(Note: I'm Greek, but writing in English for the community.
Hope that's fine!)
Disclaimer: This is a fictional but realistic thought experiment,
distilled from patterns I've observed across multiple teams and projects.
It didn't happen exactly this way to me,
but it could – and that's exactly why it's worth discussing.
---
The Scenario
Imagine a team training a large sequence model
with a weekly rolling retraining cycle. For months,
everything looks fine.
Then someone notices something strange:
the validation loss for a specific input category
isn't improving – it oscillates. Good one week,
bad the next, good again.
The full validation loss is stable,
so the pattern is hidden.
A classic period‑2 oscillation.
The team does a parameter sweep
on L2 regularization strength.
Increasing the regularization smooths out
the validation curves.
The oscillation vanishes. Success.
The Twist
Three weeks later, they evaluate the model
on an out‑of‑distribution
slice – the rare edge cases
that matter most. The model fails.
The weights critical for those edge cases
have shrunk to nothing.
The “instability” they silenced wasn’t
numerical noise. It was the system
exploring features that only the
edge cases needed.
It was in a quasi‑chaotic exploration phase.
The regularization had moved the system
from a bounded but unpredictable attractor
(dynamical family 4) to a stable but weak one
(family 1). They traded the ability
to handle edge cases for a pretty
validation curve.
Why This Matters
In my own work, I've found it useful
to think of training loops
(and any iterative system)
as belonging to one of seven
dynamical families.
Two are relevant here:
- Family 4 – Bounded but unpredictable.
The system stays within bounds
but is highly sensitive to initial conditions.
Often mistaken for noise,
but can be a crucial exploration phase.
- Family 1 – Loops that converge.
The system settles into a fixed point.
Stable, predictable
– but sometimes the wrong fixed point.
The bifurcation in the regularization
landscape is real. More stability
is not always better.
The Question for the Community
Have you ever seen
a similar trap – where stabilizing your model
killed its ability to handle edge cases
or OOD data?
How did you diagnose it?
And more generally: before you turn a knob
to make the loss curve look nice,
do you ask yourself
*which attractor you are aiming for*?
(This fictional example
is part of a broader framework
I'm developing.
Happy to share more if there's interest.)
It’s really difficult to read your message, since it’s placed inside a scrolling text box, and the lines are extremely wide.
Thanks, i edit it correctcly…
Supervised models only give reliable results on data from within their training sets. Edge cases are not well modeled.
If the edge cases are critical, here are two suggestions:
- Increase the data set to contain more edge cases.
- Try an anomaly detection method, instead of regularized regression.
It is very difficult for a model to invent completely new solutions from scratch. A model generally behaves based on patterns and examples it has already seen during training. It may extrapolate or generate variations of known patterns, but it cannot reliably produce the correct output for cases that are entirely unfamiliar to it.
For example, if a model has never encountered a certain type of problem and there is no meaningful similarity to previous examples, then it becomes very hard to expect accurate predictions or reasoning. In such situations, the model lacks the necessary reference patterns to generalize effectively.
Hello @Nick_Angelosoulis,
The way you described this problem is very interesting. I tried to search for the subject of study and was it “discrete dynamical system”? My search so far is giving me some new perspective and I am going to explore it further.
For your questions, how would you diagnose and address edge cases? Would it be not too difficult to diagnose it because we can easily set up a validation dataset of these edge cases and measure the different states of the model along training steps against it? Besides looking at the scores of these validation datasets, we may also check the directions and the magnitudes of their gradients? Addressing it sounds tricky though. To be honest, I have not paid enough attention to the edge cases before but your description is inspiring me to start thinking more of it.
Thanks for sharing and asking the questions.
Cheers,
Raymond
Thank you for this — you’re asking exactly the right questions.
On the subject: Yes, discrete dynamical systems are the right lens. The training loop is a map
but with a periodic retraining schedule
it becomes a discrete-time system with
its own bifurcation structure.
The logistic map’s period‑2 regime is the perfect analogy.
On diagnosing edge cases:
You nailed it. A dedicated edge-case validation
set tracked over time (not just at the end)
would reveal the oscillation.
In my scenario, you’d see edge‑case loss alternating
high/low before regularization,
then flatlining at mediocre after.
The autocorrelation (negative at lag‑1) is the tell.
Gradients are even more powerful:
compare gradient directions
on an edge example from week to week.
In the oscillating regime, you’ll see cosine similarity
flip sign – the system switching between
two competing internal representations.
After regularization, gradients vanish
or point the same way every time (the wrong way).
On addressing it: You’re right, it’s tricky. A few ideas:
Don’t kill the oscillation
– trace both weight configurations
(the “good week” and “bad week”
checkpoints). Often one is better for edge cases.
Then you can ensemble or route.
Use cyclic regularization
– vary L2 strength with the same period
as the oscillation, so the system
never collapses to one attractor.
And sometimes, accept that a single fixed
point is impossible.
The best model might be a bounded orbit,
not a fixed point.
Your last sentence
– “I have not paid enough attention
to edge cases before”
– honestly, that’s most of us.
The fact that you’re now thinking about them dynamically
(across training steps) puts you ahead.
That’s exactly why I shared this.
Thanks again for the thoughtful response.
If you want to dig deeper into
the seven‑family framework,
let me know – happy to share notes.
