4.5 Generalization in Deep Learning
In Chapter 2 and Chapter 3, we tackled regression and classification problems by fitting linear models to training data. In both cases, we provided practical algorithms for finding the parameters that maximized the likelihood of the observed training labels. And then, towards the end of each chapter, we recalled that fitting the training data was only an intermediate goal. Our real quest all along was to discover general patterns on the basis of which we can make accurate predictions even on new examples drawn from the same underlying population. Optimization is merely a means to an end: machine learning researchers consume optimization algorithms, and sometimes invent new ones, but always in service of a statistical goal. At its core, machine learning is a statistical discipline and we wish to optimize training loss only insofar as some statistical principle (known or unknown) leads the resulting models to generalize beyond the training set.
On the bright side, it turns out that deep neural networks trained by stochastic gradient descent generalize well across myriad prediction problems, spanning computer vision; natural language processing; time series data; recommender systems; electronic health records; protein folding; value function approximation in video games and board games; and numerous other domains. On the downside, if you were looking for a straightforward account of either the optimization story (why we can fit them to training data) or the generalization story (why the resulting models generalize to unseen examples), then you might want to pour yourself a drink. While our procedures for optimizing linear models and the statistical properties of the solutions are both described well by a comprehensive body of theory, our understanding of deep learning still resembles the wild west on both fronts.
Both the theory and practice of deep learning are rapidly evolving, with theorists adopting new strategies to explain what’s going on, even as practitioners continue to develop heuristics for training deep networks and a body of empirical knowledge that provide guidance for deciding which techniques to apply in which situations.
The summary of the present moment is that the theory of deep learning has produced promising lines of attack and scattered fascinating results, but still appears far from a comprehensive account of both (i) why we are able to optimize neural networks and (ii) how models learned by gradient descent manage to generalize so well, even on high-dimensional tasks. In many benchmark settings, optimization can drive training error close to zero, making generalization the harder unexplained part. Optimization is not universally solved: failures remain common in very deep, constrained, or poorly conditioned models. On the other hand, even absent the comfort of a coherent scientific theory, practitioners have developed a large collection of techniques that may help you to produce models that generalize well in practice. While no pithy summary can possibly do justice to the vast topic of generalization in deep learning, and while the overall state of research is far from resolved, we hope, in this section, to present a broad overview of the state of research and practice.
4.5.1 Revisiting Overfitting and Regularization
According to the “no free lunch” theorem of Wolpert (1996), any learning algorithm generalizes better on some data distributions and worse on others. Thus, given a finite training set, a model must rely on assumptions, or inductive biases. To achieve human-level performance it can help to choose inductive biases that reflect how humans think about the world. Such inductive biases show preferences for solutions with certain properties. For example, a deep MLP has an inductive bias towards building up a complicated function by the composition of simpler functions.
With machine learning models encoding inductive biases, our approach to training them typically consists of two phases: (i) fit the training data; and (ii) estimate the generalization error (the true error on the underlying population) by evaluating the model on holdout data. The difference between our fit on the training data and our fit on the test data is called the generalization gap (Section 2.6) and when this is large, we say that our models overfit to the training data. In extreme cases of overfitting, we might exactly fit the training data, even when the test error remains significant. And in the classical view, the interpretation is that our models are too complex, requiring that we either shrink the number of features, the number of nonzero parameters learned, or the size of the parameters as quantified by their norm. Recall the plot of model complexity compared with loss (Figure 2.6.1) from Section 2.6.
However deep learning complicates this picture in counterintuitive ways. First, some modern classification models are expressive enough to fit every training example in large datasets (Zhang et al. 2021). In the classical picture, we might think that this setting lies on the far right extreme of the model complexity axis, and that any improvements in generalization error must come by way of regularization, either by reducing the complexity of the model class, or by applying a penalty, severely constraining the set of values that our parameters might take. But that is where things start to get weird.
For many deep learning benchmarks, several candidate architectures can reach nearly zero training error. Their useful differences then appear in validation error, compute, or other deployment criteria. In some such regimes, despite fitting the training data perfectly, we can actually reduce the generalization error further by making the model even more expressive, e.g., adding layers, nodes, or training for a larger number of epochs. Stranger yet, the pattern relating the generalization gap to the complexity of the model (as captured, for example, in the depth or width of the networks) can be non-monotonic, with greater complexity hurting at first but subsequently helping in a so-called “double-descent” pattern (Belkin et al. 2019; Nakkiran et al. 2021), which we examine in detail below. Thus the deep learning practitioner possesses a bag of tricks, some of which seemingly restrict the model in some fashion and others that seemingly make it even more expressive, and all of which, in some sense, are applied to mitigate overfitting.
