🤖 machine learning

Why the Third Axis Is Freedom

This paper argues that the "third axis" in generative training is not explorative modeling (XM) itself, but rather "freedom"—the weakness of behavioral constraints—which XM facilitates by increasing match probability and empirically outperforms traditional selection methods in generalization and distribution shift scenarios.

Original authors: Michael Timothy Bennett

Published 2026-08-07

📖 8 min read🧠 Deep dive

CC BY 4.0

Original authors: Michael Timothy Bennett

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Great AI Guessing Game

Imagine you are teaching a robot to draw pictures. In the world of artificial intelligence, this is usually done through a process called "generative training." The robot tries to make an image, and if it looks different from the example you showed it, it gets a "penalty" or a bad score. The robot wants to minimize this penalty, so it learns to copy the example as perfectly as possible. But here's the catch: if the robot just learns to spit out the one most common answer it sees, it might become very good at copying that specific thing, but terrible at handling anything new or different. It becomes a rigid parrot rather than a creative artist.

For a long time, scientists wondered how to make these models more flexible. They discovered a trick called "Explorative Modeling." Instead of asking the robot to make just one guess, you ask it to make many guesses at once—say, ten different drawings. Then, you only punish the one that is closest to the target, ignoring the other nine. This forces the robot to keep a wider variety of ideas in its head, hoping that one of them will be the right one. This idea was recently called a new "third axis" of AI training, suggesting that the ability to hold many different "modes" or versions of an answer is the key to making AI smarter.

But is holding many ideas the same thing as being smart? This paper asks a deeper question: Is the magic in the number of guesses, or is it in the freedom the robot has to choose? The author argues that simply counting how many guesses a model makes isn't the whole story. Instead, the real secret is "freedom"—which, in this context, means how many different ways the model is allowed to be correct without breaking the rules. It's the difference between a robot that is forced to pick from a tiny list of ten options, and a robot that is allowed to pick from a vast library of possibilities, even if it currently only uses a few of them. The paper explores whether this "freedom" is what actually helps AI learn better, rather than just the raw number of guesses.

The Third Axis is Freedom, Not Just a Count

The author, Michael Timothy Bennett, takes a fresh look at this "third axis" of AI training. He argues that while the previous idea of "generative expressivity" (the ability to hold many modes) is interesting, it's actually a bit of a red herring. The true hero of the story is freedom.

Think of a model's behavior like a set of rules for a game. A "weak" rule is a rule that doesn't restrict you much. For example, a rule saying "You can wear any color shirt" is weak because it allows for millions of possibilities. A "strong" rule saying "You must wear a red shirt" is strong because it leaves you with only one choice. Bennett argues that the best AI models are the ones that keep their rules as "weak" as possible while still being correct. This "weakness" is what he calls freedom. It's the volume of compatible completions the model can still imagine. If a model is too rigid (too strong), it might fail when the world changes. If it is free (weak), it can adapt to new situations because it hasn't locked itself into a single narrow path.

The paper challenges the idea that the "third axis" is simply about the number of outputs a model can produce (its "mode count"). Bennett shows that a model could theoretically have a huge number of allowed outputs but still be terrible at generalizing if it only ever uses one of them. Conversely, a model with a smaller set of allowed outputs might be incredibly powerful if it spreads its attention evenly across them. The paper proves mathematically that freedom is a property of the model's function (what it actually does), not its form (how many parameters or layers it has). You can change the size of the model or the data it sees, but if the "freedom" of its behavior stays the same, its ability to generalize stays the same.

The Magic of "Best-of-K"

So, how do we get this freedom? The paper looks at a method called Explorative Modeling (XM), where a model generates $K$ candidates and picks the best one. The author derives a mathematical formula showing exactly how this process works.

Imagine you are looking for a specific key in a dark room.

If you only look once ( $K=1$ ), you might miss the key even if it's right there.
If you look ten times ( $K=10$ ), your chances of finding the key go up.

The paper shows that this "looking ten times" process acts like a magnifying glass on the model's freedom. It doesn't create new freedom out of thin air; instead, it rewards models that have a wide "permission profile." If a model allows many different outputs (high freedom), the "best-of-K" process is much more likely to find a good match. If the model is rigid and only allows one output, looking ten times doesn't help much because all ten guesses are likely to be the same boring thing.

