Understand Neural Nets better, post 5 of N -- Code Assistant shootout

In a series of previous blogposts [1, 2, 3, 4] I ran some experiments drawing the boundaries of the polytopes generated by a fully-connected leaky ReLU network while it was getting trained on reproducing an input image.

As I tried to scale the experiments to larger networks, I noticed a dramatic slowdown in the code, caused by the calculation of a hash of the activation pattern happening on CPU -- so each training step would be fast, but then everything would grind to a halt for the visualisation, and for each pixel the code would forward-evaluate the NN (all in all 1024*1024 times), and whenever the prediction was calculated, it'd transfer the activation pattern to CPU and then perform the hashing. This was very slow, and very non-parallel.

I had contemplated writing some custom CUDA code to speed things up - there's no reason to store the activation pattern or transfer it, the "right" way to solve the problem is computing a hash on the fly, ideally a hash with a commutative update function so the order in which the different ReLU neurons update the hash doesn't matter.

Then again, this is a hobby project, and I don't have the time to do anything overly smart for the moment. So I decided to - before doing anything sophisticated - I'll see if I can have one of the two existing coding assistant that I use regularly solve the problem for me.

So I created two different directories, checked out the same base repo into both, created branches in both, and then queried both Gemini CLI and Claude Code perform the task, using the following prompt:

The Python script in this directory trains a fully connected leaky ReLU network on an input image and tries 
to reproduce it. It also draws pictures illustrating the boundaries of the polytopes generated by the creases
that the ReLU creates in input space. Unfortunately, the code to generate the polytope visualisation is slow,
because it involves 1024*1024 evaluations of the NN forward, and then it needs to hash the activation pattern
into a hash to uniquely identify what polytope the pixel resides on.

I would like to speed up this computation, by - instead of calculating a hash of the activation pattern at the 
end - somehow embedding the calculation of a hash into the forward pass on-GPU. This might be doable with 
PyTorch hooks, but I don't know precisely. 

What I do know is that if I run 
```
python3 ./draw-poly-while-training.py  --input ./centered_ring.png --shape [100]*20 --epochs 30 --seed 12345678 --points 5050 --save-interval 10
``` 

the output looks something like this: 
```
(...)
Input size (MB): 0.01
Forward/backward pass size (MB): 16.39
Params size (MB): 0.77
Estimated Total Size (MB): 17.17
==========================================================================================
2025-07-08 15:15:25,811 - polytope_nn - INFO - Epoch 1/2000000 - Train Loss: 3.315190, Val Loss: 0.329414
2025-07-08 15:15:25,857 - polytope_nn - INFO - Epoch 2/2000000 - Train Loss: 1.045730, Val Loss: 0.065818
2025-07-08 15:15:25,901 - polytope_nn - INFO - Epoch 3/2000000 - Train Loss: 1.414065, Val Loss: 0.488735
2025-07-08 15:15:25,948 - polytope_nn - INFO - Epoch 4/2000000 - Train Loss: 0.201550, Val Loss: 0.102159
2025-07-08 15:15:26,100 - polytope_nn - INFO - Epoch 5/2000000 - Train Loss: 0.198983, Val Loss: 0.050712
2025-07-08 15:15:26,145 - polytope_nn - INFO - Epoch 6/2000000 - Train Loss: 0.255710, Val Loss: 0.060731
2025-07-08 15:15:26,189 - polytope_nn - INFO - Epoch 7/2000000 - Train Loss: 0.122960, Val Loss: 0.091274
2025-07-08 15:15:26,232 - polytope_nn - INFO - Epoch 8/2000000 - Train Loss: 0.180629, Val Loss: 0.053913
2025-07-08 15:15:26,276 - polytope_nn - INFO - Epoch 9/2000000 - Train Loss: 0.826762, Val Loss: 0.156673
2025-07-08 15:15:26,320 - polytope_nn - INFO - Epoch 10/2000000 - Train Loss: 0.211313, Val Loss: 0.117810
2025-07-08 15:16:27,853 - polytope_nn - INFO - Visualization @ epoch 10: 61.53s
2025-07-08 15:16:27,899 - polytope_nn - INFO - Epoch 11/2000000 - Train Loss: 0.174978, Val Loss: 0.053103
2025-07-08 15:16:27,943 - polytope_nn - INFO - Epoch 12/2000000 - Train Loss: 0.332561, Val Loss: 0.095801
2025-07-08 15:16:27,987 - polytope_nn - INFO - Epoch 13/2000000 - Train Loss: 0.192859, Val Loss: 0.064341
2025-07-08 15:16:28,031 - polytope_nn - INFO - Epoch 14/2000000 - Train Loss: 0.115424, Val Loss: 0.051763
2025-07-08 15:16:28,076 - polytope_nn - INFO - Epoch 15/2000000 - Train Loss: 0.362009, Val Loss: 0.128609
2025-07-08 15:16:28,122 - polytope_nn - INFO - Epoch 16/2000000 - Train Loss: 0.117143, Val Loss: 0.058641
2025-07-08 15:16:28,165 - polytope_nn - INFO - Epoch 17/2000000 - Train Loss: 0.335812, Val Loss: 0.082517
2025-07-08 15:16:28,211 - polytope_nn - INFO - Epoch 18/2000000 - Train Loss: 0.079342, Val Loss: 0.060753
2025-07-08 15:16:28,257 - polytope_nn - INFO - Epoch 19/2000000 - Train Loss: 0.104123, Val Loss: 0.047914
2025-07-08 15:16:28,304 - polytope_nn - INFO - Epoch 20/2000000 - Train Loss: 0.097466, Val Loss: 0.050452
2025-07-08 15:17:31,553 - polytope_nn - INFO - Visualization @ epoch 20: 63.25s
```