Thank you all for the thoughtful engagement
— @rmwkwok, your point about diagnosing
via a dedicated edge-case validation
set is exactly right.
That’s the simplest and most practical first step.
The oscillation would have been visible immediately
if anyone had looked at that slice over time,
rather than just the aggregate validation loss.
A few of you have asked about the
“seven dynamical families” I mentioned.
Here’s a quick sketch:
Fixed point
– converges to single attractor
(most SGD with high LR decay)
Period‑2 cycle
– oscillates between two states
(the scenario in my post)
Higher-period cycle
(4,8,…)
– rarer in deep nets, but appears
in some RNNs
Bounded chaos
– strange attractor,
sensitive to initialization
(often mistaken for noise)
Drift
– weights grow without bound
(unstable training)
Edge of chaos
– critical regime between 4 and 5,
maximal computational capacity
Stochastic resonance
– noise-driven hopping
between attractors
The trap is that family 1
(fixed point) feels safe and looks good
on a smoothed validation curve.
But families 4 and 6 are often
more capable for OOD generalization
– they keep exploring.
The L2 regularization in my example
pushed the system from family
4 across a bifurcation into family 1,
killing the edge-case features.
@rmwkwok, you also asked about
addressing it, not just diagnosing.
Here are two concrete ideas
I’ve been playing with:
Cyclic regularization
– vary the L2 strength periodically,
matching the natural oscillation period.
This lets the system keep both attractors alive.
Gradient gating
– for edge-case examples, mask updates
to features that the main distribution
relies on heavily.
Protects the rare-feature weights
from being overwritten.
And to your deeper point:
sometimes a single fixed point
is impossible given the data.
In that case, the best model is
a bounded orbit, not a fixed point.
Our evaluation pipelines aren’t built
to reward that
– but maybe they should be.
Happy to share more if there’s interest.
And thanks again – this is exactly
the kind of discussion
I was hoping for.
Hello @Nick_Angelosoulis, thanks for sharing your insights and they are all very interesting. The seven families are indeed giving me some different angles looking at problems, but I still need to dig deeper. It’s like a gift box, and I feel that I am merely unwrapping its outer layer.
Checkpoints (bounded orbits) + ensemble + route sounds like an intuitive engineering approach but takes some experience. This also reminds me of another regularization technique - dropout, which is already making a model effectively like an ensemble of multiple checkpoints, because dropout makes us train only a submodel each time and so the model itself is a combination of many submodels, but this is just a crazy thought from someone who have not dived deep enough into the topic.
I should do my study.
Cheers,
Raymond
Hi rmwkwok,
I’m glad to hear that you’re finding
the seven families useful for approaching
your ML problems.
That’s exactly why I shared the thought experiment
– not as a finished theory,
but as a lens to see training dynamics differently.
If you’re already using the families
to diagnose or design your experiments,
I’d be very curious to know:
- Which family (or families) your current models
tend to fall into? - Have you observed any bifurcations
(e.g., from bounded chaos to fixed point)
when applying regularization like L2 or dropout? - Are you tracking any metrics beyond loss/accuracy
– like spectral radius of weight matrices,
or cycle consistency over training steps
– to distinguish between families?
I’m still digging deeper myself,
so any observations from your side
would be mutual learning.
Feel free to share a specific case
where the family framework helped
(or confused) you
– I’d love to help refine it together.
Thanks again for engaging with the idea.
Thanks for raising this, gent.spah.
You’re pointing to a fundamental limitation:
models are pattern completers, not inventors.
If a problem is truly novel
– no similar examples in the training distribution
– then pure pattern matching will fail.
But I’d like to expand on that with a nuance
from dynamical systems (
the “seven families” I sketched earlier).
There’s a difference between:
- Interpolation
– the model stays in a fixed‑point attractor (
family 1). It sees something similar
to training and returns a known answer.
This is what most supervised learning does. - Extrapolation within a bounded orbit
– the model lives in family 4 (bounded chaos)
or family 6 (stochastic resonance).
Here, the model doesn’t settle. It continuously
explores the state space within a bounded region.
When faced with an input that has no exact match,
its internal dynamics can resonate
with the input’s structure in ways
that a fixed‑point model cannot.
This sometimes produces outputs that
look like invention
– not because the model has seen the answer,
but because its internal trajectory is sensitive
to subtle, non‑obvious features.
Of course, this is not true “creativity.”
The model still operates within the attractor
shaped by training. But certain families
allow the model to re‑combine
learned patterns in ways that feel novel,
while a fixed‑point model would just guess randomly
or collapse to a mean.
So your point stands:
no model invents from absolute nothing .
But the type of dynamics matters for how it handles
the unfamiliar. A model that oscillates
(family 2) will behave differently from one
that is chaotic (family 4), and both differ
from a fixed‑point model. Understanding
which family your training dynamics
fall into might help you decide when to trust extrapolation
– and when to know that you need more data
or a different architecture.
Would love to hear your thoughts on whether
you’ve observed different extrapolation behaviors
across model checkpoints or architectures.
Hello @Nick_Angelosoulis,
I have not decided yet how I am going to use what I am learning from these families. Tackling OOD samples is a good goal or it may be a subtask or it may serve some kind of verification, but it is still too soon to tell what I am going to do next. Hah, I am still letting thoughts to jump around in my mind.
Cheers,
Raymond