Read a recent article** Computer Scientists Prove why Bigger NNs do better** discussing scientific research that proved a

**. This speaks to the perturbability of AI deep learning neural networks (DNN) and how not reduce it. But also applies to many other solutions to diverse multi-dimensional data problems.**

*Universal Law of Robustness via Isoperimetry*## Mathmatical Robustness

For AI ML DNN’s, we often witnesssupposedly well trained DNN models that do very well for classifications of examples of data similar to their training data but fail miserably on data that’s outside their training data.

Mathematicians call this attribute robustness and can measure this on a mapping function using a** Lipschitz constant**. One can consider this as a measure of variability of mapping from one set to another or in the case of DNNs, lack of robustness in classifications means they fail on relatively minor changes to input data.

Most serious AI researchers have empirically discovered that bigger DNNs work better and are more robust than smaller networks. There’s been somewhat of a conundrum as to why DNNs need to get bigger to properly generalize.

## Universal Low of Robustness

What the researchers have proved is that in order to achieve some arbitrary level of robustness for a mapping function like DNNs, one needs many more parameters than expected the training data elements would indicate

For example, with the ** MNIST handwritten digit classification problem**, models with 10**5 parameters to 10**6 parameters are required to achieve a 90% and 95% accuracy, respectively. But MNIST training data is 60K examples (10**4). Why should a MNIST DNN classification model need more than 10**4 parameters to achieve 100% accurate?

From what we all learned in high school maths, to solve a function with N variables one needs N equations. This would lead one to believe that MNIST DNNs (essentially solving classification equations) should only need 60K or 10**4 parameters. But real DNNs to solve MNIST need more than that.

Looking at it in 2D. If one has two points, (x,y) for point A that maps to another (x,y) point B, one should only need to know one of the points and the slope of the line that connects them, or two parameters: point A (or B) and line slope.

Now with MNIST data that maps handwritten digits to one of 10 digits, we have essentially 10 possibilities being mapped from 60K samples. At best, we should need to know the 60K initial points in this image data space and their slope to the 10 digits they represent. Again something that approaches 60K pairs of parameters: one for the image point and one for the slope. But why doesn’t a MNIST model with 60K parameters achieve 100% accuracy.

I won’t claim to understand the math but what the researchers seem to be saying is that in order to have a **relatively smooth mapping** from the image space to the digit space one has to have 10**4 parameters X the **dimensionality of the data.** In this case, for MNIST, the dimensionality of the data is related to image size of** 28X28, 0..255** grey scale pixel images. The image space alone would be on the order of 10**5. So multiplying this by the size of the training data, the researchers estimate that the number of parameters should be 10**9 to be 100% accurate.

Although, the researchers say that the data dimensionality of the MNIST images are probably not 10**5 (how they concluded this is not evident). As such, they believe one shouldn’t need 10**9 parameters to reach 100% proper classifications. They say it’s probably 1 or 2 orders of magnitude less, because not all of the image data space is populated. So if we use 10**3 as an estimate of the effective data dimensionality, they would estimate that one would need 10**7 parameter DNN to reach 100% accuracy on MNIST data.

The author’s MNIST model achieved a 99.2% accuracy after training for 15 Epochs, batch size=5. Although 688K parameters is not quite 10**6 parameters, it’s close. Unclear why one would need another factor of 10, but getting that extra 0.8% accuracy (to 100%) can be very difficult to achieve for any DNN model.

Another example, OpenAI’s GPT-3 NLP model

And OpenAI’s ** GPT-3** NLP model has 175B parameters. Their previous version,

**only had 1.5B parameters and they say that GPT-4 will have over a 100T parameters. The chart above shows accuracy stats for 3 versions of the GPT-3 model, one with 175B, one with 13B and another with 1.3B parameters.**

*GPT-2,*According to OpenAI’s GPT-3 description, it can complete “almost any english language task” (text in ==> text out). This includes writing articles from a few prompts and text summarization.

GPT-3 was trained on almost 500B tokens (from web crawls to wikipedia dumps). Each token probably represents an english word or word phrase. According to the universal law, 175B parameters would not be sufficient. Probably why GPT-3 in the above chart didn’t reach 70%^ accuracy.

Probably would need at least another 3 orders of magnitude to get there or 175T parameters. Maybe with GPT-4, I can have it start writing my blog posts.

I don’t know about you, but I’m going to need more GPUs for my (home) AI lab.

*Photo Credit(s)*

*“Neural Network” by cesarharada.com is licensed under**Author’s MNIST model parameters**Author’s MNIST model*evaluation after 15 Epochs of training*From OpenAI’s GPT-3 Arxiv paper*