We show that a variety of modern deep learning tasks exhibit a "double-descent" phenomenon where, as we increase model size, performance first gets worse and then gets better. Moreover, we show that double descent occurs not just as a function of model size, but also as a function of the number of training epochs. We unify the above phenomena by defining a new complexity measure we call the effective model complexity and conjecture a generalized double descent with respect to this measure. Furthermore, our notion of model complexity allows us to identify certain regimes where increasing (even quadrupling) the number of train samples actually hurts test performance. Right: Test error, shown for varying train epochs. All models trained using Adam for 4K epochs. The bias-variance tradeoff is a fundamental concept in classical statistical learning theory (e.g., Hastie et al. (2005)). The idea is that models of higher complexity have lower bias but higher variance. According to this theory, once model complexity passes a certain threshold, models "overfit" with the variance term dominating the test error, and hence from this point onward, increasing model complexity will only decrease performance (i.e., increase test error). Hence conventional wisdom in classical statistics is that, once we pass a certain threshold, "larger models are worse. Such networks have millions of parameters, more than enough to fit even random labels (Zhang et al. (2016)), and yet they perform much better on many tasks than smaller models. Indeed, conventional wisdom among practitioners is that "larger models are better' ' (Krizhevsky et al. (2012), Huang et al. (2018), Szegedy et al.
The question of generalization in machine learning---how algorithms are able to learn predictors from a training sample to make accurate predictions out-of-sample---is revisited in light of the recent breakthroughs in modern machine learning technology. The classical approach to understanding generalization is based on bias-variance trade-offs, where model complexity is carefully calibrated so that the fit on the training sample reflects performance out-of-sample. However, it is now common practice to fit highly complex models like deep neural networks to data with (nearly) zero training error, and yet these interpolating predictors are observed to have good out-of-sample accuracy even for noisy data. How can the classical understanding of generalization be reconciled with these observations from modern machine learning practice? In this paper, we bridge the two regimes by exhibiting a new "double descent" risk curve that extends the traditional U-shaped bias-variance curve beyond the point of interpolation. Specifically, the curve shows that as soon as the model complexity is high enough to achieve interpolation on the training sample---a point that we call the "interpolation threshold"---the risk of suitably chosen interpolating predictors from these models can, in fact, be decreasing as the model complexity increases, often below the risk achieved using non-interpolating models. The double descent risk curve is demonstrated for a broad range of models, including neural networks and random forests, and a mechanism for producing this behavior is posited.
Neural networks are trained to exactly fit the data. Such models usually would be considered as over-fitting, and yet they have managed to obtain high accuracy on test data. It is counter-intuitive -- but it works. This has raised many eyebrows, especially regarding the mathematical foundations of machine learning and their relevance to practitioners. In order to address these contradictions, researchers at OpenAI, in their recent work, double down on this widely believed grand illusion of bigger is better.
In this expository note we describe a surprising phenomenon in overparameterized linear regression, where the dimension exceeds the number of samples: there is a regime where the test risk of the estimator found by gradient descent increases with additional samples. In other words, more data actually hurts the estimator. This behavior is implicit in a recent line of theoretical works analyzing "double-descent" phenomenon in linear models. In this note, we isolate and understand this behavior in an extremely simple setting: linear regression with isotropic Gaussian covariates. In particular, this occurs due to an unconventional type of bias-variance tradeoff in the overparameterized regime: the bias decreases with more samples, but variance increases.
The classical bias-variance trade-off predicts that bias decreases and variance increase with model complexity, leading to a U-shaped risk curve. Recent work calls this into question for neural networks and other over-parameterized models, for which it is often observed that larger models generalize better. We provide a simple explanation for this by measuring the bias and variance of neural networks: while the bias is monotonically decreasing as in the classical theory, the variance is unimodal or bell-shaped: it increases then decreases with the width of the network. We vary the network architecture, loss function, and choice of dataset and confirm that variance unimodality occurs robustly for all models we considered. The risk curve is the sum of the bias and variance curves and displays different qualitative shapes depending on the relative scale of bias and variance, with the double descent curve observed in recent literature as a special case. We corroborate these empirical results with a theoretical analysis of two-layer linear networks with random first layer. Finally, evaluation on out-of-distribution data shows that most of the drop in accuracy comes from increased bias while variance increases by a relatively small amount. Moreover, we find that deeper models decrease bias and increase variance for both in-distribution and out-of-distribution data.