Training on Test Inputs with Amortized Conditional Normalized Maximum Likelihood

Current machine learning methods provide unprecedented accuracy across a range
of domains, from computer vision to natural language processing. However, in
many important high-stakes applications, such as medical diagnosis or
autonomous driving, rare mistakes can be extremely costly, and thus effective
deployment of learned models requires not only high accuracy, but also a way to
measure the certainty in a model’s predictions. Reliable uncertainty
quantification is especially important when faced with out-of-distribution
inputs, as model accuracy tends to degrade heavily on inputs that differ
significantly from those seen during training. In this blog post, we will
discuss how we can get reliable uncertainty estimation with a strategy that
does not simply rely on a learned model to extrapolate to out-of-distribution
inputs, but instead asks: “given my training data, which labels would make
sense for this input?”.

To illustrate how this can allow for more reasonable predictions on
out-of-distribution data, consider the following example where we attempt to
classify automobiles, where all the class 1 training examples are sedans and
class 2 examples are large buses.

Figure 1: Given previously seen examples, it is uncertain what the label for
the new query point should be. Different classifiers that work well on the
training set can give different predictions on the query point.

A classifier could potentially fit the training labels correctly based on
several different explanations; for example, it could notice that buses are all
longer than sedans and classify accordingly, or it could perhaps pay attention
to the height of the vehicle instead. However, if we try to simply extrapolate
to an out-of-distribution image of a limousine, the classifier’s output could
be unpredictable and arbitrary. A classifier based on length could note that
the limousine is similar to the buses in its length and confidently predict
class 2, while a classifier utilizing the height could confidently predict
class 1. Based only on the training set, there is not enough information to
accurately decide which class a limousine should fit into, so we would ideally
want our classifier to indicate uncertainty instead of providing arbitrary
confident predictions for either class. On the other hand, if we explicitly try
to find models that explain each potential label, we would find reasonable
explanations for either label, suggesting that we should be uncertain about
predicting which class the limousine belongs to.

We can instantiate this reasoning with an algorithm that, for every possible
label, explicitly updates the model to try to explain that label for the query
point and combines the different models to obtain well-calibrated predictions
for out-of-distribution inputs. In this blog post, we will motivate and
amortized conditional normalized maximum likelihood
(ACNML), a practical instantiation of this idea that enables reliable
uncertainty estimation with deep neural networks.

Conditional Normalized Maximum Likelihood

Our method extends a prediction strategy from the minimum description length
(MDL) literature known as conditional normalized maximum likelihood (CNML),1
which has been studied for its theoretical properties, but is computationally
intractable for all but the simplest problems. We will first review CNML and
discuss how its predictions can lead to conservative uncertainty estimates. We
will then describe our method, which allows for a practical application of CNML
to obtain uncertainty estimates for deep neural networks.

The CNML distribution is derived from achieving a notion of minimax optimality,
where we define a notion of regret for each label and choose the distribution
that minimizes the worst case regret over labels. Given a training set
$D_{rm train}$, a query input $x$, and a set of potential models $Theta$, we
define the regret for each label to be the difference between the negative
log-likelihood loss for our distribution and the loss under the model that best
fits the training dataset together with the query point and label.

Intuitively, minimizing the worst case regret over labels then ensures our
predictive distribution is conservative, as it cannot assign low probabilities
to any labels that appear consistent with our training data, where consistency
is determined by the model class.

The minimax optimal distribution given a particular input $x$ and training set
$mathcal D$ can be explicitly computed as follows:

  1. For each label $y$, we append $(x,y)$ to our training set and compute the
    new optimal parameters $hat theta_y$ for this modified training set.

  2. Use $hat theta_y$ to assign probability for that label.

  3. Since these probabilities will now sum to a number greater than 1, we
    normalize to obtain a valid distribution over labels.

Daniel: no need for the image, equation above works fine.


CNML has the interesting property that it explicitly optimizes the model to
make predictions on the query input, which can lead to more reasonable
predictions than simply extrapolating using a model obtained only from the
training set. It can also lead to more conservative predictions on
out-of-distribution inputs, since it would be easier to fit different labels
for out-of-distribution points without affecting performance on the training

We illustrate CNML with a 2-dimensional logistic regression example. We compare
heatmaps of CNML probabilities with the maximum likelihood classifier to
illustrate how CNML provides conservative uncertainty estimates on points away
from the data. With this model class, CNML expresses uncertainty and assigns a
uniform distribution to any query point where the dataset remains linearly
separable (meaning there exists a linear decision boundary could correctly
classify all data points) regardless of which label was assigned for the query

Figure 2: Here, we show the heatmap of CNML predictions (left) and the
predictions of the training set MLE $hat theta_{text{train}}$ (right). The
training inputs are shown with blue (class 0) and orange (class 1) dots. Blue
shading indicates that higher probability for class 0 on that input and red
shading indicates higher probability to class 1, with darker colors indicating
more confident predictions. We note that while the original classifier assigns
confident predictions for most inputs, CNML assigns close to uniform for most
points between the two clusters of training points, indicating high uncertainty
about these ambiguous inputs.

