Functional Indirection Neural Estimator for Better Out-of-distribution Generalization
Authors: Kha Pham
Hung Le
Man Ngo
Truyen Tran
Venue: NeurIPS 2022 Accept (2022)
Keywords: functional space
analogy-making
indirection
memory
out-of-distribution generalization
IQ task

Abstract: The capacity to achieve out-of-distribution (OOD) generalization is a hallmark of human intelligence and yet remains out of reach for machines. This remarkable capability has been attributed to our abilities to make conceptual abstraction and analogy, and to a mechanism known as indirection, which binds two representations and uses one representation to refer to the other. Inspired by these mechanisms, we hypothesize that OOD generalization may be achieved by performing analogy-making and indirection in the functional space instead of the data space as in current methods. To realize this, we design FINE (Functional Indirection Neural Estimator), a neural framework that learns to compose functions that map data input to output on-the-fly. FINE consists of a backbone network and a trainable semantic memory of basis weight matrices. Upon seeing a new input-output data pair, FINE dynamically constructs the backbone weights by mixing the basis weights. The mixing coefficients are indirectly computed through querying a separate corresponding semantic memory using the data pair. We demonstrate empirically that FINE can strongly improve out-of-distribution generalization on IQ tasks that involve geometric transformations. In particular, we train FINE and competing models on IQ tasks using images from the MNIST, Omniglot and CIFAR100 datasets and test on tasks with unseen image classes from one or different datasets and unseen transformation rules. FINE not only achieves the best performance on all tasks but also is able to adapt to small-scale data scenarios.

Functional Indirection Neural Estimator for Better Out-of-distribution Generalization
Kha Pham'
Hung Le' Man Ngo² Truyen Tran'
Applied Artificial Intelligence Institute, Deakin University
2 Faculty of Mathematics and Computer Science, VNUHCM-University of Science
[phti, thai.le, truyen. trankideeainn edu.au nmman@hcmus. www.seuuuuuuuu............eeeeee vn
Abstract
The capacity to achieve -utttf-distriuution (OOD) generalization is hallmark human intelligence and yet remains out of reach for machines. This remarkable capability has been attributed to our abilities to make conceptual abstraction and analogy, and to mechanism known indirection, which binds two representations and uses one representation to refer to the other. Inspired by these mechanisms, hypothesize that OOD generalization may be achieved by performing analogy- making and indirection in the functional space instead of the data space as current methods. To realize this, we design FINE (Functional Indirection Neural Estimator), neural framework that learns to compose functions that map data input to output on-the-fly. FINE consists of backbone network and trainable semantic memory of basis weight matrices. Upon seeing a new input-output data pair, FINE dynamically constructs the backbone weights by mixing the basis weights. The mixing coefficients are indirectly computed through querying a separate corresponding semantic memory using the data pair. We demonstrate empirically that FINE can strongly improve out-of-distrion -iitt--distriuttion generalization on particular, we train FINE
IQ tasks that involve geometric transformations. and competing models on IQ tasks using images from the MNIST, Omniglot and CIFARIOM 100 datasets and test on tasks with unseen image classes from one or different datasets and unseen transformation rules. FINE not only achieves the best performance on all tasks but also is able to adapt to small-scale data scenarios.
Introduction
Every computer science problem can be solved with a higher level of indirection.
-Andrew Koenig, Butler Lampson, David J. Wheeler
Generalizing cireumstances is hallmark of intelligence [16, 4, 11]. In some Intelligence Quotient (IQ) tests-a popular benchmark for human intelligence-one must leverage their prior experience to identify the hidden abstract rules out of concrete example (e.g., transformation of an image) and then apply the rules to the next (e.g., new set of images of totally different appearance). These tasks necessitate several key capabilities, including conceptual abstraction and analogy-mane making [22]. Abstraction allows us to extend a concept to novel situations. is also driven by analogy making, which maps the current situation to the previous experience stored in the memory. Indeed, analogy-making has been argued to be one of the most important abilities of human cognition, further, "a concept is package of analogies"
[12]. The ability for humans seamlessly across concrete and abstraction levels suggests another mechanism known as indirection bind two representations and use one representation refer the other [16, 20].
traverse
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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gecmetric transformation
Q
encoder
encoder
backbone layer 1
backbone
layer
The function composer
function composition
S er ABogeu
indirection
y-x
MEMORY
Figure FINE architecture. Above: FINE uses pre-defined deep backbone architecture to approximate a function mapping given input embedding to given output embedding Below: given the imput xy and pscudo-output ye of the 7th backbone layer, FINE first computes the query W which represents the relation between 2; and y/- Then FINE performs analogy-makinn query with past experiences in the form of value memories. Finally, FINE binds the value memories with associated key memories via indirection and computes the weight W. for the tt backbone layer.
to compare the
Several deep learning models have successfully utilized analogy and indirection. The Transformer [30] and RelationNet [28] learn analogies between data, through self-attention pairwise functions. The ESBN [33] goes further by incorporating the indirection mechanism to bind an entity to symbol and then reason the symbols; and this has proved be efficient tasks involving abstract rules, similar to those aforementioned IQ tests. However, a common drawback of these approaches is that they operate the data space, and thus susceptible
-ittr--distribution samples.
In this paper, we propose to perform maalogy-makingg and indirection in functional spaces instead. We aim to learn to compose functional mapping from a given input to an output on-the-fly. By doing so, we achieve two clear advantages. First, since the class of possible mappings is often restricted, it may not require large amount of training data to learn the distribution of functions. Second, more importantly, since this approach performs indirection in functional spaces, avoids bindings between numerous entities and symbols in data spaces, thus may help improve the disttiibittttionn generalization capability.
