19.1. What Is Hyperparameter Optimization?¶ Open the notebook in SageMaker Studio Lab
As we have seen in the previous chapters, deep neural networks come with a large number of parameters or weights that are learned during training. On top of these, every neural network has additional hyperparameters that need to be configured by the user. For example, to ensure that stochastic gradient descent converges to a local optimum of the training loss (see Section 12), we have to adjust the learning rate and batch size. To avoid overfitting on training datasets, we might have to set regularization parameters, such as weight decay (see Section 3.7) or dropout (see Section 5.6). We can define the capacity and inductive bias of the model by setting the number of layers and number of units or filters per layer (i.e., the effective number of weights).
Unfortunately, we cannot simply adjust these hyperparameters by minimizing the training loss, because this would lead to overfitting on the training data. For example, setting regularization parameters, such as dropout or weight decay to zero leads to a small training loss, but might hurt the generalization performance.
Without a different form of automation, hyperparameters have to be set
manually in a trialanderror fashion, in what amounts to a
timeconsuming and difficult part of machine learning workflows. For
example, consider training a ResNet (see Section 8.6) on
CIFAR10, which requires more than 2 hours on an Amazon Elastic Cloud
Compute (EC2) g4dn.xlarge
instance. Even just trying ten
hyperparameter configurations in sequence, this would already take us
roughly one day. To make matters worse, hyperparameters are usually not
directly transferable across architectures and datasets
(Bardenet et al., 2013, Feurer et al., 2022, Wistuba et al., 2018), and need to be
reoptimized for every new task. Also, for most hyperparameters, there
are no ruleofthumbs, and expert knowledge is required to find sensible
values.
Hyperparameter optimization (HPO) algorithms are designed to tackle this problem in a principled and automated fashion (Feurer and Hutter, 2018), by framing it as a global optimization problem. The default objective is the error on a holdout validation dataset, but could in principle be any other business metric. It can be combined with or constrained by secondary objectives, such as training time, inference time, or model complexity.
Recently, hyperparameter optimization has been extended to neural architecture search (NAS) (Elsken et al., 2018, Wistuba et al., 2019), where the goal is to find entirely new neural network architectures. Compared to classical HPO, NAS is even more expensive in terms of computation and requires additional efforts to remain feasible in practice. Both, HPO and NAS can be considered as subfields of AutoML (Hutter et al., 2019), which aims to automate the entire ML pipeline.
In this section we will introduce HPO and show how we can automatically find the best hyperparameters of the logistic regression example introduced in Section 4.5.
19.1.1. The Optimization Problem¶
We will start with a simple toy problem: searching for the learning rate
of the multiclass logistic regression model SoftmaxRegression
from
Section 4.5 to minimize the validation error on the
Fashion MNIST dataset. While other hyperparameters like batch size or
number of epochs are also worth tuning, we focus on learning rate alone
for simplicity.
import numpy as np
import torch
from scipy import stats
from torch import nn
from d2l import torch as d2l
Before we can run HPO, we first need to define two ingredients: the objective function and the configuration space.
19.1.1.1. The Objective Function¶
The performance of a learning algorithm can be seen as a function \(f: \mathcal{X} \rightarrow \mathbb{R}\) that maps from the hyperparameter space \(\mathbf{x} \in \mathcal{X}\) to the validation loss. For every evaluation of \(f(\mathbf{x})\), we have to train and validate our machine learning model, which can be time and compute intensive in the case of deep neural networks trained on large datasets. Given our criterion \(f(\mathbf{x})\) our goal is to find \(\mathbf{x}_{\star} \in \mathrm{argmin}_{\mathbf{x} \in \mathcal{X}} f(\mathbf{x})\).
There is no simple way to compute gradients of \(f\) with respect to \(\mathbf{x}\), because it would require to propagate the gradient through the entire training process. While there is recent work (Franceschi et al., 2017, Maclaurin et al., 2015) to drive HPO by approximate “hypergradients”, none of the existing approaches are competitive with the stateoftheart yet, and we will not discuss them here. Furthermore, the computational burden of evaluating \(f\) requires HPO algorithms to approach the global optimum with as few samples as possible.