We illustrate how CNML computes these probabilities by illustrating the base
classifier predictions under parameters $hat theta_{text{train}}$ (the
training set MLE), as well as $hat theta_{text{0}}$, $hat
theta_{text{1}}$, the parameters computed by CNML after assigning the
labels 0 and 1 respectively to a query point.

We first consider an out-of-distribution query point far away from any of the
training inputs (shown in pink in the bottom of the leftmost image). In the
left image, we see the original decision boundary for $hat
theta_{text{train}}$ confidently classifies the query point as class 0. In
the middle, we see the decision boundary of $hat theta_0$ similarly
classifies the query point as class 0. However, we see in the rightmost image
that $hat theta_1$ is able to confidently classify the query point as class 1.
Since $hat theta_0$ confidently predicts class 0 for the query point and
$hat theta_1$ confidently predicts class 1, CNML normalizes the two
predictions to assign roughly equal probability to either label.

Figure 3: Query point shown in pink. Both $hat theta_{text{train}}$ and
$hat theta_0$ classify the query point as class 0, but $hat theta_1$ is
able to classify it as class 1.

On the other hand, for an in-distribution query point (again shown in pink) in
the middle of the class 0 training inputs, no linear classifier can fit a label
of 1 to the query point while still accurately fitting the rest of the training
data, so the CNML distribution still confidently predicts class 0 on this query

Figure 4: Query point shown in pink. All parameters are forced to classify the
query point as class 0 since it is in the middle of the class 0 training

Controlling Conservativeness via Regularization

We see in Figure 2 that CNML probabilities are uniform on most of the input
space, arguably being too conservative. In this case, the model class is in
some sense too expressive, as linear predictors with large coefficients that
can assign arbitrarily high probabilities to each label as long as the data
remains linearly separable. This problem is exacerbated with even more
expressive model classes like deep neural networks, which can potentially fit
arbitrary labels. In order to have CNML give more useful predictions, we would
need to constrain the allowed set of models to better reflect our notion of
reasonable models.

To accomplish this, we generalize CNML to incorporate regularization via a
prior term, resulting in conditional
normalized maximum a posteriori
(CNMAP) instead. Instead of computing maximum likelihood
parameters for the training dataset and the new input and label, we compute
maximum a posteriori (MAP) solutions instead, with the prior term $p(theta)$
serving as a regularizer to limit the complexity of the selected model.

Daniel Seita: above equation in LaTeX is fine.


Going back to the logistic regression example, we add different levels of L2
regularization to the parameters (corresponding to Gaussian priors) and plot
CNMAP probabilities in Figure 3 below. As regularization increases, CNML
becomes less conservative, with the assigned probabilities transitioning much
more smoothly as one moves away from the training points.

Figure 5: Heatmaps of CNMAP probabilities under varying amounts of
regularization $lambda |w|_2^2$. Increasing regularization leads to less
conservative predictions.

Computational Intractability

While we see that CNML is able to provide conservative predictions for OOD
inputs, computing CNML predictions requires retraining the model using the
entire training set multiple times for each test input, which can be very
computationally expensive. While explicitly computing CNML distributions was
feasible in our logistic regression example with small datasets, it would be
computationally intractable to compute CNML naively with datasets consisting of
thousands of images and using deep convolutional neural networks, as retraining
the model just once could already take many hours. Even initializing from the
solution to the training set and finetuning for several epochs after adding the
query input and label could still take several minutes per input, rendering it
impractical to use with deep neural networks and large datasets.

Amortized Conditional Normalized Maximum Likelihood

Since exactly computing CNML or CNMAP distributions is computationally
infeasible in deep learning settings due to the need to optimize over large
datasets for each new input and label, we need a tractable approximation. In
our method, amortized conditional normalized maximum likelihood (ACNML), we
utilize approximate Bayesian posteriors to capture necessary information about
the training data in order to efficiently compute the MAP/MLE solutions for
each datapoint. ACNML amortizes the costs of repeatedly optimizing over the
training set by first computing an approximate Bayesian posterior, which serves
as a compact approximation to the training losses.

CNMAP and Bayesian Posteriors

We note that the main computational bottleneck is the need to optimize over the
entire training set for each query point. In order to sidestep this issue, we
first show a relationship between the MAP parameters needed in CNMAP and
Bayesian posterior densities:

$$hat theta_y = arg max_theta log p_{theta}(y vert x) + log p_{theta}(D_{text{train}}) + log p(theta)$$

Daniel Seita: above equation in LaTeX is fine.


