If I present you with five examples of burritos, I don’t want you to pursue the simplest way of classifying burritos versus non-burritos. I want you to come up with a way of classifying the five burritos and none of the non-burritos that covers as little area as possible in the positive examples, while still having enough space around the positive examples that the AI can make a new burrito that’s not molecularly identical to the previous ones.

Consider the problem of designing classifiers that are able to reliably say "I don't know" for inputs well outside of their training set. In the literature, this problem is known as open-category classification [7]; a closely-related problem in AI safety is inductive ambiguity detection [4].

Solution of generalized inductive ambiguity detection could allow for agents who robustly know when to ask for clarification; in this post, we discuss the problem in the context of classification. Although narrower in scope, the solution of the classification subproblem would herald the arrival of robust classifiers which extrapolate conservatively and intuitively from their training data.

Obviously, we can't just train classifiers to "recognize" the unknown class. One, this class isn't compact, and two, it doesn't make sense - it's a map/territory confusion (the label unknown is a feature of the world model, not a meaningful ontological class). Let's decompose the concept a bit further.

Weight regularization helps us find a mathematically-simple volume in the input space which encapsulates the data we have seen so far. In fact, a binary classifier enforces a hyperplane which cleaves the entire space into two volumes in a way which happens to nearly-optimally classify the training data. This hyperplane may be relatively simple to describe mathematically, but the problem is that the two volumes created are far too expansive.

In short, we'd like our machine learning algorithms to learn explanations which fit the facts seen to date, are simple, and don't generalize too far afield. This is an example of conservatism:

Conservatism and conservative planning seems like it might directly tackle some standard concerns [in alignment] head-on and in a sufficiently-basic way to avoid loopholes, and might also be subject to those concerns. E.g.:

• Edge instantiation - if in full generality we don't go to the edge of the graph but try to stay in the center of what's already been positively classified, maybe we can avoid this.

• Unforeseen maximum - if we stick to things very similar to already-positively-classified instances, we won't automatically go into the unimagined parts of the graph.

• Context disaster - a sufficiently conservative optimizer might go on using options similar to previously-whitelisted ones even if large new sections of planning space opened up.

Prior Work

Taylor et al. provide an overview of relevant research [4]. Taylor also introduced an approach for rejecting examples from distributions too far from the training distribution [5].

Yu et al. employed adversarial sample generation to train classifiers to distinguish seen from unseen data, achieving moderate success [7]. Li and Li1 trained a classifier to sniff out adversarial examples by detecting whether the input is from a different distribution [3]. Once detected, the example had a local average filter applied to it to recover the non-adversarial original.

The Knows What It Knows framework allows a learner to avoid making mistakes by deferring judgment to a human some polynomial number of times [2]. However, this framework makes the unrealistically-strong assumption that the data are i.i.d.; furthermore, efficient KWIK algorithms are not known for complex hypothesis classes (such as those explored in neural network-based approaches).

Penalizing Volume

If we want a robust cat / unknown classifier, we should indeed cleave the space in two, but with the vast majority of the space being allocated to unknown. In other words, we're searching for the smallest, simplest volume which encapsulates the cat training data.

The smallest encapsulation of the training data is a strange, contorted volume only encapsulating the training set. However, we still penalize model complexity, so that shouldn't happen. As new examples are added to the training set, the classifier would have to find a way to expand or contract its class volumes appropriately. This is conceptually similar to version space learning.

This may be advantageous compared to current approaches in that we aren't training an inherently-incorrect classifier to prune itself or to sometimes abstain. Instead, we structure the optimization pressure in such a way that conservative generalization is the norm.2

Formalization

Let V be a function returning the proportion3 of input space volume occupied by non-unknown classes: |inputs not classified as unknown||all inputs|. We need some function which translates this to a reasonable penalty (as the proportion alone would be a rounding error in the overall loss function). For now, assume this function is ^V.

We observe a datum x whose ground truth label is y. Given loss function L, weights θ, classifier fθ, complexity penalty function R, and λ1,λ2∈R, the total loss is

L(y,fθ(x))+λ1R(θ)+λ2^V(fθ).

Depending on the formulation of ^V, fθ may elect to misclassify a small amount of data to avoid generalizing too far. If more similar examples are added, L exerts an increasing amount of pressure on fθ, eventually forcing expansion of the class boundary to the data points in question. This optimization pressure seems desirable.