Complicating things even further, while the guarantees provided by classical learning theory can be conservative even for classical models, they appear powerless to explain why it is that deep neural networks generalize in the first place. Because deep neural networks are capable of fitting arbitrary labels even for large datasets, and despite the use of familiar methods such as \(\ell_2\) regularization, worst-case complexity bounds based only on the full hypothesis class, such as basic parameter-count VC bounds, are often vacuous at modern scales. More data-dependent and algorithm-dependent bounds remain an active research area; no single account yet predicts deep-network generalization across settings. (Section 3.6 introduces these ideas; the mechanics of the classical bounds (concentration of measure, uniform convergence, and Rademacher complexity, with proofs) are developed in Section 26.6.)
4.5.1.1 Double Descent
Double descent is one observed departure from the simplest classical picture. Classical theory predicts a U-shaped test-error curve: as we add capacity, error first falls (we stop underfitting) and then rises (we begin overfitting), with a sweet spot in between (recall Figure 2.6.1 from Section 2.6). In some models, datasets, and training regimes, test error instead shows double descent. Near the interpolation threshold, the smallest capacity at which the training procedure fits the data, test error rises and may then descend a second time as capacity increases. Parameter count can locate this threshold in simple linear models, but it is not a reliable proxy for the effective capacity of a deep network. This non-monotone, two-valley shape is called double descent (Belkin et al. 2019), and it appears when we grow the model, when we train for more epochs, and when we add more data (Nakkiran et al. 2021) (Figure 4.5.1).
Why can bigger be better past the point of interpolation? In linear least-squares and random-feature models, the minimum-norm interpolant can have high variance near a rank transition and lower variance after more features create additional interpolating solutions. This mechanism gives a precise double-descent calculation, but it is a model-specific explanation rather than a theorem about all deep networks. That calculation is developed in Section 26.6; we return below to what is known about optimizer-dependent implicit bias.
Model size, moreover, is only one of three knobs that trace out this curve. Nakkiran et al. (2021) document model-wise double descent (grow the network, the flavor above and the one the appendix analyzes), epoch-wise double descent (fix the network and train longer: test error falls, rises as the model begins to interpolate noise, then falls again), and sample-wise double descent: adding training examples can hurt test performance, because more data moves the interpolation threshold and can push a fixed model back into the high-variance spike. All three are organized by a single axis that Nakkiran et al. call effective model complexity, roughly how many examples the full training procedure (model, optimizer, and budget) can fit perfectly. The error peaks wherever that quantity crosses the dataset size. This chapter only names the phenomena; the appendix proves the model-wise case, and the exercises below let you produce the epoch-wise one yourself.
4.5.2 Inspiration from Nonparametrics
Approaching deep learning for the first time, it is tempting to think of them as parametric models. After all, the models do have millions of parameters. When we update the models, we update their parameters. When we save the models, we write their parameters to disk. However, mathematics and computer science are riddled with counterintuitive changes of perspective, and surprising isomorphisms between seemingly different problems. While neural networks clearly have parameters, in some ways it can be more fruitful to think of them as behaving like nonparametric models. So what precisely makes a model nonparametric? While the name covers a diverse set of approaches, one common theme is that nonparametric methods tend to have a level of complexity that grows as the amount of available data grows.
Perhaps the simplest example of a nonparametric model is the \(k\)-nearest neighbor algorithm (we will cover more nonparametric models later, for example in Section 10.1.3). Here, at training time, the learner simply memorizes the dataset. Then, at prediction time, when confronted with a new point \(\mathbf{x}\), the learner looks up the \(k\) nearest neighbors (the \(k\) points \(\mathbf{x}_i'\) that minimize some distance \(d(\mathbf{x}, \mathbf{x}_i')\)). When \(k=1\), this algorithm is called \(1\)-nearest neighbors, and it achieves zero training error when training inputs are distinct and ties are resolved in favor of the queried example. That however, does not mean that the algorithm will not generalize. In fact, it turns out that under some mild conditions, the error of the \(1\)-nearest neighbor rule comes within a factor of two of the optimal (Bayes) error as the dataset grows (Cover and Hart 1967), and it is optimal in the noiseless case where the Bayes error is zero. (Full consistency, convergence to the optimal predictor, requires \(k\)-nearest neighbors with \(k \to \infty\) while \(k/n \to 0\).)