From this we can see that a single visualisation step takes more than a minute for a network of this size, and 
profiling shows that most of this time is spent in hashing things on the CPU, not the GPU.
I would like you to find a way to do the calculation of the hash during the forward pass on the GPU, ideally 
without storing the activation vector in memory, and instead having a hash function that can be updated
commutatively so each ReLU unit can update the final hash while it calculates the forward pass.

I want you to:

1) Create a plausible plan for improving and speeding up the code.
2) Implement that plan.
3) Re-run the script with the specified command line, and observe if a speedup indeed took place -- e.g. check
that (a) the visualisation was sped up and (b) the sum of 10 training steps and the visualisation together was
sped up.

It is frightfully easy to speed up the visualisation step but slow down the training steps so much that 10
training steps and 1 visualisation step get *slower*.

Please also verify that the image output is the same between the pre-change and post-change version, to ensure
that the changes do not break anything.

I then allowed both models to churn for a while. Both models provided changes, but Gemini failed to actually verify that the results are the same. Claude one-shotted the problem; Gemini needed the following additional prompt:

I have run your example code, and checked the output. The output images are not identical between the
pre-change and post-change version, and even the training loss changed. FWIW, none of the polytopes
are visible in your version. Could you re-check your work, and this time make sure you check whether
the outputs are the same?

With that extra prodding / prompting, the solution provided by the model worked flawlessly, and was even a tiny bit faster than the Claude version.

Let's look at the code that both models generated: The Gemini branch and the Claude branch. Reading the changes, a few things become clear:

1. Gemini shot itself in the foot on the RNG by generating a bunch of random hash coefficients, and that messed up the state of the RNG, so the training runs were no longer comparable pre/post change.
2. Gemini is using torch.matmul for the hash computation, whereas Claude is computing the hash as torch.sum( A * B ).
3. Claude has broken up the code in more smaller functions, whereas Gemini didn't. Claude's code is mildly more readable, Gemini's is the more minimal change.

Interesting stuff. Neither solution is quite what I had in mind, but they are good enough for the moment, and provide a pretty significant speedup over the (also vibe-coded) stuff that I started out with. This is the first time for me that a coding assistant helped me optimize code in a nontrivial manner, and that's ... certainly something.

Anyhow, with these optimizations I can now run my data visualisation movie generation on slightly larger NNs with millions of parameters, so more studying ahead. I now need to figure out how to upload YouTube videos programmatically, but in the meantime, here is a video of training a 100-neuron, 10 layer deep network on the "circle drawing" task from my previous posts. Vibe coding randomly changed the color of my lines, but hey, that's ok.

As per usual, there are more questions than answers in this video. The thing that puzzles me most is the relative "instability" of the training in later epochs. This is visible in "flickers" where seemingly randomly the SGD step hits on a vastly higher loss, with parts of the screen turning black and loss spiking, and then the training needs to recover. Interestingly, the geometry of the polytopes doesn't change a lot in these situations, but the linear function on many of them changes at once, in a way that is very detrimental to overall performance. Once programmatic uploading works, I'll upload many more videos, because one of the intriguing observations I have is the following:

When training diverges (for larger and deeper nets), the divergence starts by first messing up the linear functions, and only after they are gloriously messed up, the geometry of the polytopes starts to go haywire, too.

Until then!

A non-anthropomorphized view of LLMs

In many discussions where questions of "alignment" or "AI safety" crop up, I am baffled by seriously intelligent people imbuing almost magical human-like powers to something that - in my mind - is just MatMul with interspersed nonlinearities.

In one of these discussions, somebody correctly called me out on the simplistic nature of this argument - "a brain is just some proteins and currents". I felt like I should explain my argument a bit more, because it feels less simplistic to me:

The space of words

The tokenization and embedding step maps individual words (or tokens) to some \(\mathbb{R}^n\) vectors. So let us imagine for a second that we have \(\mathbb{R}^n\) in front of us. A piece of text is then a path through this space - going from word to word to word, tracing a (possibly convoluted) line.

Imagine now that you label each of the "words" that form the path with a number: The last word with 1, counting forward until you hit the first word or the maximum context length \(c\). If you've ever played the game "Snake", picture something similar, but played in very high-dimensional space - you're moving forward through space with the tail getting truncated off.

