Emergence in Neural Networks › Phase Transitions in Capability Acquisition

The key aspect of emergence is its discontinuity. Unlike the smooth loss curves predicted by scaling laws, emergent capabilities often appear through phase transitions: rapid shifts from one qualitative state to another

Recall from Part XXIII that scaling laws describe how cross-entropy loss decreases smoothly as we vary model size and the resources used to train it. The Chinchilla scaling laws give us:

$$L(N, D) = E + \frac{A}{N^{\alpha}} + \frac{B}{D^{\beta}}$$

where:

• $L(N, D)$: the cross-entropy loss as a function of model size and data
• $N$: the number of model parameters
• $D$: the amount of training data (in tokens)
• $E$: the irreducible loss (entropy of natural language)
• $A, B$: fitted scaling coefficients
• $\alpha, \beta$: power law exponents governing how quickly loss decreases with scale (typically around 0.34 and 0.28 respectively)

To understand this formula intuitively, think of it as describing three distinct contributions to model error. The first term, $E$, is a basic floor that no model can break through, the inherent unpredictability of language itself. Even if you knew everything about a speaker, you couldn't perfectly predict their next word because language contains real randomness and creative choice. This irreducible entropy sets a hard limit on how well any language model can perform.

The second and third terms capture how error decreases as we add more parameters or more training data, but with diminishing returns. The power law structure tells us something important: doubling your model size does not halve your loss. Instead, loss decreases by a factor of $2^{-\alpha} \approx 0.79$ (about 21% reduction). This means each additional order of magnitude in scale yields roughly similar absolute improvements in loss, making progress increasingly expensive. The same principle applies to data: doubling your training corpus gives increasingly smaller benefits as the corpus grows.

What makes this formula so useful is its predictive power within the domain of overall loss. Given measurements at smaller scales, researchers can extrapolate accurately to predict the loss of models orders of magnitude larger. The smooth, lawful decrease in cross-entropy loss suggests a world where model improvement is gradual and predictable.

Yet this smoothness creates a paradox. If capabilities were tied directly to loss, we would expect smooth improvements across all tasks as we scale up. But that is not what we observe for certain capabilities.

Consider a model's ability to perform three-digit addition. A small model might reach 0% accuracy. We scale up by 10x, and accuracy remains at 0%. We scale up another 10x, and suddenly the model reaches 95% accuracy. The cross-entropy loss decreased smoothly throughout, but task performance showed a phase transition. The model's overall language modeling improved gradually, but its arithmetic ability jumped discontinuously. This disconnect between smooth loss and discontinuous capabilities lies at the heart of the emergence phenomenon.

Let's create a visualization that contrasts smooth scaling with emergent behavior:

The contrast is clear. Perplexity improves smoothly and predictably with scale, while task accuracies remain flat until crossing a threshold, then rapidly rise to high performance. Different tasks emerge at different scales, suggesting that the computational complexity of each task determines when it becomes solvable.

We can characterize emergent behavior mathematically using a sigmoid function in log-parameter space:

$$\text{Accuracy}(N) = \frac{1}{1 + \exp\left(-k \cdot (\log N - \log N_{\text{threshold}})\right)}$$

where:

• $N$: the number of model parameters
• $N_{\text{threshold}}$: the necessary scale where emergence occurs (the inflection point of the sigmoid)
• $k$: the steepness parameter controlling how sharp the transition is
• $\log N - \log N_{\text{threshold}}$: the distance from threshold in log-space, equivalent to $\log(N/N_{\text{threshold}})$

This sigmoid formulation captures the characteristics of emergent phenomena. Let's unpack why each component matters for understanding emergence.

The sigmoid function is a natural choice here because it smoothly interpolates between 0 and 1, capturing the transition from "capability absent" to "capability present." The sigmoid maps any input into a value between 0 and 1. This makes it ideal for modeling probabilities or accuracy scores that are bounded by definition. When the argument inside the exponential is a large negative number (meaning $N$ is much smaller than $N_{\text{threshold}}$), the exponential term dominates and accuracy approaches zero. When the argument is a large positive number (meaning $N$ is much larger than $N_{\text{threshold}}$), the exponential term vanishes and accuracy approaches one.

The transition happens in log-space. A model needs to be a certain multiple of the threshold size, not a certain absolute number of parameters larger. This shows the multiplicative nature of neural network scaling. Going from 1 billion to 10 billion parameters has roughly the same effect as going from 10 billion to 100 billion. The logarithmic relationship means that equal ratios produce equal effects, which aligns with our empirical observations about how model capabilities scale.

When $k$ is large, the transition is sharp: near-zero performance below threshold, near-perfect above. You can think of $k$ as controlling the "width" of the transition zone in log-parameter space. A large $k$ means the model goes from incapable to capable over a narrow range of scales, perhaps just a factor of 2 or 3 in parameter count. When $k$ is small, the transition is gradual, spreading over orders of magnitude. The claim of emergence is in effect that certain capabilities have large $k$ values, meaning they appear suddenly rather than gradually.