as potential academic research projects.)

Below, we provide examples of how we might apply the language of physics to rethink our understanding of LLMs.

LLMs Training

Using terminology from thermal physics, we can conceptualize LLMs as tunable physical systems, where the training process is akin to applying thermal pressure to adjust system parameters. This concept has been elaborated in my other article, "The Thermodynamics of Machine Learning," so I won't delve deeply into it here.

Emergence of Intelligence?

While there is extensive debate regarding the intelligence of systems like ChatGPT, I will refrain from contributing to this contentious topic, as defining intelligence remains unclear. Nevertheless, it is evident that ChatGPT consistently generates sophisticated and engaging outputs.

If we adhere to our physics analogy, this should come as no surprise. From snowflakes to tornadoes, we understand that even simple laws can produce highly complex behaviors, and such behaviors can lead to structures that exhibit intelligent characteristics.

Complexity is a challenging concept to define. To advance further, we can examine key attributes of complex systems, one of which is phase transition.

Phase Transition

Many complex physical systems exhibit distinct phases, each characterized by a unique set of properties. Therefore, it stands to reason that within LLMs, there could also be distinct phases, each optimized for specific tasks (such as coding versus proofreading).

How might we validate or refute this assertion? This is where the intrigue lies. In physics, phases emerge when interactions give rise to interesting structures. Examples include:

• When water cools, the attractive forces between molecules strengthen, leading to solid formation.
• When metals are cooled to extremely low temperatures, phonon interactions may cause electrons to attract, forming Type-I superconductors.

Could something analogous occur in LLMs? For instance, in ChatGPT, one might speculate that certain token groupings related to "code" or "proofreading" could trigger a cascade of specific interactions that generate particular outputs.

Another technical aspect of phase transitions involves modifying symmetries. This relates to the formation of structures, such as ice crystal patterns from water vapor. While LLMs do not possess physical symmetries, they should contain some permutation symmetries regarding model weights. Model performance should remain consistent as long as they are initialized with similar statistics and trained under the same conditions. The specific values of a weight only become critical during training, akin to the freezing-in of weights. However, delving deeper into spontaneous symmetry breaking requires technical exploration, which will be addressed later.

Are LLMs Efficient?

Critiques often arise regarding the apparent inefficiency of LLMs due to their vast number of parameters, especially when compared to physical models. However, these criticisms may not be entirely justified.

Why? It boils down to the technical limitations of our computers, which lead to significant disparities between physics and LLMs:

• Physical laws possess infinite precision, while LLMs function with finite precision.
• Physics showcases vast hierarchies, with some forces being negligible and others substantial. In LLMs, we strive to normalize outputs and weights to be similar in scale.
• In physics, tiny effects can lead to substantial consequences (like Earth's gravity). Conversely, small effects in LLMs are often rounded off and disregarded.
• Nature operates as an extraordinarily efficient computer, executing interactions instantaneously across all scales with infinite precision. LLMs, on the other hand, are relatively slow computers constrained by finite precision.

Thus, while we aim to enhance LLMs to better mimic physical processes and develop more advanced models, practical limitations of computers prevent a complete simulation of our reality (as discussed in "Why We Don't Live In a Simulation"). Consequently, relying on a large number of parameters may be a last-resort strategy to address these inadequacies.

It's even conceivable that finite precision imposes an upper limit on the complexity achievable with standard computers, making it challenging to significantly reduce the parameter count (though advancements in quantum computing could alter this landscape in the future).

Improvements to LLMs

Could our physics analogy offer insights for the next generation of LLMs? I believe it’s possible. Logically, we can explore two potential directions based on our assumptions:

1. Desirable Physics-like Features: We should draw inspiration from physics to design improved model structures.
2. Undesirable Physics-like Features: Such features may constrain LLM capabilities due to inherent computational limits, necessitating avoidance.

Focusing on the first possibility, how might we address the limitations of LLMs like ChatGPT?

1. Preserving Hierarchies: Rather than solely normalizing weights and reducing precision, we should investigate alternative methods to accommodate diverse interactions with varying strengths and scales. We could draw parallels from how electromagnetism (strong) and gravity (weak) are intertwined in nature.
2. Accommodating Different Phases: Using the same basic molecular equations for both ice and water is inefficient. Describing different phases (like sound waves versus water waves) can lead to more efficient models. We could develop structures that inherently accommodate macroscopic differences within the model.
3. Advanced Physics Techniques: In physics, emergent phenomena aren't studied solely through fundamental equations. Employing techniques like thermodynamics, mean-field theory, and renormalization can streamline complex problems. Incorporating these concepts into LLM building blocks could enhance efficiency. For instance, recent developments in linear attention (A. Katharopoulos et al.) may represent a mean-field approach.

By exploring these strategies, we might amplify the capabilities and efficiencies of LLMs, leveraging insights from physics to propel the field forward.

Epilogue

In summary, we have illustrated how the mathematics governing LLMs parallels those in physics, enabling us to utilize our understanding of familiar physical systems to comprehend these new emergent phenomena, such as ChatGPT. I hope this demystifies the underlying mechanisms of LLMs.

More broadly, I aspire to convey how physics can shed light on complex subjects like LLMs. I firmly believe that interdisciplinary approaches enhance scientific understanding.

If you found this article engaging, you may also appreciate my other writings that explore the intersections of physics and AI.

Please feel free to share your thoughts or feedback, as it motivates me to produce more insightful content! 👋

A discussion on the trade-offs between larger language models like GPT and smaller alternatives, highlighting their respective strengths and weaknesses.

An overview of breakthroughs in large language models including GPT, ChatGPT, and BERT, examining their impact on the field of AI.