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This is: Understanding “Deep Double Descent”, published by Evan Hubinger on the AI Alignment Forum.
If you're not familiar with the double descent phenomenon, I think you should be. I consider double descent to be one of the most interesting and surprising recent results in analyzing and understanding modern machine learning. Today, Preetum et al. released a new paper, “Deep Double Descent,” which I think is a big further advancement in our understanding of this phenomenon. I'd highly recommend at least reading the summary of the paper on the OpenAI blog. However, I will also try to summarize the paper here, as well as give a history of the literature on double descent and some of my personal thoughts.
Prior work
The double descent phenomenon was first discovered by Mikhail Belkin et al., who were confused by the phenomenon wherein modern ML practitioners would claim that “bigger models are always better” despite standard statistical machine learning theory predicting that bigger models should be more prone to overfitting. Belkin et al. discovered that the standard bias-variance tradeoff picture actually breaks down once you hit approximately zero training error—what Belkin et al. call the “interpolation threshold.” Before the interpolation threshold, the bias-variance tradeoff holds and increasing model complexity leads to overfitting, increasing test error. After the interpolation threshold, however, they found that test error actually starts to go down as you keep increasing model complexity! Belkin et al. demonstrated this phenomenon in simple ML methods such as decision trees as well as simple neural networks trained on MNIST. Here's the diagram that Belkin et al. use in their paper to describe this phenomenon:
Belkin et al. describe their hypothesis for what's happening as follows:
All of the learned predictors to the right of the interpolation threshold fit the training data perfectly and have zero empirical risk. So why should some—in particular, those from richer functions classes—have lower test risk than others? The answer is that the capacity of the function class does not necessarily reflect how well the predictor matches the inductive bias appropriate for the problem at hand. [The inductive bias] is a form of Occam’s razor: the simplest explanation compatible with the observations should be preferred. By considering larger function classes, which contain more candidate predictors compatible with the data, we are able to find interpolating functions that [are] “simpler”. Thus increasing function class capacity improves performance of classifiers.
I think that what this is saying is pretty magical: in the case of neural nets, it's saying that SGD just so happens to have the right inductive biases that letting SGD choose which model it wants the most out of a large class of models with the same training performance yields significantly better test performance. If you're right on the interpolation threshold, you're effectively “forcing” SGD to choose from a very small set of models with perfect training accuracy (maybe only one realistic option), thus ignoring SGD's inductive biases completely—whereas if you're past the interpolation threshold, you're letting SGD choose which of many models with perfect training accuracy it prefers, thus allowing SGD's inductive bias to shine through.
I think this is strong evidence for the critical importance of implicit simplicity and speed priors in making modern ML work. However, such biases also produce strong incentives for mesa-optimization (since optimizers are simple, compressed policies) and pseudo-alignment (since simplicity and speed penalties will favor simpler, faster proxies). Furthermore, the arguments for the universal prior and minimal circuits being malign suggest that such strong simplicity and speed priors could...
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