The training of neural networks is stochastic (e.g., weights are randomly initialized, minibatches are randomly sampled), so that our observations will be noisy: \(y \sim f(\mathbf{x}) + \epsilon\), where we usually assume that the \(\epsilon \sim N(0, \sigma)\) observation noise is Gaussian distributed.
Faced with all these challenges, we usually try to identify a small set of well performing hyperparameter configurations quickly, instead of hitting the global optima exactly. However, due to large computational demands of most neural networks models, even this can take days or weeks of compute. We will explore in Section 19.4 how we can speedup the optimization process by either distributing the search or using cheapertoevaluate approximations of the objective function.
We begin with a method for computing the validation error of a model.
class HPOTrainer(d2l.Trainer): #@save
def validation_error(self):
self.model.eval()
accuracy = 0
val_batch_idx = 0
for batch in self.val_dataloader:
with torch.no_grad():
x, y = self.prepare_batch(batch)
y_hat = self.model(x)
accuracy += self.model.accuracy(y_hat, y)
val_batch_idx += 1
return 1  accuracy / val_batch_idx
We optimize validation error with respect to the hyperparameter
configuration config
, consisting of the learning_rate
. For each
evaluation, we train our model for max_epochs
epochs, then compute
and return its validation error:
def hpo_objective_softmax_classification(config, max_epochs=8):
learning_rate = config["learning_rate"]
trainer = d2l.HPOTrainer(max_epochs=max_epochs)
data = d2l.FashionMNIST(batch_size=16)
model = d2l.SoftmaxRegression(num_outputs=10, lr=learning_rate)
trainer.fit(model=model, data=data)
return trainer.validation_error().detach().numpy()
19.1.1.2. The Configuration Space¶
Along with the objective function \(f(\mathbf{x})\), we also need to define the feasible set \(\mathbf{x} \in \mathcal{X}\) to optimize over, known as configuration space or search space. For our logistic regression example, we will use:
config_space = {"learning_rate": stats.loguniform(1e4, 1)}
Here we use the use the loguniform
object from SciPy, which
represents a uniform distribution between 4 and 1 in the logarithmic
space. This object allows us to sample random variables from this
distribution.
Each hyperparameter has a data type, such as float
for
learning_rate
, as well as a closed bounded range (i.e., lower and
upper bounds). We usually assign a prior distribution (e.g, uniform or
loguniform) to each hyperparameter to sample from. Some positive
parameters, such as learning_rate
, are best represented on a
logarithmic scale as optimal values can differ by several orders of
magnitude, while others, such as momentum, come with linear scale.
Below we show a simple example of a configuration space consisting of typical hyperparameters of a multilayer perceptron including their type and standard ranges.
Name 
Type 
Hyperparameter Ranges 
logscale 

learning rate 
float 
\([10^{6},10^{1}]\) 
yes 
batch size 
integer 
\([8,256]\) 
yes 
momentum 
float 
\([0,0.99]\) 
no 
activation function 
categorical 
\(\{\text{tanh}, \text{relu}\}\) 

number of units 
integer 
\([32, 1024]\) 
yes 
number of layers 
integer 
\([1, 6]\) 
no 
In general, the structure of the configuration space \(\mathcal{X}\) can be complex and it can be quite different from \(\mathbb{R}^d\). In practice, some hyperparameters may depend on the value of others. For example, assume we try to tune the number of layers for a multilayer perceptron, and for each layer the number of units. The number of units of the \(l\mathrm{th}\) layer is relevant only if the network has at least \(l+1\) layers. These advanced HPO problems are beyond the scope of this chapter. We refer the interested reader to (Baptista and Poloczek, 2018, Hutter et al., 2011, Jenatton et al., 2017).
The configuration space plays an important role for hyperparameter optimization, since no algorithms can find something that is not included in the configuration space. On the other hand, if the ranges are too large, the computation budget to find well performing configurations might become infeasible.