Note that \(1\)-nearest neighbor requires that we specify some distance function \(d\), or equivalently, that we specify some vector-valued basis function \(\phi(\mathbf{x})\) for featurizing our data. The zero-training-error statement assumes distinct training inputs and a rule for ties. The limiting Cover–Hart guarantee also requires a suitable metric space and regularity conditions connecting nearby inputs to their label distributions; an arbitrary or degenerate distance need not satisfy it. Under those conditions, \(1\)-nearest neighbor approaches its near-optimal limit, but different distance metrics \(d\) encode different inductive biases and with a finite amount of available data will yield different predictors. Different choices of the distance metric \(d\) represent different assumptions about the underlying patterns and the performance of the different predictors will depend on how compatible the assumptions are with the observed data.
In a sense, because neural networks are over-parametrized, possessing many more parameters than are needed to fit the training data, they tend to interpolate the training data (fitting it perfectly) and thus behave, in some ways, more like nonparametric models. More recent theoretical research has established deep connection between large neural networks and nonparametric methods, notably kernel methods. In particular, Jacot et al. (2018) demonstrated that in the limit, as multilayer perceptrons with randomly initialized weights grow infinitely wide, they become equivalent to (nonparametric) kernel methods for a specific choice of the kernel function (essentially, a distance function), which they call the neural tangent kernel. While current neural tangent kernel models may not fully explain the behavior of modern deep networks, their success as an analytical tool shows how a nonparametric limit can help analyze for understanding the behavior of over-parametrized deep networks.
4.5.3 Early Stopping
While deep neural networks are capable of fitting arbitrary labels, even when labels are assigned incorrectly or randomly (Zhang et al. 2021), this capability only emerges over many iterations of training. A line of work (Arpit et al. 2017; Rolnick et al. 2017) has shown that in the setting of label noise, neural networks tend to fit cleanly labeled data first and only subsequently to interpolate the mislabeled data. Moreover, this phenomenon can be turned into a generalization bound: if a model fits the clean examples but not deliberately mislabeled ones added to the training set, one can certify (with high probability) that its population error is small (Garg et al. 2021).
Together these findings help to motivate early stopping, a classic technique for regularizing deep neural networks. Here, rather than directly constraining the values of the weights, one constrains the number of epochs of training. The most common way to determine the stopping criterion is to monitor validation error throughout training (typically by checking once after each epoch) and to cut off training when the validation error has not decreased by more than some small amount \(\epsilon\) for some number of epochs. This is sometimes called a patience criterion. As well as the potential to lead to better generalization in the setting of noisy labels, another benefit of early stopping is the time saved. Once the patience criterion is met, one can terminate training. For large models that might require days of training simultaneously across eight or more GPUs, well-tuned early stopping can save researchers days of time and can save their employers many thousands of dollars.
Early stopping often helps when continued optimization begins to fit label noise. Its effect is not determined by label noise alone: model mismatch, optimization dynamics, augmentation, and the validation criterion also matter. Natural image tasks such as distinguishing cats from dogs should not be assumed realizable. Treat the stopping epoch as a hyperparameter selected on validation data, and retain the parameters from the best validated epoch rather than the last epoch examined.
4.5.4 Classical Regularization Methods for Deep Networks
In Chapter 2, we described several classical regularization techniques for constraining the complexity of our models. In particular, Section 2.7 introduced a method called weight decay, which consists of adding a regularization term to the loss function in order to penalize large values of the weights. Depending on which weight norm is penalized this technique is known either as ridge regularization (for \(\ell_2\) penalty) or lasso regularization (for an \(\ell_1\) penalty). In the classical analysis of these regularizers, they are considered as sufficiently restrictive on the values that the weights can take to prevent the model from fitting arbitrary labels.