The author proves that as you increase $K$ (the number of guesses), the training process naturally pushes the model toward being "weaker" (more free). It stops trying to be a perfect copycat of just one thing and starts learning to cover a broader range of possibilities. This is especially true when the targets (the things the model is trying to learn) are not all the same. If some targets are rare and some are common, a small $K$ will make the model focus only on the common ones. But as $K$ gets larger, the model is forced to pay attention to the rare ones too, eventually spreading its "permission" across all valid options.

The Experiments: Putting Theory to the Test

The paper doesn't just stay in the realm of math; it runs two major experiments to see if this holds up in the real world of neural networks.

Experiment 1: Does more guessing make the model freer?
The researchers trained AI models with different values of $K$ (from 1 to 32). They measured the "freedom" of the trained models by counting how many different valid outputs the models were actually allowed to produce.

The Result: When they used a higher $K$ , the models became significantly freer. For example, under uniform targets, increasing $K$ from 1 to 8 increased the "mean log freedom" by about 8.3. The models didn't just get better at guessing; they actually expanded their "permission profile" to include more valid possibilities. This confirmed that the training method successfully converts the budget of "guesses" into functional freedom.

Experiment 2: Does picking the "freer" model make it better?
Here, the researchers trained a bunch of models and then had to choose the best one to use in a new situation (where the rules of the game changed slightly). They compared two ways of choosing:

Child-Validation: Picking the model that did best on the original training data (the "child" task).
Freedom Selection: Picking the model that had the highest "freedom" score, even if it wasn't the absolute best on the original data.

The Result: The "Freedom Selection" method won 29 out of 30 times. The models chosen for their freedom performed much better on the new, balanced data (improving the hit rate from roughly 0.317 to 0.389). This suggests that by selecting for freedom, you are selecting for a model that is more robust and adaptable to change, even if it looks slightly less perfect on the specific data it was trained on.

The Big Picture: Means vs. Ends

The paper concludes with a clear distinction between the tool and the goal.

Exploration (the $K$ guesses) is the means. It's the mechanism, the budget, the way we force the model to look around.
Freedom is the end. It's the actual property of the model that makes it good at generalizing.

The author argues that "generative expressivity" (the idea that having many modes is the goal) is just a proxy for freedom. It's a way to count the possibilities, but it misses the nuance of how those possibilities are used. A model can have a high mode count but still be rigid if it doesn't use them well. True generalization comes from freedom—the ability of the model to remain open to many compatible futures.

In the end, the paper suggests that the "third axis" of AI training has always been freedom. The new methods like Explorative Modeling are just a clever way to train models to be less constrained, less rigid, and more ready for whatever the future throws at them. By training with more guesses, we aren't just teaching the AI to guess better; we are teaching it to be more free.

Technical Summary: Why the Third Axis Is Freedom

Problem Statement
Recent work on Explorative Modeling (XM) proposes "generative expressivity" as a third pretraining axis, defined as the maximum number of modes a loss-minimizing model can retain over a class of data distributions. XM trains models by generating $K$ candidates per step and updating only on the closest match to the target. While XM reports performance gains scaling with $K$ , model size, and data, this paper argues that "generative expressivity" is an inferior metric for understanding generalization. Specifically, it cannot rank models trained under the same objective, does not describe the specific model selected by training, and lacks a formal theorem connecting it to optimal generalization. The paper posits that the true underlying mechanism driving XM's success is not mode count, but freedom—a property of the function defined as the "weakness of constraint" implied by a model's behavior.