We then try to minimize expected total loss over the true distribution X:

argminθEx,y∼X[L(y,fθ(x))+λ1R(θ)+λ2^V(fθ)].

Tractable Approximations

Exhaustively enumerating the input space to calculate V is intractable - the space of 256×256 RGB images is of size 2563256×256. How do we measure or approximate the volume taken up by the non-unknown classes?

Black Box

Random image sampling wouldn't be very informative (beyond answering "is this class extending indefinitely along arbitrary dimensions?").

Yu et al.'s results weakly suggest that adversarial approaches can approximate the hypersurface of the class boundaries [7].

Bayesian optimization could deduce the hypersurface of the class boundaries, giving a reasonable estimate of volume.

The continuous latent space of a variational autoencoder [1] could afford new approximation approaches for ^V (more on this later).

Analytic

We do know the details of fθ - how can we exploit this?

Decision trees offer a particularly simple solution. Consider a decision tree on a one grayscale pixel input space; the tree outputs unknown if pixel value x>127 and dog otherwise. Then without having to run a single input through the tree, we find the proportion of the space taken up by non-unknown classes to be 1−128256=.5.

This result generalizes to any decision tree over discrete variables; simply calculate in closed form how many possible inputs conform to the conditions for each unknown leaf, and add them up. For an input space S classified by a decision tree with m nodes, we reduce the time complexity from O(|S|) to O(m); this is great, as it is almost always the case that |S|≫m.

Neural networks are a bit trickier. Work has been done on extracting decision trees from neural networks [6], but this requires training a new network to do so and the resulting trees may not have sufficient fidelity.

Since the output of a one hidden-layer neural network can be represented as the linear equation WX=Z, we can, perhaps, simply fix W (θ) and Z (the layer's output values) to solve for X (the input). Given such a solution, we might, in theory, be able to calculate the volume by integrating over the possible Z values for each classification. The universal approximation theorem shows that a one hidden-layer neural network (with perhaps an exponential number of nodes) can approximate any function.

All this is getting a bit silly. After all, what we really care about is the proportion of the input space taken up by non-unknown classes; we don't necessarily need to know exactly which inputs correspond to each output.

White Box

Let's treat this as a prediction problem. Take n randomly-generated images and run them through the first layer of the network, recording the values for the activation of node k in the first layer (a1k); assume we incur some approximation error ϵ due to having insufficiently sampled the true distribution. Derive a PDF over the activation values for each node.

Repeat this for each of the l network layers, sampling input values from the PDFs for the previous layer's nodes. At the end of the network, we have a probabilistic estimate of the output class proportions.

However, this may only yield results comparable to random image sampling, as many node activations may not occur often for arbitrary data. While this would be descriptive of the input space as a whole, it would be unlikely to sample any cat images, for example.

Blessing of Dimensionality

In my opinion, the most promising approach involves abusing the properties of high-dimensional volumes; in particular, the well-known Curse of Dimensionality.

## Introduction

Consider the problem of designing classifiers that are able to reliably say "I don't know" for inputs well outside of their training set. In the literature, this problem is known as

open-category classification[7]; a closely-related problem in AI safety isinductive ambiguity detection[4].Solution of generalized inductive ambiguity detection could allow for agents who robustly know when to ask for clarification; in this post, we discuss the problem in the context of classification. Although narrower in scope, the solution of the classification subproblem would herald the arrival of robust classifiers which extrapolate conservatively and intuitively from their training data.

Obviously, we can't just train classifiers to "recognize" the unknown class. One, this class isn't compact, and two, it doesn't make sense - it's a map/territory confusion (the label unknown is a feature of the world model, not a meaningful ontological class). Let's decompose the concept a bit further.

Weight regularization helps us find a mathematically-simple volume in the input space which encapsulates the data we have seen so far. In fact, a binary classifier enforces a hyperplane which cleaves the

entirespace into two volumes in a way which happens to nearly-optimally classify the training data. This hyperplane may be relatively simple to describe mathematically, but the problem is that the two volumes created are far too expansive.In short, we'd like our machine learning algorithms to learn explanations which fit the facts seen to date, are simple, and don't generalize too far afield. This is an example of

conservatism:## Prior Work

Taylor et al. provide an overview of relevant research [4]. Taylor also introduced an approach for rejecting examples from distributions too far from the training distribution [5].