To this end, we introduce new class of problems that requires functional analogy-making and indirection, which are deemed be challenging for current data-oriented approaches. The tasks are similar to popular IQ tasks in which the model is given hints about the hidden rules, then it has to predict the missing answer following the rules. One reasonable approach is that models should be able to compare the current task to what they saw previously to identify the rules between appearing entities, and thus has to search on functional spaces instead of data spaces. More concretely, we construct the IQ tasks by applying geometric transformations to images from MNIST dataset, hand- written Omniglot dataset and real- real-image CIFARION 100 dataset, where the training set and test set contain disjoint image classes from the same different datasets, and possibly disjoint transformation rules
Second, we present novel framework named Functional Indirection Neural Estimator (FINE) to solve this class of problems (see Fig. 1 for the overall architecture of FINE). FINE consists of (a) neural backbone to approximate the functions and (b) a trainable key-value memory module that stores the basis of the network weights that spans the space of possible functions defined by the backbone. The weight basis memories allow FINE to perform amalogy-making and indirection in function space. More concretely, when a new IQ task arrives, FINE first (1) takes the hint images to make analogies with value memories, then (2) performs indirection to bind value memories with key memories and finally (3) computes the approximated functions based on key memories. Throughout

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Rotation
Black White
S
B
?
+ D
B
Figure 2: Examples of two IQ tasks involving geometric transformations. Choices with green circles the correct solutions. Left. 90-degree rotation. Right. syntactic black-white transformation: a part of the image is transformed to the opposite colors.
a comprehensive suite of experiments, we demonstrate that FINE outperforms the competing methods and adapts more effectively in small data scenarios.
Tasks
For concreteness, will focus Intelligence Quotient (IQ) tests, which have been widely accepted as one of the reliable benchmarks to measure human intelligence [26]. We will study a popular class of IQ tasks that provides hints following some hidden rules and requires the player to choose among given choices to fill in a placeholder so that the filled-in entity obeys the rules of the currett task. In order succeed in these tasks, the player must be able figure the hidden rules and perform amalogy-making to select the correct choice. Moreover, once figuring out the rules for the current task, a human player can almost always solve tasks with similar rules regardless of the appearing entities given in the tasks. This remarkable ability of cuu-oo-distrrbutie humans treat objects and relations abstractly instead of relying on the raw sensory data.
generalization indicates that
We aim to solve IQ tasks that involve geometric transformations (e.g., see Fig. for an example), which include affine transformations (translation, rotation, shear, scale, and reflection), non-linear transformations (fisheye, horizontal wave) and syntactic transformations (black-white, swap). Details of transformations are given in Supplementary. In task, the models are given 3 images x, y and v', where is the result of æ after applying a geometric transformation. The models are then asked to select y' among 4 choices y1, y2, y3: Y1 so that (x', ./) follows the same rule (x, y) (i.e. if = f()) then y' = f(x') for transformation f). The 4 choices include (i) one with correct object/image and correct transformation (which the solution), (ii) one with correct object/image and incorrect transformation, (iii) one with incorrect object/image and correct transformation, and (iv) one with both incorrect object/image and transformation.
Inspired by human capability, a reasonable approach to solve these tasks is that models should be able figure out the transformation (or relation) between objects/images and apply the transformation to novel objects/images. The datasets can be classified into two main categories: single-rrnsformatiin datasets and multi-transformation datasets. Single-transtee trangooatansooonition datasets a particular transformation, e.g. rotation. Note that the individual transformations of the same type vary, e.g., rotations by different angles. Multi-transs cultiitraasfoommtiion datasets, on the other hand, consist of several transformation types. To test the generalization capability of the models, we build testing sets including classes of images that have never been seen during training (see Section 4. 1 and Section 4. 4.2), or even more challenging tasks including unseen rules and unseen datasets (see Section 4.3). Models must be able leverage knowledge and memory gained from the training dataset solve new task.
that only include
Method
3.1 Functional Hypothesis
Let X' and ) be the data input and output spaces, respectively. Denote by (Numin Prain) C (X'))) the training set, and (Zest Mes) C (X, ()) the non-overlapping test set. Classical ML assumes that Xmain and Yes are drawn from the same distribution. Under this hypothesis, it is reasonable (for frequentists) to find function f: X - y the functional space F that fits both the train and the test sets (i.c., the discrepancy between fzz and corresponding y is small for all (æ, y) in (Xuain Vanin)
3

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and (Rist Vea)). However, when Keest is drawn from a different data distribution from that of Xuain- has been widely reported that current deep learning models fail drastically [2, 8. 13, 32, 33). This is because the models are inferred exclusively from the observed data distribution. Moreover, it could be the case that relations between x and y testing samples are unseen during training, which raises questions on the feasibility of learning single function when dealing with out-of- A natural solution for this problem would be to train the model to learn the functions adaptively. Formally speaking, the model will learn a function composer Ø: X' x 3 + F that maps each pair of (x E X x y (where y is the associated output of x) to a function dx.y in F so that &n,y approximates the true relation between I and y. As discussed, training models this way leads two clear advantages: (1) it can help to handle the cases when there are multiple (and possibly disjoint) relations between inputs and outputs within the training and testing datasets; and (2) models are less dependent on data and thus can achieve more stable results on different training and testing sets. Empirical evidence for these points will be given in Section 4.
tasks.
Since there are an infinite number of functions that can map an input to an output with any given degree of precision, the key is to design & so that it can map given input-output pair to "good enough" function. We humans may draw analogies between the current situation and our experiences and then work out the most suitable options [12]. For example, we can recognize math problem in exam to be similar to a previous exercise in class with different variable names. We may find the presented idea in paper related to that in another paper read before, someone even says "new ideas are just re-distribution of old ideas". All these examples illustrate that analogy equip the function composer 6 with a semantic memory to store past knowledge, on which analogy-making is performed. The memory also plays the role of constraining the searching region for Φ that only looks for functions in the subspaces spanned by the memories instead of the whole functional space F. Further analysis is given in Supplementary.
powerful strategy in human thinking process. Inspired by this mechanism, analogy-making making is a
remaining question is how to design the memory. Let us be inspired again by human thinking process: When we see an animal, we compare its face, legs or tail with things we know and finally conclude it's a "dog". Here "dog" is an abstract concept bound with primary characteristics (e.g. face, legs, tail, etc.); once new entity arrives, compare its properties with these characteristies, and if they are similar, we utilize the indirection mechanism to infer it is indeed the concept we are consid- ering. In our case, the concepts functions, more specifically, the geometric transformations. We thus maintain a key-value memory structure, in which the keys represent abstract concepts and values the associated characteristics of the concepts. A new input-output pair matches with some values, and the indirection enables us to compute the functions based on corresponding keys.