Methodology
The paper employs a theoretical framework based on Stack Theory, which measures the "freedom" of a policy by counting the number of compatible completions (refinements) it allows within an embodied language, rather than counting parameters or modes. The methodology proceeds through three main stages:

Theoretical Derivation:
- The author derives a Best-of- $K$ Coverage Identity for arbitrary measurable output spaces. This identity shows that the risk of hard best-of- $K$ training depends on the probability that a candidate misses an acceptable region, raised to the power $K$ .
- It is proven that under deployment consistency (where training candidates are drawn from the same law used at inference), the training process applies a transform $q \mapsto 1 - (1-q)^K$ to the mass of acceptable regions.
- A finite generative language is constructed where a policy's freedom is defined by the cardinality of its extension (the set of compatible completions). The paper proves that for $K > 1$ , the probability of hitting a target rises strictly with the local freedom of the policy, provided the policy distributes mass evenly across its permitted outputs.
- The paper distinguishes between deterministic $K$ -head models (which have a hard support ceiling of $K$ ) and stochastic Forward XM (where the support can exceed $K$ ).
Experimental Design:
- Experiment 1 (Training Dynamics): A latent-conditioned continuous generator was trained using sample-based Forward XM with varying $K$ ( $1, 2, 4, \dots, 32$ ). The study measured the "deployed permission profile" (the set of outputs the model can emit with positive probability) and calculated the resulting log-freedom. This was tested under both uniform and context-dependent target distributions.
- Experiment 2 (Model Selection): A pool of XM-trained models was generated. Two selection strategies were compared: standard "child-validation" (selecting based on performance on the training distribution) and a "freedom selector" (selecting models that maintain high child performance but maximize measured freedom on unlabelled parent contexts). The goal was to test if freedom selection improves generalization under a distribution shift (from long-tail child frequencies to balanced parent frequencies).

Key Contributions

Reframing the Third Axis: The paper formally argues that the "third axis" in generative modeling is freedom (the volume of compatible completions), not generative expressivity (mode count). Freedom is a property of the function, whereas expressivity is an objective-level capacity.
Formal Connection between Exploration and Freedom: The author proves that Best-of- $K$ training acts as a mechanism to amplify access to rare permissions. For $K > 1$ , the objective strictly prefers policies with higher local freedom (more permitted outputs) over those with fewer, provided the mass is balanced.
Generalization Theorem: It is proven that under independent nonempty demands on unseen contexts, the probability of a policy remaining compatible with future requirements is exactly proportional to its total freedom.
Empirical Validation:
- The paper demonstrates that increasing $K$ in Forward XM training directly increases the measured freedom of the deployed policy.
- It shows that selecting models based on freedom (rather than just child-task performance) significantly improves performance on a shifted parent distribution, even when the selected models have slightly lower performance on the original child task.

Results

Theoretical: The derived coverage identity confirms that exploration ( $K$ ) amplifies the value of rare permissions. The paper proves that for $K \ge 2$ , the unique population optimum for uniform targets is the uniform distribution over the full support, which maximizes local freedom. For nonuniform targets, the optimum trades frequency for support breadth as $K$ increases, eventually converging to uniform mass over the full support.
Experiment 1: Increasing $K$ from 1 to 32 increased the measured mean log-freedom. Under uniform targets, freedom saturated at $K=16$ . Under context-dependent targets, freedom continued to rise, demonstrating that larger $K$ converts exploration budgets into broader permission profiles.
Experiment 2: In 29 out of 30 test worlds, the freedom selector outperformed the child-validation selector on the balanced-parent task. The freedom-selected models achieved a balanced-parent hit rate of 0.389 compared to 0.317 for the child-validation baseline (a +0.072 improvement). These models retained broader valid repertoires (8.53 active outputs vs. 5.85) and higher log-freedom, accepting a marginal reduction in long-tail child performance.

Significance and Claims
The paper claims to provide a formal division of labor for generative modeling:

Parameters and Data determine the size of the permitted region a model can embody.
Exploration ( $K$ ) controls how strongly training rewards sampled access to that region.
Freedom is the resulting property of the function that determines future compatibility.

The author asserts that "generative expressivity" is merely a proxy for freedom that discards the extension structure essential for generalization. The paper concludes that XM is a means to an end: it is an effective mechanism for "free-maxing" (selecting the weakest correct policy). The "third axis" has always been freedom; XM simply operationalizes the training pressure to care about it. The results suggest that selecting for freedom, rather than just minimizing loss or counting modes, is a superior strategy for improving generalization under distribution shifts.

The Great AI Guessing Game

The Third Axis is Freedom, Not Just a Count

The Magic of "Best-of-K"

The Experiments: Putting Theory to the Test

The Big Picture: Means vs. Ends

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