The LLM takes your previous path into account, calculates probabilities for the next point to go to, and then makes a random pick into the next point according to these probabilities. An LLM instantiated with a fixed random seed is a mapping of the form \((\mathbb{R}^n)^c \mapsto (\mathbb{R}^n)^c\).

In my mind, the paths generated by these mappings look a lot like strange attractors in dynamical systems - complicated, convoluted paths that are structured-ish.

Learning the mapping

We obtain this mapping by training it to mimic human text. For this, we use approximately all human writing we can obtain, plus corpora written by human experts on a particular topic, plus some automatically generated pieces of text in domains where we can automatically generate and validate them.

Paths to avoid

There are certain language sequences we wish to avoid - because the sequences these models generate try to mimic human speech in all it's empirical structure, but we feel that some of the things that humans have empirically written are very undesirable to be generated. We also feel that a variety of other paths should ideally not be generated, if - when interpreted by either humans or other computer systems - undesirable results arise.

We can't specify strictly in a mathematical sense which paths we would prefer not to generate, but we can provide examples and counterexamples, and we try to hence nudge the complicated learnt distribution away from them.

"Alignment" for LLMs

Alignment and safety for LLMs mean that we should be able to quantify and bound the probability with which certain undesirable sequences are generated. The trouble is that we largely fail at describing "undesirable" except by example, which makes calculating bounds difficult.

For a given LLM (without random seed) and sequence, it is trivial to calculate the probability of the sequence to be generated. So if we had a way of somehow summing or integrating over these probabilities, we could say with certainty "this model will generate an undesirable sequence once every N model evaluations". We can't, currently, and that sucks, but at the heart, this is the mathematical and computational problem we'd need to solve.

The surprising utility of LLMs

LLMs solve a large number of problems that could previously not be solved algorithmically. NLP (as the field was a few years ago) has largely been solved.

I can write a request in plain English to summarize a document for me and put some key datapoints from the document in a structured JSON format, and modern models will just do that. I can ask a model to generate a children's book story involving raceboats and generate illustrations, and the model will generate something that is passable. And much more, all of which would have seemed like absolute science fiction 5-6 years ago.

We're on a pretty steep improvement curve, so I expect the number of currently-intractable problems that these models can solve to keep increasing for a while.

Where anthropomorphization loses me

The moment that people ascribe properties such as "consciousness" or "ethics" or "values" or "morals" to these learnt mappings is where I tend to get lost. We are speaking about a big recurrence equation that produces a new word, and that stops producing words if we don't crank the shaft.

To me, wondering if this contraption will "wake up" is similarly bewildering as if I was to ask a computational meteorologist if he isn't afraid of his meteorological numerical calculation will "wake up".

I am baffled that the AI discussions seem to never move away from treating a function to generate sequences of words as something that resembles a human. Statements such as "an AI agent could become an insider threat so it needs monitoring" are simultaneously unsurprising (you have a randomized sequence generator fed into your shell, literally anything can happen!) and baffling (you talk as if you believe the dice you play with had a mind of their own and could decide to conspire against you).

Instead of saying "we cannot ensure that no harmful sequences will be generated by our function, partially because we don't know how to specify and enumerate harmful sequences", we talk about "behaviors", "ethical constraints", and "harmful actions in pursuit of their goals". All of these are anthropocentric concepts that - in my mind - do not apply to functions or other mathematical objects. And using them muddles the discussion, and our thinking about what we're doing when we create, analyze, deploy and monitor LLMs.

This muddles the public discussion. We have many historical examples of humanity ascribing bad random events to "the wrath of god(s)" (earthquakes, famines, etc.), "evil spirits" and so forth. The fact that intelligent highly educated researchers talk about these mathematical objects in anthropomorphic terms makes the technology seem mysterious, scary, and magical.

We should think in terms of "this is a function to generate sequences" and "by providing prefixes we can steer the sequence generation around in the space of words and change the probabilities for output sequences". And for every possible undesirable output sequence of a length smaller than \(c\), we can pick a context that maximizes the probability of this undesirable output sequence.

A much clearer formulation, which helps more clearly articulate the problems to solve.

Why many AI luminaries tend to anthropomorphize

Perhaps I am fighting windmills, or rather a self-selection bias: A fair number of current AI luminaries have self-selected by their belief that they might be the ones getting to AGI - "creating a god" so to speak, the creation of something like life, as good as or better than humans. You are more likely to choose this career path if you believe that it is feasible, and that current approaches might get you there. Possibly I am asking people to "please let go of the belief that you based your life around" when I am asking for an end to anthropomorphization of LLMs, which won't fly.

Why I think human consciousness isn't comparable to an LLM

The following is uncomfortably philosophical, but: In my worldview, humans are dramatically different things than a function \((\mathbb{R}^n)^c \mapsto (\mathbb{R}^n)^c\). For hundreds of millions of years, nature generated new versions, and only a small number of these versions survived. Human thought is a poorly-understood process, involving enormously many neurons, extremely high-bandwidth input, an extremely complicated cocktail of hormones, constant monitoring of energy levels, and millions of years of harsh selection pressure.