Note that we have swapped the role of keys and values of the traditional memory (where the query is matched against the key, not value). This is to emphasize that we perform indirection to map concrete functional values to abstract functional keys.
3.2 Functional Indirection Neural Estimator
In this section, we present the Functional Indirection Neural Estimator (FINE), an neural architecture to realize the general idea laid out in Section 3.1. FINE learns to compose function mapping an input embedding x to an output embedding y on-the-fly. The function is drawn from a parametric family specified by a backbone neural network. Coupled with the backbone is a function composer o, which is trained to compute the parameters of the backbone. More specifically, o maps a data input-output pair to the weights of the neural networks.
FINE solves the proposed IQ tasks as follows: (1) Given two images 2; from the hint, FINE uses o to produce the function transforming & to y; denoted by Φr.y. (2) Then, we feed the third image x' to Φz.g and get output y" = Φ (x.), ). (3) We define a similarity metric to compare y" with given choices. The choice with the closest distance from y² is model's answer. The function composer and the similarity metric are specified as follows.
Images are encoded by a trainable encoder (see Fig. 1). To effectively solve IQ tasks introduced in Section 2, we use the p4-CNN [7] encoder to serve as an inductive bias for geometric transformations. Throughout this paper, we refer to the images by their embeddings.
Encoder:

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The functional memory
Let us focus a weight matrix We E R4"x4" at the f-th layer of the backbone network. In practice, We belongs the huge di x di di -dimensional real matrix space May.qu. To reduce the complexity of W, we assume that W, only belongs to a s-dimensional subspace of Maa where 8 < d'dd This subspace has a basis of & matrices which are trainable and stored in FINE'S memory, and W will be written
linear combination of these matrices.
Denote by Mt the memory for the f-th backbone layer. M₁ includes two sub-memories: the key mem- ory M = {M M and the corresponding value memory Mq = {M ...., My }. Elements of M and Mq are trainable matrices of size dfth x ditt, We further let x E Rd and Vi E "R be the associated input and output, respectively, where 2; is output of the (t - 1 layer and Yt is the pseudo-output computed by trainable 1-layer neural network yt (v))
With this design, we control the complexity of functional hypothesis space by either constraining the form of the backbone or the capacity of the functional memory.
Memory reading
By the virtue of simplicity, we aim find simple query that demonstrate the relation between Iy and yt- Although the exact relation may be non-linear, we found that query induced from linear relation is enough efficiently read from memory. Formally, we want to find query W such that W72,, y/- The best-approx solution is W y where 2 is the pseudo-inverse of z and can be efficiently approximated by the iterative Ben-Israel and Cohen algorithm [3]. This way of query computing requires no parameter opposed to other methods, where the input is often fed into a trainable neural network to compute the query.
With the query in hand, the next step is to perform analogy -mkiing. In FINE, the query W represents for the current situation and the value memory My consists of past experiences. The concrete query interacts with value memories to measure how close the current situation is to each of the experiences. The similarities are computed as dot products between the query and value memories and normalized by a factor of / di dqu:
0 fconcat (aatattMallaa Myalue) flatten (W?)))
v /dpac
where the flatten operator flattens the matrix W of size di x d: into vector of size d.u, while the fconcat operator first flattens all matrices in Ma then concatenates them together to form the value matrix of size (dt d(t) X 8. The resulting d, is s-dimensional vector measuring the similarities between the query and the entries in the value memory. Here we omit the softmax operator as in usual attention to allow the similarities with more freedom. The same idea is shared in the ESBN [33], where the softmax similaritics are scaled by a sigmoidal factor.
Finally, the value memories are bound with their associated key memories via indirection. This can be understood as moving forward from the concrete space of data and value memories to the abstract space of functions and key memories. With the key memories and the similarity vector the weight W, of current backbone layer can be computed as the linear combination of key memories:
W = reshape (fconcat( M.
(2)
where the reshape operator reshapes the vector of size d/mdqtm to a matrix of size di x diu. Since the softmax omitted when calculating the similarities. W, not constrained be in the convex hull of the key memories and indeed can lie anywhere in the subspace spanned by those key memories.
Memory update
The key and value memories are updated using gradient descent:
M,
key/valuc
-M
keyivaluc
aL AM key/valu
=i 1,
(3)
where À > 0 is the learning rate and L is the loss of the training step.
5

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The similarity metric
After determining &r.y. the model is given new input x' and being asked to select the correct associated output y among 4 choices Vi V/, V y/4 so that (a', follows the same transformation rule as (x, v). This problem can be cast as finding the choice that is the most similar with y* &n (z'). We consider weighted Euclidean metric that measures the distance between two vectors u, E IRd
nu v)
Σ aus u;)²
where {a 2 0 are trainable scalars, i.e., each component of u and e contributes with different importance. Finally, the probability to pick choice is computed as:
exp(-n v))
p (3// x²)
for i= 1,2, 2,
exp( - n1//. Σ j=1
4 Experiments
We conduct experiments to show the out-of-distribution generalization capability of FINE when performing tasks introduced in Section 2. For non-invertible transformations (c.g. reflection), a simple 2-layer MLP as the backbone. For invertible transformations, we use the NICE architecture inductive bias for invertibility. Since in each NICE layer, only half of
[9] backbone the input is transformed, we use the same memory for two consecutive layers, i.e., M2 = M2 To balance between the backbone complexity and computational cost, use 4 NICE layers in all experiments.