Yu et al. employed adversarial sample generation to train classifiers to distinguish seen from unseen data, achieving moderate success [7]. Li and Li1 trained a classifier to sniff out adversarial examples by detecting whether the input is from a different distribution [3]. Once detected, the example had a local average filter applied to it to recover the non-adversarial original.

The Knows What It Knows framework allows a learner to avoid making mistakes by deferring judgment to a human some polynomial number of times [2]. However, this framework makes the unrealistically-strong assumption that the data are

i.i.d.; furthermore, efficient KWIK algorithms are not known for complex hypothesis classes (such as those explored in neural network-based approaches).## Penalizing Volume

If we want a robust cat / unknown classifier, we should indeed cleave the space in two, but with the vast majority of the space being allocated to unknown. In other words, we're searching for the smallest, simplest volume which encapsulates the cat training data.

The

smallestencapsulation of the training data is a strange, contorted volume only encapsulating the training set. However, we still penalize model complexity, so that shouldn't happen. As new examples are added to the training set, the classifier would have to find a way to expand or contract its class volumes appropriately. This is conceptually similar to version space learning.This may be advantageous compared to current approaches in that we aren't training an inherently-incorrect classifier to prune itself or to sometimes abstain. Instead, we structure the optimization pressure in such a way that conservative generalization is the norm.2

## Formalization

Let V be a function returning the proportion3 of input space volume occupied by non-unknown classes: |inputs not classified as unknown||all inputs|. We need some function which translates this to a reasonable penalty (as the proportion alone would be a rounding error in the overall loss function). For now, assume this function is ^V.

We observe a datum x whose ground truth label is y. Given loss function L, weights θ, classifier fθ, complexity penalty function R, and λ1,λ2∈R, the total loss is

Depending on the formulation of ^V, fθ may elect to misclassify a small amount of data to avoid generalizing too far. If more similar examples are added, L exerts an increasing amount of pressure on fθ, eventually forcing expansion of the class boundary to the data points in question. This optimization pressure seems desirable.

We then try to minimize expected total loss over the true distribution X:

## Tractable Approximations

Exhaustively enumerating the input space to calculate V is intractable - the space of 256×256 RGB images is of size 2563256×256. How do we measure or approximate the volume taken up by the non-unknown classes?

## Black Box

## Analytic

We do know the details of fθ - how can we exploit this?

Decision treesoffer a particularly simple solution. Consider a decision tree on a one grayscale pixel input space; the tree outputs unknown if pixel value x>127 and dog otherwise. Then without having to run a single input through the tree, we find the proportion of the space taken up by non-unknown classes to be 1−128256=.5.This result generalizes to any decision tree over discrete variables; simply calculate in closed form how many possible inputs conform to the conditions for each unknown leaf, and add them up. For an input space S classified by a decision tree with m nodes, we reduce the time complexity from O(|S|) to O(m); this is great, as it is almost always the case that |S|≫m.

Neural networksare a bit trickier. Work has been done on extracting decision trees from neural networks [6], but this requires training a new network to do so and the resulting trees may not have sufficient fidelity.Since the output of a one hidden-layer neural network can be represented as the linear equation WX=Z, we can, perhaps, simply fix W (θ) and Z (the layer's output values) to solve for X (the input). Given such a solution, we might, in theory, be able to calculate the volume by integrating over the possible Z values for each classification. The universal approximation theorem shows that a one hidden-layer neural network (with perhaps an exponential number of nodes) can approximate any function.

All this is getting a bit silly. After all, what we really care about is the proportion of the input space taken up by non-unknown classes; we don't necessarily need to know exactly

whichinputs correspond to each output.## White Box

Let's treat this as a prediction problem. Take n randomly-generated images and run them through the first layer of the network, recording the values for the activation of node k in the first layer (a1k); assume we incur some approximation error ϵ due to having insufficiently sampled the true distribution. Derive a PDF over the activation values for each node.

Repeat this for each of the l network layers, sampling input values from the PDFs for the previous layer's nodes. At the end of the network, we have a probabilistic estimate of the output class proportions.

However, this may only yield results comparable to random image sampling, as many node activations may not occur often for arbitrary data. While this would be descriptive of the input space as a whole, it would be unlikely to sample any cat images, for example.

## Blessing of Dimensionality

In my opinion, the most promising approach involves abusing the properties of high-dimensional volumes; in particular, the well-known Curse of Dimensionality.

To illustrate, the volume of a ball in