Rather than computing optimal parameters for the new query point and the
training set, we can reformulate CNMAP as optimizing over just the query point
and a posterior density. With a uniform prior (equivalent to having no
regularizer), we can recover the maximum likelihood parameters to perform CNML
if desired.2

ACNML now utilizes approximate Bayesian inference to replace the exact Bayesian
posterior density with a tractable density $q(theta)$. As many methods have
been proposed for approximate Bayesian inference in deep learning, we can
simply utilize any approximate posterior that provides tractable densities for
ACNML, though we focus on Gaussian approximate posteriors for simplicity and
computational efficiency. After computing the approximate posterior once during
training, the test-time optimization procedure becomes much simpler, as we only
need to optimize over our approximate posterior instead of the training set.
When we instantiate ACNML and initialize from the MAP solution, we find that it
typically takes only a handful of gradient updates to compute new (approximate)
optimal parameters for each label, resulting in much faster test-time inference
than a naive CNML instantiation that fine tunes using the whole training set.

In our paper, we analyze the
approximation error incurred by using a particular Gaussian posterior in place
of the exact training data likelihoods, and show that under certain
assumptions, the approximation is accurate when the training set is large.


We instantiate ACNML with two different Gaussian posterior approximations,
SWAG-Diagonal and
and train models on the CIFAR-10 image classification dataset. To evaluate
out-of-distribution performance, we then evaluate on the CIFAR-10 Corrupted
datasets, which apply a diverse range of common image corruptions at different
intensities, allowing us to see how well methods perform under varying levels
of distribution shift. We compare against methods using Bayesian
marginalization, which average predictions across different models sampled from
the approximate posterior. We note that all methods provide very similar
accuracy both in-distribution and out-of-distribution, so we focus on comparing
uncertainty estimates.

Figure 7: Reliability Diagrams comparing ACNML against the corresponding
Bayesian model averaging method (SWAGD) and the MAP solution (SWA). ACNML
generally predicts with lower confidence than other methods, leading to
comparatively better uncertainty estimation as the data becomes more

We first examine ACNML’s predictions using reliability diagrams, which
aggregate the test data points into buckets based on how confident the model’s
predictions are, then plot the average confidence in a bucket against the
actual accuracy of the predictions. These diagrams show the distribution of
predicted confidences and can capture how effectively a model’s confidence
reflects the actual uncertainty over the prediction.

As we would expect from our earlier discussion about CNML, we find that ACNML
reliably gives more conservative (less confident) predictions than other
methods, to the point where its predictions are actually underconfident on
the in-distribution CIFAR10 test set where all methods provide very accurate
predictions. However, on the out-of-distribution CIFAR10-C tasks where
classifier accuracy degrades, ACNML’s conservativeness provides much more
reliable confidence estimates, while other methods tend to be severely

Figure 8: ECE comparisons: We compare instantiations of ACNML with two
different approximate posteriors against their Bayesian counterparts.

We quantitatively measure calibration using the
Expected Calibration Error,
which uses the same buckets as the reliability diagrams and
computes the average calibration error (absolute difference between average
confidence and accuracy within the bucket) over all buckets. We see that ACNML
instantiations provide much better calibration than their Bayesian counterparts
and the deterministic baseline as the corruption intensities increase and the
data becomes more out-of-distribution.


In this post, we discussed how we can obtain reliable uncertainty estimates on
out-of-distribution data by explicitly optimizing on the data we wish to make
predictions on instead of relying on trained models to extrapolate from the
training data. We then showed that this can be done concretely with the CNML
prediction strategy, a scheme that has been studied theoretically but is
computationally intractable to apply in practice. Finally we presented our
method, ACNML, a practical approximation to CNML that enables reliable
uncertainty estimation with deep neural networks. We hope that this line of
work will help enable broader applicability of large scale machine learning
systems, especially in safety-critical domains where uncertainty estimation is
a necessity.

We thank Sergey Levine and Dibya Ghosh for providing valuable feedback on this post.

This post is based on this following paper:

  1. For additional background on MDL and CNML, we refer to the following
    The particular variant of CNML used here is referred to as CNML-3 in the text, and as sNML-1 in
    Roos et al

  2. Despite referring to maximum likelihood in the name, ACNML actually
    approximates the CNMAP procedure, with CNML being a special case of CNMAP
    by using a uniform prior as regularizer. In practice, due to the
    over-conservativeness of CNML mentioned before, we will use non-uniform
    priors (as used by the Bayesian deep learning methods we build off of). 

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