We understand essentially nothing about it. In contrast to an LLM, given a human and a sequence of words, I cannot begin putting a probability on "will this human generate this sequence". 

To repeat myself: To me, considering that any human concept such as ethics, will to survive, or fear, apply to an LLM appears similarly strange as if we were discussing the feelings of a numerical meteorology simulation.

The real issues

The function class represented by modern LLMs are very useful. Even if we never get anywhere close to AGI and just deploy the current state of technology everywhere where it might be useful, we will get a dramatically different world. LLMs might end up being similarly impactful as electrification.

My grandfather lived from 1904 to 1981, a period which encompassed moving from gas lamps to electric, the replacement of horse carriages by cars, nuclear power, transistors, all the way to computers. It also spanned two world wars, the rise of Communism and Stalinism, almost the entire lifetime of the USSR and GDR etc. The world on his birth looked nothing like the world when he died.

Navigating the dramatic changes of the next few decades while trying to avoid world wars and murderous ideologies is difficult enough without muddying our thinking.

Some experiments to help me understand Neural Nets better, post 4 of N

After the previous blog posts here, here, and here, a friend of mine pointed me to some literature to read, and I will do so now :-).

The papers on my reading list are:

1. https://proceedings.mlr.press/v80/balestriero18b.html - Randall Balestrieros paper on DNNs as splines.
2. https://arxiv.org/abs/1906.00904 - ReLU networks have surprisingly few activation patterns (2019)
3. https://arxiv.org/abs/2305.09145 - Deep ReLU networks have surprisingly simple polytopes (2023)
4. https://www.frontiersin.org/journals/big-data/articles/10.3389/fdata.2023.1274831/full

I'll blog more once I get around to reading them all.

Some experiments to help me understand Neural Nets better, post 3 of N

What is this? After my first post on the topic, 9 months elapsed before I posted again, and now I am posting within days of the last post?

Anyhow, after my last post I could not resist and started running some experiments trying to see whether I could induce "overfitting" in the neural networks I had been training - trying to get a heavily overparametrized neural network to just "memorize" the training points so it generalizes poorly.

In the experiments I ran in previous posts, one of the key advantages is that I know the "true distribution" from which we are drawing our training data -- the input image. An overfit network would hence find ways to color the points in the training data correctly, but somehow not do so by drawing a black ring on white background (so it would be correct on the training data but fail to generalize).

So the experiment I kicked off was the following: Start with a network that has many times more parameters than we have training points: Since we start with 5000 training points, I picked 30 layers of 30 neurons for a total parameter count of approximately 27000 parameters. If von Neumann said he can draw an elephant with 4 parameters and make it wriggle it's trunk with 5, he'd certainly manage to fit 5000 training points with 27000 parameters?

Anyhow, to my great surprise, there was no hint of overfitting:

The network very clearly learns to draw a circle instead of fitting individual points. That is somewhat surprising, but perhaps this is just an artifact of our training points being relatively "dense" in the space, 5000 training points out of 1024*1024 is still 0.4%, that's a good chunk of the total space.

As a next step, I trained the same network, but with ever-reduced quantities of training data: 2500 points, 1250 points, 625 points, and 312 points. Surely training on 312 data points using 27000 parameters should generate clear signs of overfitting?

At 2500 points, while there is a noticeable slowdown in the training process, the underlying concept seems to be learnt just fine:

As we drop much lower, to 625 points, we can see how the network is struggling much more to learn the concept, but ... it still seems to have a strong bias toward creating a geometric shape resembling the ring instead of overfitting on individual points?

It appears that the learning process is slowed down - by epoch 6000 the network hasn't managed to reproduce the entire circle yet - and training seems to be less stable - but it looks as if the network is moving into the right direction. What happens if we halve the training points once more?

It's a bit of a mystery - I would have expected that by now we're clearly in a regime where the network should try fit individual points, we gave it just 0.02% of the points in the space. The network is clearly struggling to learn, and by epoch 6000 it is far from "ready" -- but it's certainly working towards a ring shape.

These experiments raise a number of questions for me:

1. It seems clear to me that the networks have some form of baked-in tendency to form contiguous areas - perhaps even a geometric shape - and the data needs to become very very sparse in order for true overfitting to occur. It's really unclear to me why we see the emergence of shapes here -- it would certainly be easy for the network to just pick the 312 polytopes in which the training points reside, and their immediate neighbors, and then have a steep linear function with big parameters to color just the individual dots black. But that's not what is happening here; there's some mechanism or process that leads to the emergence of a shape.

2. It almost seems like there is a trade-off -- if you have less data, you need to train longer, perhaps much longer. But it's really not clear to me that we will not arrive at comparatively good approximations even with 312 data points.

As a next step, I am re-running these experiments with 20000 epochs instead of 6000, to see if the network trained on very sparse training data catches up with the networks that have more data over time.

