Applying this to NNs seems to mean that we should expect (groups of) parameters to specialize for different functions if their “production curve” is convex and (groups of) parameters should be reused for multiple functions if their production curve is concave. That insight may help with interpretability. The question is if this is already known under different terminology among ML folks.
I’m not a deep ML researcher, but here is what ChatGPT says about how different parts of the training lead to more “convex” or “concave” effects:
ChatGPT 5.4 Long Reasoning
only when the current representation and gradient geometry make shared use of parameters costly. In SGD terms, the cleanest local signal is usually gradient interference.
For a shared parameter block θ\thetaθ, let and . For a small step, the first-order improvement from using the same parameters for both is roughly driven by , while splitting capacity lets you get something closer to . The difference is the cross-term:
So, locally, specialization is favored when : the two functions are trying to push the same parameters in conflicting directions. This is exactly the multitask “negative transfer” picture, and methods that enforce more orthogonal gradients are motivated by reducing that competition. [Regularizing Deep Multi-Task Networks using Orthogonal Gradients]
That gives a useful phase picture for SGD.
Very early training: specialization is usually weakest. In wide nets near initialization, training can be close to the lazy / kernel regime, where the network mostly reweights random features instead of strongly reorganizing them. In that regime, hidden units are still largely interchangeable, and the “shared mixed unit” often wins because there is not yet enough learned structure for durable task-specific interference to appear. Feature learning, which is the regime where internal specialization can really emerge, is precisely the regime beyond that lazy behavior. [Disentangling feature and lazy training in deep neural networks]
Mid training: this is where specialization most plausibly appears. Once hidden features begin to move, two things happen: first, symmetry between nominally equivalent units can break; second, some units become slightly better at one subfunction than another, and further SGD updates reinforce that asymmetry. In teacher–student analyses of layered networks, this shows up as a specialization transition, i.e. a move from an unspecialized symmetric phase to a specialized phase where hidden units take on different roles. For ReLU networks this transition is reported as continuous rather than abrupt in the recent statistical-physics analyses. [The Implicit Bias of Gradient Noise: A Symmetry Perspective]
Late training: specialization often stops increasing in the same sense. In classification settings, there is evidence for a terminal phase of training where last-layer representations undergo neural collapse: within-class variation shrinks and class means become arranged in a highly symmetric geometry. That is a kind of sharpening and consolidation, but not necessarily further functional diversification of internal parts. So late training often looks less like “wood vs leaves keep splitting” and more like “the learned class geometry is being compressed into a cleaner final arrangement.”
We could also ask the other way around: If early training is more concave and mid-training is convex, what does this imply for markets?
Presumably, in early, concave markets, traders offer multiple goods.
In mid, convex markets, traders specialize in few or a single product.
Applying this to NNs seems to mean that we should expect (groups of) parameters to specialize for different functions if their “production curve” is convex and (groups of) parameters should be reused for multiple functions if their production curve is concave. That insight may help with interpretability. The question is if this is already known under different terminology among ML folks.
I’m not a deep ML researcher, but here is what ChatGPT says about how different parts of the training lead to more “convex” or “concave” effects:
ChatGPT 5.4 Long Reasoning
For a shared parameter block θ\thetaθ, let
So, locally, specialization is favored when
That gives a useful phase picture for SGD.
Very early training: specialization is usually weakest.
In wide nets near initialization, training can be close to the lazy / kernel regime, where the network mostly reweights random features instead of strongly reorganizing them. In that regime, hidden units are still largely interchangeable, and the “shared mixed unit” often wins because there is not yet enough learned structure for durable task-specific interference to appear. Feature learning, which is the regime where internal specialization can really emerge, is precisely the regime beyond that lazy behavior. [Disentangling feature and lazy training in deep neural networks]
Mid training: this is where specialization most plausibly appears.
Once hidden features begin to move, two things happen: first, symmetry between nominally equivalent units can break; second, some units become slightly better at one subfunction than another, and further SGD updates reinforce that asymmetry. In teacher–student analyses of layered networks, this shows up as a specialization transition, i.e. a move from an unspecialized symmetric phase to a specialized phase where hidden units take on different roles. For ReLU networks this transition is reported as continuous rather than abrupt in the recent statistical-physics analyses. [The Implicit Bias of Gradient Noise: A Symmetry Perspective]
Late training: specialization often stops increasing in the same sense.
In classification settings, there is evidence for a terminal phase of training where last-layer representations undergo neural collapse: within-class variation shrinks and class means become arranged in a highly symmetric geometry. That is a kind of sharpening and consolidation, but not necessarily further functional diversification of internal parts. So late training often looks less like “wood vs leaves keep splitting” and more like “the learned class geometry is being compressed into a cleaner final arrangement.”
We could also ask the other way around: If early training is more concave and mid-training is convex, what does this imply for markets?
Presumably, in early, concave markets, traders offer multiple goods.
In mid, convex markets, traders specialize in few or a single product.
And in late markets?