In deep learning implementations, weight decay remains a popular tool. However, researchers have noted that typical strengths of \(\ell_2\) regularization are insufficient to prevent the networks from interpolating the data (Zhang et al. 2021). Their effects cannot be reduced to preventing interpolation. Depending on the model and optimizer, weight decay changes parameter norms, margins, effective learning rates, and the trajectory through parameter space. Early stopping is a separate control on that trajectory. Like the number of layers or the distance metric in 1-nearest neighbor, these choices may improve generalization by encoding inductive biases compatible with the patterns found in datasets of interest. Thus, classical regularizers remain popular in deep learning implementations, even if the theoretical rationale for their efficacy may be radically different.
4.5.4.1 Implicit Regularization
A learning algorithm imposes an implicit bias: among many interpolating parameters, its initialization and update rule make some solutions more likely than others. The bias is understood sharply in a few model classes. For linearly separable data, gradient descent on the logistic loss provably converges in direction to the \(\ell_2\) maximum-margin separator, even with no explicit penalty (Soudry et al. 2018). This theorem concerns a linear predictor and does not establish that SGD finds small-norm or well-generalizing solutions in an arbitrary deep network. Likewise, raw parameter-space flatness changes under function-preserving reparameterizations, so it cannot by itself explain generalization (Dinh et al. 2017). Sharpness-aware minimization is a useful training method (Foret et al. 2021), but its success does not turn flatness into a parameterization-invariant theory. Weight decay and early stopping interact with these algorithmic biases; their effects must be measured rather than inferred from the linear theorem. Grokking illustrates this: on small algorithmic tasks, networks first memorize the training set and only much later, after many further steps of training, suddenly generalize, a reminder that optimization dynamics govern generalization as much as architecture (Power et al. 2022). Figure 4.5.2 shows the signature: training accuracy saturates almost immediately, while validation accuracy sits at chance for orders of magnitude more steps before snapping to near-perfect, long after any conventional early-stopping rule would have given up.
Notably, deep learning researchers have also built on techniques first popularized in classical regularization contexts, such as adding noise to model inputs. In the next section we will introduce the famous dropout technique (invented by Srivastava et al. (2014)), which has become a mainstay of deep learning, even as the theoretical basis for its efficacy remains similarly mysterious.
4.5.5 Summary
Unlike classical linear models, which tend to have fewer parameters than examples, deep networks tend to be over-parametrized, and for most tasks are capable of perfectly fitting the training set. This interpolation regime challenges many hard-and-fast intuitions. Functionally, neural networks look like parametric models. But thinking of them as nonparametric models can sometimes be a more reliable source of intuition. Because it is often the case that all deep networks under consideration are capable of fitting all of the training labels, nearly all gains must come by mitigating overfitting (closing the generalization gap). Paradoxically, the interventions that reduce the generalization gap sometimes appear to increase model complexity and at other times appear to decrease complexity. However, these methods seldom decrease complexity sufficiently for classical theory to explain the generalization of deep networks, and why certain choices lead to improved generalization remains for the most part a massive open question despite the concerted efforts of many brilliant researchers.
4.5.6 Exercises
- In what sense do traditional complexity-based measures fail to account for generalization of deep neural networks?
- Why might early stopping be considered a regularization technique?
- How do researchers typically determine the stopping criterion?
- What important factor seems to differentiate cases when early stopping leads to big improvements in generalization?
- Beyond generalization, describe another benefit of early stopping.
- Epoch-wise double descent. Take the MLP of Section 4.2 on Fashion-MNIST, randomly relabel 15% of the training examples, and train far past convergence (several hundred epochs), recording test error after every epoch. Plot test error against the epoch count on a log axis. Do you observe a second descent after the initial overfitting rise? How does the curve change with the label-noise fraction, and how does the epoch of the peak relate to when the model starts fitting the noisy labels? What does this imply for choosing an early-stopping patience?
- () Grokking.* Reproduce the setup of Power et al. (2022): train a small network (they use a two-layer transformer, but a wide MLP on one-hot pairs also works) to predict \(c = (a + b) \bmod 97\) from the pair \((a, b)\), using a random 50% of all pairs for training, with weight decay, for \(10^5\) or more steps. Plot training and validation accuracy against the logarithm of the step count, and compare with Figure 4.5.2. How does the delay before generalization change with the training fraction and with the weight-decay strength?