Some experiments to help me understand Neural Nets better, post 2 of N

In this post, I will explain my current thinking about neural networks. In a previous post I explained the intuition behind my "origami view of NNs" (also called the "polytope lens" in some circles). In this post, I will go a little bit into the mathematical details of this.

The standard textbook explanation of a layer of a neural network looks something like this: 

\[ \sigma( \overline{W}x + b )\]

where \(\sigma : \mathbb{R} \to \mathbb{R}\) is a nonlinearity (either the sigmoid or the ReLU or something like it), \(\overline{W}\) is the matrix of weights attached to the edges coming into the neurons, and \(b\) is the vector of "biases". Personally, I find this notation somewhat cumbersome, and I prefer to pull the bias vector into the weight matrices, so that I can think of an NN as "matrix multiplications alternating with applying a nonlinearity".

I really don't like to think about NNs with nonlinearities other than ReLU and leaky ReLU - perhaps over time I will have to accept that these are a thing, but for now all NNs that I think about are either ReLU or leaky ReLU. For the moment, we also assume that the network outputs a real vector in the end, so it is not (yet) a classifier.

Assume we have a network with \(k\) layers, and the number of neurons in each layer are \(n_1, \dots, n_k\). The network maps between real vector spaces (or an approximation thereof) of dimension \(i\) and \(o\).

\[
    NN : \mathbb{R}^i \to \mathbb{R}^o
\]
I would like to begin by pulling the bias vector into the matrix multiplications, because it greatly simplifies notation. So the input vector \(\overline{x}\) gets augmented by appending a 1, and the bias vector \(b\) gets appended to \(\overline{W}\):
\[
    W' = [\overline{W}b], x = \left[\begin{array}{c}\overline{x}\\1\end{array}\right]
\]

Instead of \(\sigma(\overline{W}x + b)\) we can write \(\sigma(W'x)\).
In our case, \(\sigma\) is always ReLU or leaky ReLU, so a "1" will be mapped to a "1" again. For reasons of being able to compose things nicely later, I would also like the output of \(\sigma(W'x)\) to have a 1 as last component, like our input vector \(x\). To achieve this, I need to append a row of all zeroes terminated in a 1 to \(W'\). Finally we have:

\[
    W = \left[\begin{array}{cc}\overline{W}& b\\0, \dots & 1\end{array}\right], x = \left[\begin{array}{c}\overline{x}\\1\end{array}\right]
\]

The previous post explained why the NN divides the input space into polytopes on which the approximated function will be entirely linear. Consider the data point \(x_1\). If you evaluate the NN on \(x_1\), a few of the ReLUs will light up (because their incoming data sums to more than 0) and a few will not. For a given \(x_1\), there will be \(k\) boolean vectors representing the activation (or non-activation) of each ReLU in the NN. Which means we have a function which for a given input vector, layer, and neuron number in the layer returns either \(0\) or \(1\) in the ReLU case, or \(0.01\) and \(1\) in the leaky ReLU case.

We call this function \(a\). We could make it a function with three arguments (layer, neuron index, input vector), but I prefer to move the layer and the neuron index into indices, so we have:

\[
    a_{l, n} : \mathbb{R}^i \to \{ 0, 1 \} \textnormal{ for ReLU }
\]
and

\[
    a_{l, n} : \mathbb{R}^i \to \{ 0.01, 1 \} \textnormal{ for leaky ReLU }
\]
This gives us a very linear-algebra-ish expression for the entire network:

\[
    NN(x) = W_1 A_1 \dots W_k A_k x = \prod_{i=0}^k (W_i A_i)x
\]

Where the \(A_k\) are of the form

\[
    A_k = \left( \begin{array}{cccc} a_{k, 1}(x) & \dots & 0 & 0\\ \dots & \dots & \dots & \dots \\ 0 & \dots & a_{k, n_k}(x) & 0\\ 0 & 0 & 0 & 1\end{array}\right)
\]

So we can see now very clearly that the moment that the activation pattern is determined, the entire function becomes linear, and just a series of matrix multiplications where every 2nd matrix is a diagonal matrix with the image of the activation pattern on the diagonal.

This representation shows us that the function remains identical (and linear) provided the activation pattern does not change - points on the same polytope will have an identical activation pattern, and we can hence use the activation pattern as a "polytope identifier" -- for any input point \(x\) I can run it through the network, and if a second point \(x'\) has the same pattern, I know it lives on the same polytope.

So from this I can take the sort of movies for single-layer NNs that were created in part 1 - where we can take an arbitrary 2-dimensional image as the unknown distribution that we wish to learn and then visualize the training dynamics: Show how the input space is cut up into different polytopes on which the function is then linearly approximated, and show how this partition and approximation evolves through the training process for differently-shaped networks.

We take input images of size 1024x1024, so one megabyte of byte-sized values, and sample 5000 data points from them - a small fraction, about 0.4% of the overall points in the image. We specify a shape for the MLP, and train it for 6000 steps, visualizing progress.