19.1.2. Random Search¶
Random search is the first hyperparameter optimization algorithm we will consider. The main idea of random search is to independently sample from the configuration space until a predefined budget (e.g maximum number of iterations) is exhausted, and to return the best observed configuration. All evaluations can be executed independently in parallel (see Section 19.3), but here we use a sequential loop for simplicity.
errors, values = [], []
num_iterations = 5
for i in range(num_iterations):
learning_rate = config_space["learning_rate"].rvs()
print(f"Trial {i}: learning_rate = {learning_rate}")
y = hpo_objective_softmax_classification({"learning_rate": learning_rate})
print(f" validation_error = {y}")
values.append(learning_rate)
errors.append(y)
validation_error = 0.17659997940063477
The best learning rate is then simply the one with the lowest validation error.
best_idx = np.argmin(errors)
print(f"optimal learning rate = {values[best_idx]}")
optimal learning rate = 0.24202220709278127
Due to its simplicity and generality, random search is one of the most frequently used HPO algorithms. It does not require any sophisticated implementation and can be applied to any configuration space as long as we can define some probability distribution for each hyperparameter.
Unfortunately random search also comes with a few shortcomings. First, it does not adapt the sampling distribution based on the previous observations it collected so far. Hence, it is equally likely to sample a poorly performing configuration than a better performing configuration. Second, the same amount of resources are spent for all configurations, even though some may show poor initial performance and are less likely to outperform previously seen configurations.
In the next sections we will look at more sample efficient hyperparameter optimization algorithms that overcome the shortcomings of random search by using a model to guide the search. We will also look at algorithms that automatically stop the evaluation process of poorly performing configurations to speed up the optimization process.
19.1.3. Summary¶
In this section we introduced hyperparameter optimization (HPO) and how we can phrase it as a global optimization by defining a configuration space and an objective function. We also implemented our first HPO algorithm, random search, and applied it on a simple softmax classification problem.
While random search is very simple, it is the better alternative to grid search, which simply evaluates a fixed set of hyperparameters. Random search somewhat mitigates the curse of dimensionality (Bellman, 1966), and can be far more efficient than grid search if the criterion most strongly depends on a small subset of the hyperparameters.
19.1.4. Exercises¶
In this chapter, we optimize the validation error of a model after training on a disjoint training set. For simplicity, our code uses
Trainer.val_dataloader
, which maps to a loader aroundFashionMNIST.val
.Convince yourself (by looking at the code) that this means we use the original FashionMNIST training set (60000 examples) for training, and the original test set (10000 examples) for validation.
Why could this practice be problematic? Hint: Reread Section 3.6, especially about model selection.
What should we have done instead?
We stated above that hyperparameter optimization by gradient descent is very hard to do. Consider a small problem, such as training a twolayer perceptron on the FashionMNIST dataset (Section 5.2) with a batch size of 256. We would like to tune the learning rate of SGD in order to minimize a validation metric after one epoch of training.
Why cannot we use validation error for this purpose? What metric on the validation set would you use?
Sketch (roughly) the computational graph of the validation metric after training for one epoch. You may assume that initial weights and hyperparameters (such as learning rate) are input nodes to this graph. Hint: Reread about computational graphs in Section 5.3.
Give a rough estimate of the number of floating point values you need to store during a forward pass on this graph. Hint: FashionMNIST has 60000 cases. Assume the required memory is dominated by the activations after each layer, and look up the layer widths in Section 5.2.
Apart from the sheer amount of compute and storage required, what other issues would gradientbased hyperparameter optimization run into? Hint: Reread about vanishing and exploding gradients in Section 5.4.
Advanced: Read (Maclaurin et al., 2015) for an elegant (yet still somewhat unpractical) approach to gradientbased HPO.
Grid search is another HPO baseline, where we define an equispaced grid for each hyperparameter, then iterate over the (combinatorial) Cartesian product in order to suggest configurations.
We stated above that random search can be much more efficient than grid search for HPO on a sizable number of hyperparameters, if the criterion most strongly depends on a small subset of the hyperparameters. Why is this? Hint: Read (Bergstra et al., 2011).