We compare FINE with three major classes of models: (a) models that make analogies in the data space, including Transformer [30], PrediNet [29] and RelationNet [28]; (b) models that leverage indirection to bind feature vectors with associated symbols and reason the symbols, including the ESBN [33]; and (c) models that aim to learn a mapping from data to the functional space, including the HyperNetworks [14]. For HyperNetworks,
still use the NICE backbone and just apply their fast-weight generation method for fair comparisons. Except for FINE and HyperNetworks, all models trained with context normalization [32], which has been proved to be effective in improving the generalization ability.
Datasets & implementation: We generate data for IQ tasks described in Section 2 using images from Omniglot dataset [5], which includes 1,623 handwritten characters, and real-image CIFAR 100 dataset [17]. If not specified, models are trained with p4-CNN encoder [7]. Experiments are conducted using PyTorch on a single GPU with Adam optimizer. Reported results are averages of 10 runs.
4.1 Results on Omniglot Dataset
Single-transformnn
Multi
Affine
Non- linear
Syntactic
afline
Trans.
Rot.
Refl.
Shear
Scale
Fish.
H. Wave
B&W
Swap
RelationNet
26.2
25.5
27
27.5
30.2
49.7
26.0
25.3
PrediNet
68.5
43.9
32.9
62.4
65.7
36.2
46.1
60.5
57.5
34.9
HyperNet
88.9
62.0
94.0
74.5
81.8
63.2
80.4
88.6
90.1
54.0
Transformer
89.5
64.8
44.3
86.3
84.0
41.4
91.0
97.6
49.9
59.4
ESBN
79.8
58.6
50.1
84.5
83.4
67.1
86.4
90.5
71.6
63.1
FINE-MLP
96.1
73.9
95.1
84.5
86.0
70.4
84.8
94.9
87.5
72.0
FINE TICE
94.3
77.6
57.7
87.2
86.6
78.5
95.9
98.4
96.2
69.1
Table
Test accuracy (%) on Omniglot dataset.
We use 100 characters for training and other 800 characters for testing. The train and test set size 10,000 and 20,000, respectively. For FINE, use 4 NICE layers with 48 memories for each pair

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of NICE layers. Experimental results for single-trane transffrtonoonono tasks are shown in Table 1. Overall, FINE dominates others with large margins. For example, the gap the rumner-up on the Rotation task is nearly 13%. With test accuracy over 75% on all tasks, FINE shows a strong capability of out-of-distribution generalization.
We further conduct multi-affine- moult--fffie--trasfoommm transformation experiments. In this case, the training set includes multiple types of affine transformations, while other settings similar to the tinlletransformation case. Results are also reported in Table 1. FINE continues to outperform other models. This is because only FINE explicitly assumes the existence of multiple good functions that represent the transformation from input to output data. We note that although HyperNet also makes similar assumption by generating data-specific weights, does not utilize analogy-making and indirection and thus, fails to generalize to unseen images.
4.2
Results on CIFAR100 Dataset
Affine
Non-linear
Syntactic
Trans.
Rot.
Refl.
Shear
Scale
Fish.
H. Wave
B& W
Swap
RelationNet
59.9
49.6
29.9
45.3
66.2
28.7
39.5
26.2
29.7
PrediNet
72.4
65.6
40.6
74.3
76.1
37.1
53.9
32.7
39.6
HyperNet
94.8
86.8
46.6
91.3
85.2
46.8
80.5
47.8
46.0
Transformer
98.4
86.3
47.5
95.4
84.9
47.2
95.1
51.6
47.6
ESBN
96.6
81.9
50.6
90.1
81.5
57.7
95.7
68.8
50.5
FINE-MLP
98.9
89.7
80.6
95.7
86.8
59.6
95.2
83.1
50.8
FINE-NICE
99.2
91.3
51.1
95.6
87.0
76.8
98.3
89.1
51.6
Table 2: Test accuracy (%) on CIFARI00 dataset of single-transfee tragel-tranfoomatiin tasks. For readability
report only the mean values here. Full table reported in Supplementary.
RelationNet
PrediNet
Transformer
ESBN
FINE
FINE
-MLP 79.6+0..5 7.5 7.19
-NICE
Group CNN
32.5 1.1
46 + 1.0
67.8
0.5
8.5 1.6
3-layer ResNet MLP
31.1 + 2.7
2.5
6.7
59.3
1.4
5..7
70.0
Table 3: Test accuracy (%) on CIFAR100 dataset of mult-affinnetransffrraatoo tasks.
For CIFAR100 dataset, we follow similar settings as in experiments on the Omniglot dataset, except that use 50 classes for training and 50 remaining classes for testing. We also conduct experiments and multi-affine- transformation tasks. Results for single-transformatte tasks reported in Table 2. Again, FINE outperforms all other models all tasks, especially Reflection where the gap is nearly 30%. Although FINE does not show good performance on Swap task in Omniglot experiments, it is still slightly better than other models. On the remaining tasks, FINE achieves test accuracy of more than 80%.
on
For the with different encoder architectures, including the p4-CNN, 3-layer ResNet and 2-layer MLP, in Table 3. The results show two superior characteristics of FINE: first, FINE is consistently better than other models across different encoders; second, FINE is more stable with small standard deviations. This empirical result supports the functional hypothesis stated in Section 3.1, where suggest that focusing on functions distribution instead of data distribution can boost model's generalization capability and stability.
task, report the performances of the models when trained
4.3 More Extreme OOD Tasks
Previous tasks only include unseen classes of objects during testing. In this section, we further test FINE and related models on more challenging OOD tasks: tasks with unseen rules during training and even ones with images from unseen datasets. The training and testing sets are either images from CIFAR100, Omniglot or MNIST datasets, while the hidden rules are either translation, rotation or shear. For translation, training problems consist of translation vectors (a, b) with [a), [b] s 3.