For simplicity, we try to learn a black ring on white ground, with sharply-delineated edges - first with a network that has 14 neurons per layer, and is 6 layers deep. 

On the left-hand side, we see the evaluated NN with the boundaries of the polytopes that it has generated to split the input space. In the center, we only see the output of the NN - what the NN has "learnt" to reproduce so far. And on the right hand side we see the original image, with the tiny, barely perceptible red dots the 5000 training points, and the blue dots a validation set of 1000 points. 

Here is a movie of the dynamics of the training run:

This is pretty neat, how about a differently-shaped NN? What happens if we force the NN through a 2-neuron bottleneck during the training process?

This last network has 10 layers of 10 neurons, then one layer of 2 neurons, then another 3 layers of 10 neurons. By number of parameters it is vaguely comparable to the other network, but it exhibits noticeably different training dynamics.

What happens if we dramatically overparametrize a network? Will it overfit our underlying data, and find a way to carve up the input space to reduce the error on the training set without reproducing a circle?

Let's try - how about a network with 20 neurons, 40 layers deep? That should use something like 20k floating point parameters in order to learn 5000 data points, so perhaps it will overfit?

Turns out this example doesn't, but it offers particularly rich dynamics as we watch it: Around epoch 1000 we can see how the network seems to have the general shape of the circle figured out, and most polytope boundaries seem to migrate to this circle. The network wobbles a bit but seems to make headway. By epoch 2000 we think we have seen it all, and the network will just consolidate around the circle. Between epoch 3000 and 4000 something breaks, loss skyrockets, and it seems like the network is disintegrating and training is diverging. By epoch 4000 it has re-stabilized, but in a very different configuration for the input space partition. This video ends around epoch 5500.

This is quite fascinating. There is no sign of overfitting, but we can see how the as the network gets deeper, training gets less stable: The circle seems to wobble much more, and we have these strange catastrophic-seeming phase changes after which the network has to re-stabilize. It also appears as if the network accurately captures the "circle" shape in spite of having only relatively few data points and more than enough capacity to overfit on them.

I will keep digging into this whenever time permits, I hope this was entertaining and/or informative. My next quest will be building a tool that - for a given point in input space - extracts a system of linear inequations that describe the polytope that this point lives on. Please do not hesitate to reach out if you ever wish to discuss any of this!

The German debt brake is stupid!

Welcome to one of my political posts. This blog post should rightfully be titled "the German debt brake is stupid, and if you support it, so are you (at least in the domain of economics)". Given that a nontrivial number of Germans agree with the debt brake, and given that there is a limit on the sensible number of characters in the title, I chose a shorter title - for brevity and to reduce offense. I nonetheless think that support for the debt brake, and supporters of the debt brake, are stupid.

In the following, I will list the reasons why I think the debt brake is stupid, and talk about a few arguments I have heard in favor of the debt brake, and why I don't buy any of them.

Reason 1: The debt brake is uniquely German, and I think the odds that Germany has somehow uncovered a deeper economic truth than anyone else is not high.

If you engage with economists a bit, you'll hear non-German economists make statements such as "there is economics, and there is German economics, and they have little in common" or "the problem with German economics is that it's really a branch of moral philosophy and not an empirical science". Pretty much the entire world stares in bewilderment at the debt brake law, and I have yet to find a non-German economist of any repute that says the German debt brake is a sensible construct.

The Wikipedia page is pretty blatant in showing that pretty much the only group supporting the debt brake are ... 48% of a sample of 187 German university professors for economics, in a poll conducted by an economic research think tank historically associated with the debt brake.

Now, I am not generally someone that blindly advocates for going with the mainstream majority opinion, but if the path you have chosen is described by pretty much the entire world as bizarre, unempirical, and based on moral vs. scientific judgement, one should possibly interrogate one's beliefs carefully.

If the German debt brake is a sensible construct, then pretty much every other country in the world is wrong by not having it, and the German government has enacted something unique that should convey a tangible advantage. It should also lead to other countries looking at these advantages and thinking about enacting their own, similar, legislation.

The closest equivalent to the German debt brake is the Swiss debt brake - but Switzerland has a lot of basis-democratic institutions that allow a democratic majority to change the constitution; in particular, a simple double-majority - majority of voters in the majority of cantons - is sufficient to remove the debt brake again. Switzerland can still act in times of crisis provided most voters in most cantons want to.

Germany, with the 2/3rds parliamentary majority required for a constitutional change, cannot. As such, the German debt brake is the most stringent and least flexible such rule in the world.

I don't see any evidence that the debt brake is providing any benefits to either Germans or the world. I see no other country itching to implement a similarly harsh law. Do we really believe that Germany has uncovered a deeper economic truth nobody else can see?

Reason 2: The debt brake is anti-market, and prevents a mutually beneficial market activity

While I am politically center-left, I am fiercely pro-market. I think markets are splendid allocation instruments, decentralized decision-making systems, information processors, and by-and-large the primary reason why the West out-competed the USSR when it came to producing goods. Markets allow the many actors in the economy to find ways how they can obtain mutual advantage by trading with each other, and interfering with markets should be done carefully, usually to correct some form of severe market failure (natural monopolies, tragedy-of-the-common, market for lemons etc. -- these are well-documented).