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Train set
CIFAR 100
CIFARTO0
Omniglot MNIST
Test set
CIFAR 100
Omniglot
Trans.
Rot.
Shear
Trans.
Rot.
Shear
Trans.
Rot.
Shear
RelationNet
25.4
26.5
26.4
25
24.9
25
24.8
25
25.2
PrediNet
25.5
39.4
36.2
26.2
26.1
26.3
23.8
26.4
25.8
HyperNet
31.6
67.2
58.6
22.2
26.4
22.9
22.7
29
28.2
Transformer
30.9
64.2
55
30
31.7
33.8
28.4
27
25.1
ESBN
16.4
81.3
42.8
15.4
39.2
33.3
17.7
41.3
36.6
FINE
62
85.6
77.8
22.1
43.2
39.2
39.4
44.7
37.9
Table 4: Test accuracy (%) on more extreme OOD tasks with unseen objects, unseen rules and
(possibly) unseen datasets. Translation: train with translation vectors (a, b) with |al, [b] < 3. test
with either |a| > 3 or [6] > 3. Rotation: train with rotation angles a I 1800, test with a > 180%.
Shear: train with shear angles (a, 33 with |al, |8] < 30% test with either |a| >> 30 or [3] >> 30"
and models are tested with either la| > 3 or b] > 3; for rotation, model are trained with angles S 180 and tested with > 180%; for shear, training angles (a, .9) are ones with |a], [8] S 30" while testing ones are either |ax] > 30 or [3] > 30". All tasks have 5,000 data points for training and 10,000 for testing. We report results of FINE with NICE backbone and related models in Table 4. As expected, FINE continues to outperform other models on most of the tasks, even on extreme OOD tasks with unseen datasets and unseen rules where performances of all models drop significantly. This demonstrates FINE is capable of effectively learning the basis weights, which are stored in the memory, to represent novel rules.
4.4 Model Analysis and Ablation Study
Clustering on functional spaces
translation
rotation
shear
16.
66,
045, 63)
(40065)
225 270
(45
(5.45)
182
6-50-60)
1.25
ventiaal
0.5
0.75
shwar
rotution
trans ont
scale refeecioon
scale
reflection
Figure 3: Clusters on functional space. We project weights produced by the function composer ø to see whether Φx locates closely to Øz2.02 on the functional space if the relation between .1 and y1 similar that between 22 and y2. Except for Reflection, the weights are nicely clustered.
We study how the transformations are distributed in the functional space. We use the FINE model trained on multi-affine transformation tasks. For an input-output pair (z, v). we flatten and concatenate all weights of NICE layers to form vector representing 2a.ay. We then use the UMAP [21] to project &z,y's 's vectors onto the 2D plane. Results are shown in Fig. 3. It is interesting to see that shears with the same horizontal clusters, one for smaller scale and one for bigger scale. In contrast, reflection representations seem not to be clustered properly, which is worth further investigation in future work.
vertical angles positioned closely; and scales separated into 2 "big"
Number of memories and backbone layers
We do an ablation study to see the effect of number of memories and backbone layers in FINE, For the limit case with 0 memory, assign the query matrices to be the weights for NICE layers without
8

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the saalogy-making and indirection process. For the limit case with O NICE layer, we replace the NICE backbone by 2-layer MLP.
Results are shown in Fig. 4(a). Overall, we can observe clear improvements when we increase the number of memories number of NICE layers. More interestingly, the more numher of memories is, the more stable the results will be. Increasing the number of memories is equivalent to enlarging the range of c, and increasing the number of NICE layers equivalent enlarging the hypothesis space F. Enlarging F may help FINE approach the true functions while still being sufficiently constrained by the number of memories, thus still being able to maintain its generalization capability.
Test accurary (9)
80
80-
EZAN
Transformer
70-
50
60-
O NICE layers
2 NICE layers
Test
20
NICE layers
0
32
1,000
10,000
100,000
Memories (a)
Training points
(b)
Figure 4: (a) Performance of FINE with different number of memories and backbone layers. Overall, the test accuracy increases with more memories and backbone layers. (b) Performance with different number of training points. FINE works fairly well on small datasets while others may fail significantly.
Number of training data points
We train models are reported in Fig. 4(b). FINE can adapt with small datasets of sizes 100 or 1000 and achieve fair accuracy (47.8% and 64.4%, respectively). Moreover, FINE achieves average test accuracy of 81.1 % on training set of size 10,000, which is quite close to the 85.3% accuracy on 100,000 data points training sets. This shows FINE may obtain near-optimal solution even with small number of training data points. In contrast, ESBN and Transformer need 100,000 training data points to achieve 80% or higher test accuracy, while only achieving roughly 70% test accuracy on smaller datasets.
transformation task with training sets of different sizes. Results
5 Related work
In recent years, there has been strong interest in designing models that capable of generalizing systematically, The Module Networks [1], which dynamically compose neural networks out of train- able modules, have been shown to possess some degree of systematic generalization [2]. Parascandolo et al. [24] proposed a method to train multiple competing experts to explain image transformations on MNIST and Omniglot datasets, yet the transformations are simpler than ones in our FINE dataset and it is not clear whether the proposed method can deal with unseen transformations. The Neural Interpreter [25] uses attention mechanism to recompose functional modules for each input-output pair and test their method on abstract reasoning tasks, yet tends to require a large amount of data to learn. Switch Transformer [10] mitigates the communication and computational cost in mixture of experts models. Recently, ESBN [33], which utilizes the indirection mechanism, shows a great promise on OOD tasks. Both methods achieve their degree of systematic generalization by injecting symbolic biases into the models. Our model FINE follows the same strategy, but it performs analogy-making and indirection functional spaces instead of data spaces, and this has proved to boost both the performance and stability.