The market for government debt is a market like any other. Investors that believe that the government provides the best risk-adjusted return when compared to all other investment opportunities wish to lend the government money to invest it and provide the return. The government pays interest rate to these investors, based on the risk-free rate plus a risk premium.

Capital markets exist in order to facilitate decentralized resource allocation. If investors think that the best risk-adjusted returns are to be had by loaning the government money to invest in infrastructure or spend on other things, they should be allowed to offer lower and lower risk premia.

The debt brake interferes in this market by artificially constraining the government demand for debt. Even if investors were willing to pay the German government money to please please invest it in the broader economy, the German government wouldn't be allowed to do it.

In some sense, this is a deep intervention in the natural signaling of debt markets, and the flow of goods. It is unclear what market failure is being addressed here. 

Reason 3: The debt brake prevents investments with positive expected value

Assuming an opportunity arises where the government can invest sensibly in basic research or other infrastructure investments with strongly positive expected value for GDP growth and hence governmental income. Why should an arbitrary debt brake prohibit investments that are going to be net good for the whole of society?

Reason 4: The debt brake is partially responsible for the poor handling of the migration spike in 2015

Former Chancellor Merkel is often criticised for her "Wir schaffen das" ("We can do it") during the 2015 migration crisis. My main criticism, even back then, was that a sudden influx of young refugees has the potential for providing a demographic dividend, *provided* one manages to integrate the refugees into the society, the work force, and the greater economy rapidly. This necessitates investment, though: German language lessons, housing in economically non-deprived areas, German culture lessons, and much more -- and that sticking to the debt brake in an exceptional situation such as the 2015 migrant crisis is a terrible idea, because a sudden influx of refugees can have a destabilizing and economically harmful effect if the integration is botched. Successfully integrated people pay taxes and strengthen society, failure of integration leads to unemployment, potentially crime, and social disorder.

My view is that Merkel dropped the entire weight of the integration work on German civil society (which performed as best as they could, and admirably) because she was entirely committed to a stupid and arbitrary rule. I also ascribe some of the strength of Germany's far right on the disappointment that came from this mishandling of a crisis-that-was-also-an-opportunity.

Reason 5: The debt brake is based on numbers that economists agree are near-impossible to estimate correctly

It is extremely challenging to estimate the "structural deficit" of a given government, and most economists agree that there's no proper objective measurement of it, particularly when not done in retrospect. A law that prohibits governments from acting based on an unknowable quantity appears to be a bad law to me.

Reason 6: The debt brake is fundamentally based on a fear that politicians act too much in their own interest - but does not provide a democratic remedy

The underlying assumption of the debt brake is that politicians will act with their own best interest in mind, running long-term structural deficits that eventually bankrupt a country. In some sense, the notion is that "elected representatives cannot be trusted to handle the purse string, because they will use it to bribe the electorate to re-elect them".

We can discuss the extent to which this is true, but in the end a democracy should adhere to the sovereign, which is the voters. If we are afraid of a political caste abusing their position as representatives to pilfer the public's coffers, we should give the public more direct voting rights in budgetary matters, not artificially constrain what may be legitimate and good investments.

There is a deep anti-democratic undercurrent in the debt brake discussion: Either that the politicians cannot be trusted to behave in a fiscally responsible manner, or that the voters cannot be trusted to behave in a fiscally responsible manner, or that the view of politicians, voters and markets about what constitutes fiscal responsibility are somehow incorrect.

Reason 7: A German debt brake would be terrible policies for any business, why is it a good idea for a country?

Imagine for a second a company would pass bylaws that prevent issuing any additional debt, only to be bypassable by a shareholder meeting where 2/3rds of all shareholders agree that the debt can be issued. This would essentially give minority shareholders a fantastic way of taking the company hostage and demand concessions because taking on debt is a standard part of doing business. If we don't think that a majority of elected politicians can be trusted to not abuse the purse strings to extract benefits for themselves, why do we think it's a good idea to give a smaller group of elected politicians the right to block the governments ability to react in a crisis?

Reason 8: A lot of debt-brake advocacy is based in the theory of "starving the beast"

Debt-brake advocates are often simultaneous advocates of lower taxes. The theory is that by lowering taxes (and hence revenues) while creating a hard fiscal wall (the debt brake) one can force the government to cut popular programs to shrink the government - in other situations, cutting popular programs would be difficult as voters would not support it.

This idea was called "starving the beast" among US conservatives in the past. There's plenty of criticism of the approach, and all empirical evidence points to it being a terrible idea. It's undemocratic, too, as one is trying to create a situation of crisis to achieve a goal that would - assuming no crisis and democracy - not achievable.