IQ tests are powerful testbeds for visual and abstract reasoning. Inspired by Raven's Progressive Matrices (RPM), the RAVEN dataset [34] was proposed as a testbed for visual reasoning models. However, this dataset does not focus on testing the ability of OOD generalization. Webb et al. [33] propose a series of IQ tasks with Unicode characters to show the effectiveness of indirection in tasks involving abstract rules, however these tasks are relatively simple since they only require the models to understand the diffedifferent relation. The ARC dataset [6] aims to serve as benchmark for general intelligence and includes various psychometric IQ tasks in the form of grid structures. this paper, we propose IQ tasks involving geometric transformations as introduced in Section 2. These tasks are not only flexible so that we can include images from different datasets or create/combine numerous transformations, but also challenging to test OOD generalization abilities of models.

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The weight composition feature of FINE links back to the concept of fast weights [19]. the idea of computing data-specific network weights on-the-fly. HyperNetworks [14] stylizes this idea by computing the fast weights using a separate trainable (slow weight) network. The Meta-learned Neural Memory [23] uses the pseudo-target technique (which we also leverage in FINE) and appropriately Memory (NSM) [18] proposes a hybrid approach between slow-weight and fast-weight to compute network weights on-the-fly based on slow-weight key and value memories. However, NSM only performs sequential learning tasks, while FINE aims to solve OOD IQ tasks requiring abstract cognition. Moreover, FINE computes the query based on the input and the (pseudo-) output, while NSM's query is computed based on the input only. Memories have been to be versatile in meta- meta-learning and few-shot learning [15, 27, 31] due to the ability to rapidly store past examples and adapt to new situations. In case, an IQ task can be thought of as a one-shot learning scenario in which FINE has to make use of the long-term key-value memory to adapt to the current task.
updates the short-term memory once new input arrives. The Neural
Limitations
Operating in functional spaces requires FINE's memory to store several trainable weight matrices to infer the data-specific weights for the backbone. Moreover, each layer of the backbone requires its own memory, thus FINE may need large number of parameters for very deep backbones. This could be addressed by parameter-sharing across layers and limiting the rank of the weight matrices.
It remains to design the backbone architecture optimally. We have used the NICE architecture as backbone for invertible transformations and a 2-layer MIP backbone for non-invertible cases. We further investigated our model's performances with larger number of training points (up to IM) and observed that FINE with the NICE backbone peaks at 100,000 training points with around 85% test accuracy, while FINE with MLP backbone continues to improve and achieves around 95% test accuracy at IM training points. This calls for further theoretical analysis to guide the architectural design of the backbone network.
Finally, testing FINE on IQ tasks in the visual space may limit its potential on IQ tasks involving other modalities. For example, one may consider textual IQ problem: "if abc - abd, then mmnnnp -+"". is worth to emphasize that the concepts of analogy-making and indirection in functional spaces are indeed general and thus the idea of FINE should be applicable to various scenarios.
7
Conclusion
To study the out-of distriibitiiiooinn (OOD) generalization capability of models, we have proposed IQ tasks that require models to rapidly recognize the hidden rules of geometric transformations between a pair of images and transfer the rules to new pair of different image classes. Such tasks would necessitate human- like abilities for conceptual abstraction, analogy-making and utilizing indirection. We put forward a hypothesis that these mechanisms should be performed in the functional space instead of data space as in current deep learning models. To realize the hypothesis, we then proposed FINE, cemmmrr--augmented neural architecture that learns to compose functions mapping an input to an output on-the-fly. The memory has two trainable components: the value sub-memory and the binding key sub-memory, where the keys are basis weight matrices that span the space of functions. For an IQ task, when given a hint in the form of an input-output pair, and FINE estimates the analogy between the pair and the values mixing coefficients. These coefficients are then used to mix the binding keys via indirection to generate the weights of the backbone neural net which computes the intended function. For test input of different class, the function is used to estimate the most compatible output, thus solving the 10 task. Through an extensive suite of experiments using images from the Omniglot and CIFAR datasets to construct the IQ tasks, FINE is found be reliable in figuring out the hidden relational pattern in each IQ task and thus is able to solve new tasks, even with unseen image classes. Importantly, FINE outperforms other models in all experiments, and can generalize well in small data regimes.
Future works will include making FINE robust against OOD in transformations without catastrophic forgetting when new transformations are continually introduced.
10

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References
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[20] Gary Mareus. The algebraic mind, 2001.
[21] Leland McInnes, John Healy, and James Melville. Umap: Uniform manifold approximation and projection for dimension reduction. arXiv preprint 11X2:::800.03426 2018.

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[22] Melanie Mitchell. Abstraction and analogy-made making in artificial intelligence. Annals of the New York Academy of Sciences, 1505(1):7--00,, 2021.
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[33] Taylor Whittington Webb, Ishan Sinha, and Jonathan Cohen. Emergent symbols through binding in external memory. In International Conference on Learning Representations, 2020.
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12

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Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes] Our main contribution to perform analogy-making and indirection functional spaces for better OOD generalization.
(b) Did you describe the limitations of your work? [Yes] see Section
(c) Did you discuss any potential negative societal impacts of your work? [No]
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full set of assumptions all theoretical results? [N/A]
(b) Did you include complete proofs of all theoretical results? [N/A]
3. If you experiments..
(a) Did you include the code, data, and instructions needed to reproduce the main experi-
mental results (either in the supplemental material or URL)? [Yes] see Supplemen- tary
(b) Did you specify all the training details (e.g.. data splits, hyperparameters, how they were chosen)? [Yes] see Section
(c) Did you report error bars (e.g., with respect to the random seed after running experi- ments multiple times)? [Yes] see Section 4 and Appendix
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, cloud provider)? [Yes] see Section 4
4. If you are using existing assets (e.g., code, data, models) or curating/reelasing new assets...
(a) your work uses existing assets, did you cite the creators? [Yes] We use codes of [33] to run our experiments for the RelationNet, PrediNet, Transformer and ESBN.
(b) Did you mention the license of the assets? [Yes] Supplementary
(c) Did you include any new assets either in the supplemental material or as URL? [Yes] We include codes for our tasks and models.