Reason 9: Germany has let it's infrastructure decay to a point where the association of German industry is begging for infrastructure investments

The BDI is hardly a left-leaning tax-and-spend-happy group. They're historically very conservative, anti-union etc. - yet in recent years the decay of German infrastructure, from roads to bridges to the train system, has sufficiently unsettled them that we now have an alliance of German Unions and the German Employer Association call for much-needed infrastructure investments and modernisation.

The empirical evidence seems to be "when presented with a debt brake, politicians make necessary investments, and instead prefer to hollow out existing infrastructure".

Reason 10: Europe needs rearmament now, which requires long-time commitments to defense spending, but also investment in R&D etc.

The post-1945 rules-based order has been dying, first slowly in the GWOT, then it convulsed with the first Trump term; it looked like it might survive when Biden got elected, but with the second Trump term it is clear that it is dead. Europeans have for 20 years ignored that this is coming, in spite of everybody that made regular trips to Washington DC having seen it. The debt brake now risks paralyzing the biggest Eurozone economy by handing control over increased defense spending to radical fringe parties that are financed and supported by hostile adversaries.

Imagine a German parliament where the AfD and BSW jointly hold 1/3rd of the seats, and a war breaks out. Do we really want an adversary to be able to decide how much debt we can issue for national defense?

But the debt brake reassures investors and hence drives down Germany's interest rate payments!

Now, this is probably the only argument I have heard in favor of the debt brake that may merit some deeper discussion or investigation. There is an argument to be made that if investors perceive the risk of a default or the risk of inflation to be lower, they will demand a lesser coupon on the debt they provide. And I'm willing to entertain that thought. Something either I or someone that reads it should do is:

1. Calculate the risk premium that Germany had to pay over the risk-free rate in the past.
2. Observe to what extent the introduction of the debt brake, or the introduction of the COVID spending bills etc. impacted the spread between the risk-free rate and the yield on German government debt.

There are some complications with this (some people argue that the yield on Bunds *is* the risk-free rate, or at least the closest approximation thereof), and one would still have to quantify what GDP shortfall was caused by excessive austerity, so the outcome of this would be a pretty broad spectrum of estimates. But I will concede that this is worth thinking about and investigating.

At the same time, we are in a very special situation: The world order we all grew up in is largely over. The 1990s belief that we will all just trade, that big countries don't get to invade & pillage small countries, and that Europe can just disarm because the world is peaceful now is dead, and only a fool would cling to it.

I know that people would like to see a more efficient administration, and a leaner budget. These are good goals, and should be pursued - but not by hemming in your own government to be unable to react to crises, be captured by an aggressive minority, and reduce democratic choice.

Apologies for this rant, but given the fact that Europe has squandered the last 20 years, and that I perceive the German approach to debt and austerity to be a huge factor in this, it is hard for me to not show some of my frustration.

What I want for Christmas for the EU startup ecosystem

Hey all,

I have written about the various drags on the European tech industry in the past, and recently been involved in discussions on both X and BlueSky about what Europe needs.

In this post, I will not make a wishlist of what concrete policy reforms I want, but rather start "product centric" -- e.g. what "user experience" would I want as a founder? Once it is clear what experience you want as a founder, it becomes easier to reverse-engineer what policy changes will be needed.

What would Europe need to make starting a company smoother, easier, and better?

Let's jointly imagine a bit what the world could look like.

Imagine a website where the following tasks can be performed:

1. Incorporation of a limited liability company with shares. The website offers a number of standardized company bylaws that cover the basics, and allows the incorporation of a limited liability company on-line (after identity verification etc.).
   
2. Management of simple early-stage funding rounds on-line: Standardized SAFE-like instruments, or even a standardized Series A agreement, and the ability to sign these instruments on-line, and verify receipt of funds.
   
3. Management of the cap table (at least up to and including the Series A).
   
4. Ability to employ anyone in the Eurozone, and run their payroll, social security contributions, and employer-side healthcare payments. Possibly integrated with online payment.
5. Ability to grant employee shares and manage the share grants integrated with the above, with the share grants taxed in a reasonable way (e.g. only tax them on liquidity event, accept the shares themselves as tax while they are illiquid, or something similar to the US where you can have a lightweight 409a valuation to assign a value to the shares).
   
6. Integration with a basic accounting workflow that can be managed either personally or by an external accountant, with the ability to file simplified basic taxes provided overall revenue is below a certain threshold.
7. Ways of dealing with all the other paperwork involved in running a company on-line.

This is a strange mixture of Carta, Rippling, Docusign, Cloud Atlas, a Notary, and Intuit -- but it would make the process of starting and running a company much less daunting and costly.

Ideally, I could sign up to the site, verify my identity, incorporate a basic company with standardized bylaws, raise seed funding, employ people, run their payroll, and file basic taxes and paperwork.

In the above dream, what am I missing?

My suspicion is that building and running such a website would actually be not difficult (if the political will in Europe existed), and would have a measurable impact on company formation and GDP. If we want economic growth like the US, Europe needs to become a place where building and growing a business is easier and has less friction than in the US.

So assuming the gaps that I am missing are filled in, the next step is asking: What policy reforms are necessary to reach this ideal?