(d) Did you discuss whether and how consent was obtained from people whose data you're 're using/curating? [Yes] see Supplementary
(e) Did you discuss whether the data you are using/curating contains personally identifiable information offensive content? [Yes] see Supplementary
5. If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, applicable? [N/A]
(b) Did you describe any potential participant risks, with links Institutional Review Board (IRB) approvals, if applicable? [N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
13


Reviews:
[{"review_id": "s6DLck9nQ_2", "reviewer": "Reviewer_otSZ", "summary": "This paper addresses the out-of-distribution generalization of deep learning models in IQ visual tasks involving extracting geometric transformation between a pair of images and applying the extracted transformation to a new image. It presents a memory-augmented neural architecture and on-the-fly model parameter retrieval from the memory to achieve OOD generalization in functional spaces.\n", "questions": "It's not clear whether the proposed method would work for other transformations.", "limitations": "Yes, the authors have addressed the limitations of their work.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n+ Unlike previous work that performs indirection and analogy-making in the data spaces, this paper proposes a mechanism for OOD in functional spaces.\n\n+ The weights of backbone can be determined on-the-fly using (input, output) query to retrieve from a memory.\n\nWeaknesses:\n- The proposed method may need a large memory to store trainable weights for very deep backbones.\n\n- The authors tested the proposed method only in functional spaces related to geometrical transformation. It's not clear whether the proposed method would work\nfor other transformations.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "LQTFhq0IHCt", "reviewer": "Reviewer_kR5Y", "summary": "This paper introduces a mechanism for functional indirection (FINE) in neural networks to achieve OOD generalization in abstract reasoning tasks. FINE proposes to dynamically select weights for a neural network backbone for a particular data input-output pair and use those weights to make prediction for an input that share similar hidden rule. The weights are selected from pre-defined key-value memory which is comprised of the weights that spans the space of possible functions. FINE is like a constrained form of hypernetworks where weights of the main (backbone) network are determined by another network. In this paper, this role is played by function composer $\\phi$ which finds optimal weights and their arrangement for the main network using a limited basis of weights in the memory.\n\nThe paper further introduces a new abstract reasoning dataset based on Omniglot and CIFAR100 for evaluation.", "questions": "- Why create new set of IQ tasks when a similar benchmark of PGMs studying different forms of rule based OOD generalization exists already? It fulfils the criterion of providing hints of the hidden rule with few images and then using that to infer the predictions.\n- Why are tasks restricted to identifying image transformations only? Would the method not be able to infer complex relational reasoning tasks as demonstrated in PGMs?\n- How are the basis of network weights restricted such that it only spans to a limited set of functions that can be used by the backbone?\n- Wouldn’t the limited basis of network weights restrict generalization to only observed rules and their combinations thereof? How would you scale to unseen rules?\n- Some of the references in the related work section are missing for e.g. Neural Interpreters [3] uses attention mechanism to recompose functional modules for each input-output pair and test their method on abstract reasoning tasks. Similarly, Switch transformers[4] switch modules based on relevance to inputs.\n- The model and the evaluation share similarities with meta-learning framework where a few input-output pairs are provided to adapt the network weights (e.g. in MAML) and the resulting model is used to make inference on the unknown inputs coming from the same classes. Analogous to the FINE datasets, the classes would be the hidden rule that input-output share. Curious to know what authors think about this and would it be possible to make a small experiment with Omniglot?\n- Why is similarity metric chosen to identify correct output $y’$ instead of using normal cross-entropy?\n- Do the newly introduced datasets test generalization only for unseen characters or also unseen rules? For e.g. the model could be trained for detecting character rotation from 0 to 90 degrees and tested on the characters rotated from 90 -270 degrees.\n\n[3] https://arxiv.org/abs/2110.06399\n[4] https://arxiv.org/abs/2101.03961", "limitations": "The main limitations seem to be around experimental evaluation. I would be happy to increase my score if authors address those concerns.\n\n\nThere doesn't seem to be any obvious negative societal impact.", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strength:**\n- The proposed methodology of achieving functional indirection is novel and interesting.\n- The paper is well written and easy to follow.\n- The method is shown to achieve good performance on OOD generalization across unseen categories of the datasets used for training.\n\n\n**Weaknesses:**\nMy main concerns revolve around the evaluation of method.\n- The paper goal is to solve OOD abstract reasoning and analogy making. However, the method is not evaluated on the known abstract reasoning benchmark of PGMs. Instead authors propose similar but less complex tasks for evaluation.\n-  The newly introduced datasets test generalization only for the unseen characters of the training dataset but not for unseen rules that could be extrapolated or interpolated. See me related comment in the questions section.\n- The proposed method, FINE, implicitly selects the weights based on the transformation (hidden rule) of the input-output pair. This is highly similar to [1] where a mixture of expert (networks) explicitly compete to explain image transformations on MNIST and Omniglot. [1] also showed huge benefits of mechanism-specific function selection for OOD generalization. I believe a comparison or even explanation on the similarities among the method would be good to have in the paper.\n- Since this framework focuses on the generalization of image transformation mechanisms across unseen classes (of Omniglot and CIFAR100), I would encourage authors to test FINE model trained with Omniglot transformations on MNIST data with transformation, which was shown in [1]. \n\n[1] https://arxiv.org/abs/1712.00961\n[2] https://arxiv.org/abs/1807.04225", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "KEcjMZU5PJ", "reviewer": "Reviewer_JbkE", "summary": "This paper introduces FINE, a method for achieving out of distribution generalization through analogy making and indirection in the space of functions. FINE makes an analogy from a given input and output example to infer the function that ties them then uses indirection to approximate the function by composing a set of functions saved in memory. The paper introduces a visual reasoning dataset for evaluating out of distribution generalization. The dataset is based on standard vision benchmarks, CIFAR100 and Omniglot. Each sample consists of an input-output pair used as a cue for the transformation, an input image and 4 choices. FINE performs better than several comparable models over all functions and function combinations in terms of accuracy and sample efficiency. Ablation experiments highlight the importance of the number of layers and the memory size.", "questions": "These questions are directly related to the weaknesses mentioned above.\n- The paper should clarify that the OOD generalization claim concerns the input images that undergo the transformations, not the transformations that are learned by the model. Evaluations on unseen transformations would be a valuable addition to the paper. \n- The training setup for baseline models should be described in the supplementary material. Ideally, all models should be capable of inferring the transformation.\n- As mentioned above, It would be a fair comparison to provide results on all models without NICE layers.\n- Can this framework be adapted and evaluated on OOD generalization in image classification or other tasks in ML ?\n- The first 2 paragraphs of section 3.1 could reformulated for clarification. As mentioned above, it would be more concise to write this section without using hypotheses. It’s simpler to explain FINE’s advantage when the task requires learning many functions with a single backbone.\n \n", "limitations": "The authors address certain limitations of the paper but address no negative social impact.", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The paper takes inspiration from features of human intelligence and combines them in a novel way.\n- It proposes a formal framework that uses this technique to mitigate the lack of OOD generalization in neural networks. It shows that the proposed method can abstract image-level transformations in the latent space and that the abstraction allows the model to generalize transformations to novel images.\n- The paper proposes a novel use case in visual reasoning that highlights the new framework’s advantage. The method is compared to several relevant baselines.\n- The paper is clearly written overall. The code is provided for reproducing the results.\n- The authors discuss the certain limitations of their approach.\n\nWeaknesses:\n- To learn the proposed task, the model has to identify and build the transformation. Thus, the OOD samples should be different functions, not only different input images (for example, can the model learn rotations of angles between 0 and 90 degrees and extrapolate to angles of 90-180 degrees ?)\n- To provide a fair comparison to baselines, all models need to be adapted to this framework. The training setup and hyperparameters used for training these models are not explained in the main paper or the supplementary material. This information is crucial for explaining their performance.\n- The paper doesn’t discuss other applications for this framework. OOD generalization is an active research topic in ML and image classification is among the main applications (adversarial examples and domain shifts and noise corruptions are OOD examples). How can this framework benefit OOD generalization in this task?\n- Although the use of NICE layers is intuitively motivated, it is not a necessary building block for FINE. Its use is motivated by the reversibility of certain transformations, which is a design choice in the dataset. Since most baselines (excluding hypernetwork) are not equipped with NICE layers, it would be a fair comparison to provide results on all models without NICE layers.\n- The choice of using 2 NICE layers instead of one layer is not fully explained.\n- The $y_t$ vectors used in the analogy step are output from $\\gamma_t(y)$. The motivation and significance of this choice are not explained.\n- In the first 2 paragraphs of subsection 3.1, H1 considers a unique output for each input while H2 considers the possibility of several outputs based on the transformation. The second hypothesis is not different from H1 if the target transformation is supplied with the input. This is the choice made indirectly in the dataset by providing an input-output pair as a hint for the transformation. It would be more concise to write this section without using hypotheses. It’s simpler to explain FINE’s advantage when the task requires learning many functions with a single backbone.\n\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "JqP5oIrk1Q2", "reviewer": "Reviewer_WCRb", "summary": "The problem of generalization to out-of-distribution (OOD) samples is key to most problems in AI. The authors address this challenge in the context of IQ-like tasks by introducing a method called FINE, for Functional indirection neural estimator. The authors are inspired by the idea of indirection, trying to connect two different representations and using one to learn or interpret the other. The authors show that the proposed architecture does well in those IQ tasks when considering images that are different from those in the training set upon applying the same rules as those in the training set. ", "questions": "The paper starts with the assertion that humans can generalize to OOD. I am curious about the evidence for this. Sure, everybody says this sort of thing. But what kind of real quantitative evidence do the authors have for this statement?\n\nOn line 85, the authors assert that the models are required to recognize the objects and figure out the relation between them. What is the evidence for any of this? Sure, from introspection, humans may reason about the task in this way, but this does not mean that this is what the models are *required* to do. This is an example of anthropomorphizing.\n(In contrast, lines 92/93 are trivially true. Yes, models must use the training data, there is no magic! )\n\nThe conclusions are similarly based on introspection only. The authors state that FINE reliably figures out the hidden relational pattern in each IQ task and is able to solve new tasks but the authors do not show any of this. The authors show that the model works better than a few other models when considering somewhat different images between the training set and the test set, which is pretty cool in and of itself. But there is nothing less and nothing more than that here. Other than somewhat different images, where does the paper show \"new tasks\" or \"discovering relational pattern\"?  \n\n\n\n", "limitations": "The authors spell out some limitations mostly to highlight other interesting problems that could be addressed. The key challenges mentioned above are not discussed.", "rating": 3, "confidence": 5, "soundness": 2, "presentation": 4, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths\n\nThe question of generalization to OOD is fundamental to intelligence problems. \n\nThe idea of indirection is very interesting and the authors propose a very nice implementation of this idea. \n\nFigure 4 is interesting in showing clusters in weight space. It would be interesting to show whether other models reveal the same property or whether this is specific to the proposed architecture.\n\nWeaknesses\n\nThe term OOD is often used in a very loose manner. This study is a good example. To really define OOD, one should define D (i.e., the distribution). What I understand the authors are doing is selecting some “classes” in Omniglot and testing on other classes. Or selecting some “classes” in CIFAR and testing on other classes. This seems pretty standard. But the question is how much OOD is really being tested here. Imagine that your training class is letter “i” and your test class is letter “l”. Those letters are really similar. Sure, they are different “classes”. The issue is how similar the training set is to the test set. Can the authors train with Omniglot classes and test with CIFAR classes or vice versa? That would be impressive OOD! \n\nThe word generalization here only applies to testing with somewhat different images. The hard challenge in IQ tests is to generalize to novel rules.  \n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept