Learning Task-General Representations with Generative Neuro-Symbolic Modeling
Authors: Reuben Feinman
Brenden M. Lake
Venue: ICLR 2021 Poster (2021)
Keywords: few-shot concept learning
neuro-symbolic models
probabilistic programs
generative models

Abstract: People can learn rich, general-purpose conceptual representations from only raw perceptual inputs. Current machine learning approaches fall well short of these human standards, although different modeling traditions often have complementary strengths. Symbolic models can capture the compositional and causal knowledge that enables flexible generalization, but they struggle to learn from raw inputs, relying on strong abstractions and simplifying assumptions. Neural network models can learn directly from raw data, but they struggle to capture compositional and causal structure and typically must retrain to tackle new tasks. We bring together these two traditions to learn generative models of concepts that capture rich compositional and causal structure, while learning from raw data. We develop a generative neuro-symbolic (GNS) model of handwritten character concepts that uses the control flow of a probabilistic program, coupled with symbolic stroke primitives and a symbolic image renderer, to represent the causal and compositional processes by which characters are formed. The distributions of parts (strokes), and correlations between parts, are modeled with neural network subroutines, allowing the model to learn directly from raw data and express nonparametric statistical relationships. We apply our model to the Omniglot challenge of human-level concept learning, using a background set of alphabets to learn an expressive prior distribution over character drawings. In a subsequent evaluation, our GNS model uses probabilistic inference to learn rich conceptual representations from a single training image that generalize to 4 unique tasks, succeeding where previous work has fallen short.

Published as a conference paper at ICLR 2021
LEARNING TASK-GENERAL REPRESENTATIONS WITH GENERATIVE NEURO-SYMBOLA MODELING
Reuben Feinman & Brenden M. Lake
New York University
(reuben. feinman, arredden)@nyu.. edu
ABSTRACT
People can learn rich, sameral-purpose conceptual representations from only raw perceptual inputs. Current machine learning approaches fall well short of these human standards, although different modeling traditions often have complementary strengths. Symbolic models can capture the compositional and causal knowledge that enables flexible generalization, but they struggle to learn from raw inputs, rely- ing on strong abstractions and simplifying assumptions. Neural network models can learn directly from raw data, but they struggle to capture compositional and causal structure and typically must retrain tackle new tasks. We bring together these two traditions to learn generative models of concepts that capture rich compo- sitional and causal structure, while learning from raw data. We develop generative neuro-syme couuo-symbolic (GNS) model of handwritten character concepts that uses the con- trol flow of a probabilistic program, coupled with symbolic stroke primitives and a symbolic image renderer, to represent the causal and compositional processes by which characters are formed. The distributions of parts (strokes), and correlations between parts, modeled with neural network subroutines, allowing the model to learn directly from raw data and express nonparametric statistical relationships. We apply our model the Omniglot challenge of human-level concept learning, using background set of alphabets to learn expressive prior distribution over character drawings. In a subsequent evaluation, our GNS model uses probabilistic inference to learn rich conceptual representations from a single training image that generalize to unique tasks, succeeding where previous work has fallen short.
INTRODUCTION
Human conceptual knowledge supports many capabilities spanning perception, production and reasoning [37] A signature of this knowledge is its productivity and generality: the internal models and representations that people develop can be applied flexibly to new tasks with little or no training experience [30]. Another distinctive characteristic of human conceptual knowledge is the way that interacts with raw signals: people learn new concepts directly from raw, nigh-dimensional sensory data, and they identify instances of known concepts embedded in similarly complex stimuli. A central challenge is developing machines with these human-like conceptual capabilities.
Engineering efforts have embraced two distinct paradigms: symbolic models for capturing structured knowledge, and neural network models for capturing nonparametric statistical relationships. Symbolic models are well-suited for representing the causal and compositional processes behind perceptual observations, providing explanations akin to people's 's intuitive theories (38 Quintessential examples
include accounts of concept learning as program induction
13 46 29
28 Symbolic programs
provide a language for expressing causal and compositional structure, while probabilistic modeling offers a means of learning programs and expressing additional conceptual knowledge through priors. The Bayesian Program Learning (BPL) framework [29), for example, provides dictionary of simple sub-part primitives for generating handwritten character concepts, and symbolic relations that specify how to combine sub- hartspars into parts (strokes) and parts into whole character concepts. These abstractions support inductive reasoning and flexible generalization to a range of different
tasks, utilizing single conceptual representation 29

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BPL model (centered)
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Figure 1: Character drawings produced by the BPL model (left), GNS model (middle), and humans (right),
Symbolic models offer many useful features, but they come with important limitations. Foremost, symbolic probabilistic models make simplifying and rigid parametric assumptions, and when the assumptions are wrong- as is common in complex, nigh-dimensional data create bias [[]] The BPL character model, for example, assumes that parts are largely independent a priori, an assumption that is not reflective of real human-drawn characters. As consequence, characters generated from the raw BPL prior lack the complexity of real characters (Fig[]] left), even though the posterior samples can appear much more structured. Another limitation of symbolic probabilistic models is that the construction of structured hypothesis spaces requires significant domain ledgledge (2]. Humans, meanwhile, build rich internal models directly from raw data, forming hypotheses about the conceptual features and the generative syntax of a domain. As one potential resolution, previous work has demonstrated that the selection of structured hypotheses can itself be attributed to learning in a Bayesian framework 47
Although more flexible than a priori structural decisions, models of this kind still make many assumptions, and they have not yet tackled the types of raw, high-dimensional stimuli that are distinctive of the neural network approach.
13 14
24 40
The second paradigm, neural network modeling, prioritizes powerful nonparametric statistical learn- ing over structured representations. This modeling tradition emphasizes emergence, the idea that
conceptual knowledge arises from interactions of distributed sub-symboli processes 336 32 Neural networks are adept at learning from raw data and capturing complex patterns. However, they can struggle to learn the compositional and causal structure in how concepts are formed 30 even when this structure is salient in the data, they may have no obvious means of incorporating it. These limitations have been linked to shortcomings in systematic generalization (35] 27 and creative abilities (31) An illustrative example is the Omniglot challenge: in 4 years of active research, neural network models do not yet explain how people quickly grasp new concepts and use them a variety of ways, even with relatively simple handwritten characters [3I). Surveying over 10 neural models applied to Omniglot, Lake et al. [31] found that only two attempted both classification and generation tasks, and they were each outperformed by the fully-symbolic, probabilistic BPI.. Moreover, neural generative models tended to produce characters with anomalous characteristics, highlighting their shortcomings in modeling causal and compositional structure (see Fig. A13 and [3] Fig. 2a]).
In this paper, we introduce a new approach that leverages the strengths of both the symbolic and neural network paradigms by representing concepts as probabilistic programs with neural network subroutines. We describe an instance of this approach developed for the Omniglot challenge [29] of task-gen cask-general representation learning and discuss how we see our Omniglot model fitting into a broader class of Generative Neuro-Symte process. As with traditional probabilistic programs, the control flow of a GNS program is an explicit representation of the causal generative process that produces new concepts and new exemplars. Moreover, explicit re-use of parts through repeated calls to procedures such as GeneratePa P att (Fig. [ ensures representation that is compositional, providing appropriate inductive bias for compositional generalization. Unlike fully-symbolic probabilistic programs, however, the distribution of parts and correlations between parts in GNS are modeled with neural networks. This architectural choice allows the model to learn directly from raw data, capturing nonparametric statistics while requiring only minimal prior knowledge.
(GNS) models that seek to capture the data-generatiin
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Table I: Attempted Omniglot tasks by model. Attempt does not imply successful completion.
BPL 29
RCN 12
VHE 21
SG 43
SPIRAL 8
Matching
MAML
Graph
Prototypical
ARC
Task
Net 4
i
Net
Net HS
One-shot classification
Parsing
X
Generale exemplars
Generate concepui (type)
Generate concepts
We develop GNS model for the Omniglot challenge of learning flexible, task-general representations
of handwritten characters. We report results on 4 Omniglot challenge tasks with single model: 1) one-shot classification, 2) parsing/segmentationn 3) generating new exemplars, and 4) generating new concepts (without constraints); the 5th and final task of generating new concepts (from type) is left for future work. We also provide log-likelihood evaluations of the generative model. Notably, our goal
chase state-of-le the-art performance on one task across many datasets (e.g., classification). Instead we build model that learns deep. task-general knowledge within a single domain and evaluate on a range of different tasks. This "deep expertise" is arguably just as important as "broad expertise" in characterizing human-level concept learning [7 machines that seek human- like abilities will need both. Our work here is one proposal for how neurally-gon gerrallygrounded approaches can move beyond pattern recognition toward the more flexible model-building abilities needed for deep expertise [30]. Appendix E we discuss how to extend GNS to another conceptual domain, providing proposal for a GNS model of 3D object concepts.
RELATED WORK
The Omniglot dataset and challenge has been widely adopted in machine learning, with models such
as Matching Nets 44). MAML 7.. and ARC 44 applied to just one-shot classification, and others
such DRAW 19, SPIRAL and VHE [1 applied one or more generative tasks. In their "3-year progress report," Lake et al. [3 reviewed the current progress on Omniglot, finding that although there was considerable progress one-shot classification, there had been little emphasis placed on developing task-general models to match the flexibility of human learners (Table [ Moreover, sourally-groundee models that attempt more creative generation tasks were shown to produce characters that either closely mimicked the training examples or that exhibited anomalous variations, making for easy identification from humans (see Fig. A13 and [3T] Fig. 2a]). Our goal distinct in that aim to learn single neuro-symbolic generative model that can perform variety of unique tasks, and that generates novel yet structured new characters.
Neuro-symbolic modeling has become an active area of research, with applications to learning input-output programs [72 3 39 48), question answering [50]34] and image description 26 4
GNS modeling distinguishes itself from prior work through its focus hybrid generative modeling, combining both structured program execution and neural networks directly in the probabilistic generative process. Neuro-symbolic VQA models (50] 34 neither generative nor task-general; they are trained discriminatively to answer questions. Other neuro-symbolic systems use neural networks help perform inference in fully-symbolic generative model [26] or to parameterize a prior over fully-symbolic hypotheses (22]. In order to capture the dual structural and statistical characteristics of human conceptual representations, we find important to include neural nets directly in the forward generative model. As applied to Omniglot, our model bears some resemblance to SPIRAL []: however, SPIRAL does not provide a density function, and has no hierarchical structure, limiting its applications to image reconstruction and unconditional generation.
Another class of models on the neuro- syro-symmolic spectrum aims to learn "object representations" with neural networks 18 25), which add minimal object-like symbols to support systematic reasoning and generalization. Although these models have demonstrated promising results in applications such scene segmentation and unconditional generation, they have yet demonstrated the type of rich inductive capabilities that we are after: namely, the ability to learn "deep" conceptual representations few examples that support variety of discriminative and generative tasks.
from just
Other works (e.g. 16 20) have used autoregressive models like ours with similar stroke primitives to model the causal generative processes of handwriting. We develop novel architecture for generating
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procedure GENERAIE
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Figure 2: A generative neuro-symbolic (GNS) model of character concepts. The type model Generatellpe (P(+)) produces character types stroke at time, using an image canvas C as memory. At cach step, the current canvas C is fed to procedure ceneeatePaare and strokc sample is produced. The canvas first processed by the location model, a CNN-MLP architecture that samples starting location y. and next by the stroke model, a CNN-LSTM architecture that samples trajectory I while attending to the encoded canvas. Finally, symbolic renderer updates the canvas according to and y. and termination model decides whether terminate the type sample. Unique exemplars produced from character type by sampling from the token model conditioned v, adding motor noise the drawing parameters and performing a random affine transfrmation.
handwriting, which represents explicit compositional structure by modeling parts and relations with separate modules and applying intermediate symbolic rendering. Most importantly, these prior models have not made a connection to the image: therefore while they can generate handwriting as symbolic coordinates, they cannot explain how people use their causal knowledge to learn new characters from visual presentations, how they infer the strokes of character seen on paper, or how they generate a new example of an observed character. By combining a powerful autoregressive model of handwriting with a symbolic image renderer and algorithms for probabilistic inference, we seek to replicate spectrum of unique human concept learning abilities.
GENERATIVE MODEL
which offers useful scaffolding for conceptual models (Fig. 2, left) The type-level model P()) defines a prior distribution over character concepts, capturing overarching principles and regularities that tie together characters from different alphabets and providing procedure to generate new character concepts unconditionally in latent stroke format (Fig. 2, right). A token-level model p(0]u)) captures the within-class variability that arises from motor noise and drawing styles, and an image distribution P(1 provides an explicit model of how causal stroke actions translate to image pixels. All parameters of our model are learned from the Omniglot background set of drawings (Appendix A The full joint distribution over type v,
Our GNS model leverages the type- token hierarchy of BPL 229].
token ((m) and image ((m) factors as
P( ((m) [(m) P(W) ((g(m)) (P)((m)))))(((TT)))) P(T(m) p((m).
(bb)
Although sharing common hierarchy, the implementation details of each level in our GNS model differ from BPL in critical ways. The GNS type prior (()) is a highly expressive generative model that uses an external image canvas, coupled with symbolic rendering engine and attentive
Aspects of our model were recently published in non non-archival conference proceedings ( The previous manuscript presents only a prior distribution that alone performs just one task (generating new concepts); our new developments include a full hierarchical model, a differentiable image renderer / likelihood, and procedure for approximate probabilistic inference from image data. These ingredients together enable GNS to perform unique conceptual tasks.
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recurrent neural network, to condition future parts the previous parts and model sophisticated causal and correlational structure. This structure is essential to generating new character concepts in realistic, human-lite like ways (Sec. [5). Moreover, whereas the BPL model is provided symbolic relations for strokes such as "attach start" and "attach along." GNS learns implicit relational structure from the data, identifying salient patterns in the co-occurrences of parts and locations. Importantly, the GNS generative model is designed to be differentiable at all levels, yielding log-likelihood gradients that enable powerful new inference algorithms (Sec. and estimates of marginal image likelihood (Sec. 5)
Type prior. The type prior P(+) captured by a seuro-symbolic generative model of character drawings. The model represents character as a sequence of strokes (parts), with each stroke i decomposed into starting location yi E TR2 and variable-length trajectory x; E R x2, Rather than use raw pen trajectories as our stroke format, we use minimal spline representation of strokes, obtained from raw trajectories by fitting cubic b-splines with a residual threshold. The starting location y/ therefore conveys the first spline control point, and trajectory x; = {Ail..... ...,, Aud, conveys the offsets between subsequent points of a (d,+1)-lengll) spline. These offsets are transformed into a sequence of relative points x; { ..... +id.+1). with 241 0, specifying locations relative yr-
The model samples a type one stroke at a time, using an image canvas C' as memory to convey the sample state. At each step, starting location for the next stroke is first sampled from the location model, followed by a trajectory from the stroke model. The stroke is then rendered to the canvas C, and termination model decides whether to terminate three model components is expressed by a neural network, using a LSTM as the stroke model to generate trajectories as in [16]. The details of these neural modules are provided in Appendix A The type model (()) specifies an auto-regressive density function that can evaluate exact likelihoods of character drawings, and its hyperparameters (the three neural networks) are learned from the Omniglot background set of 30 alphabets using maximum likelihood objective. A full character type & includes the random variables &a {N. Vlue, Tise)- where и E Z+ is the number of strokes. The density function (()) is also fully differentiable w.r.t. the continuous random variables in tb.
continue the sample. Each of the
Token model. A character token g(m) (v(m)) _ A(m) represents unique instance of (m)
a character concept, where (m) are the token-level locations, Fiss the token-level parts, and A (m) E IRd1 the parameters of an affine warp transformation. The token distribution factorizes as
p(O(m)))) (b) ((A(m)) II P( (m) (y)) (m) xx4).
Here, P(u, (m)
w:) represents simple noise distribution for the location of each stroke, and P(a. 1x) for the stroke trajectory. The first two dimensions of affine warp ((m) control a global re-scaling of the token drawing, and the second two global translation of its center of mass. The distributions and pseudocode of our token model are given in Appendix[A)
Image model. The image model ((7(m) | ((m)) based on [2 and composed of two pieces. First, a differentiable symbolic engine f receives the token g(m) and produces an image pixel probability map Ping = (f(m), e) by evaluating each spline and rendering the stroke trajectories. Here, a E R+ is a parameter controlling the rendering blur around stroke coordinates, and e E (0, 1) controlling pixel noise, each sampled uniformly at random. The result then parameterizes an image distribution ((((m)) gim)) Bernoulli (pimg), which differentiable w.r.t. g(m)) o, and
4 PROBABILISTIC INFERENCE
Given an image 1 of a novel concept, our GNS model aims to infer the latent causal, compositional process for generating new exemplars. We follow the high-level strategy of BPL for constructing discrete approximation Q A I) to the desired posterior distribution (29].
K
P( 0 1) ~ Q(0,0000000 0 11) = Σ THŌ(0 - 0x)8(v) -x)).
(3)
heuristic search algorithm is used to find K good parses, {eb 0) 1h that explain the underlying image with high probability. These parses are weighted by their relative posterior probability,
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Figure 3: Classification fits and parsing. (a) Posterior parses from two training images were refit to the same test image. The first row of each grid shows the training image and its top-3 predicted parses (best emboldened). The second shows the test image and its re-fitted training parses. Reconstructed test images shown in the final row. The correct training image reports a high forward score, indicating that ((T) is well-explained by the motor programs for this f(e) (b) character images from 3 classes are shown alongside their ground truth human parses, predicted parses from the GNS model, and predicted parses from the BPL model.
Ta X म.. P( Oz, I) such that Σx Tr 1. To find the K good parses, search fast bottom- up methods to propose many variants of the discrete variables, filtering the most promising options, before optimizing the continuous variables with gradient descent (we K 5). See Appendix B for further details about inference, and Appendix for generating new exemplars of a concept.
Inference for one-shot classification. In one-shot classification, models are given single training image ((() from each of e L,... C' classes, and asked to classify test images according to their corresponding training classes. For each test image / ( compute approximation of the Bayesian score log P(I(T) I(e)) ) for every example [(e), using our posterior parses (v), B(e)). and corresponding weights TI.K from ((c) (Eq. 3). The approximation is formulated as
| ((0)) se log /
log P(
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Th max P(I(T) ((T)) ww). (
(4)
where the maximum over (TT) is determined by refitting token-level parameters ((s) to image (T) with gradient descent. Following Lake et al. [29), we use a two-way version of the Bayesian score that also considers parses of I(T) refit to [(e), The classification rule is therefore
P(](e) ((T))
ch = arg max log P(I(T) z(0)22 = arg max log
P(I(T) r(o)),
(5)
P(T(e))
where P(I())) En tt is approximated from the unnormalized weights of ((c) parses.
EXPERIMENTS
GNS evaluated four concept learning tasks from the Omniglot challenge [1]: one-shot classification, parsing, generating new exemplars, and generating new concepts. All evaluations use novel characters from completely held-out alphabets in the Omniglot evaluation set. As mentioned

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earlier, our goal is provide single model that captures deep knowledge of a domain and performs strongly in a wide range of tasks, rather than besting all models on every task. Our experiments include mixture of quantitative and qualitative evaluations, depending on the nature of the task.
One-shot classification. GNS was compared with alternative models on the one-shot classification task from Lake et al. [29] The task involves a series of 20-way within-alphabet classification episodes, with each episode proceeding as follows. First, the machine is given one training example from each of 20 novel characters. Next, the machine must classify 20 novel test images, each corresponding to one of the training classes. With 20 episodes total, the task yields 400 trials. Importantly, all character classes in an episode come from the same alphabet as originally proposed [29]. requiring finer discriminations than commonly used between- between-alphabet tests [31].
As illustrated in Fig. 3al GNS classifies a test image by choosing the training class with the highest Bayesian score (Eq. 5) A summary of the results shown in Table [ GNS was compared with other machine learning models that have been evaluated on the within- alphabets classification task (31] GNS achieved an overall test error rate of 5.7% across all 20 episodes (N=400). This result is very close to the original BPL model, which achieved 3.3% error with signif- icantly more hand-design. The symbolic relations in BPL's token model provide rigid constraints that are key to its strong classification performance [29]. GNS achieves strong classification performance while emphasizing the nonparametric statistical knowledge needed for creative generation in subsequent tasks. Beyond BPL, our GNS
Table 2: Test error within- alphahet one-shot classification.
Model
Error
GNS
5.7%
BPL
29
3.3%
RCN 12
7.3%
VHE
21
18. 18.7%
Proto.
Net
45
13.7%
ARC
44
1.5%
Pused 4x training classes
model outperformed all other models that received the same background training. The ARC model 44 achieved an impressive 1.5% error, although it was trained with four-fold class augmentation and many other augmentations, and it can only perform this one task. In Appendix Fig. [AIO] we show a larger set of classification fits from GNS, including examples of miselassified trials
Parsing. In the Omniglot parsing task, machines must segment a novel character into an ordered set of strokes. These predicted parses can be compared with human ground-truth parses for the same images. The approximate posterior of GNS yields a series of promising parses for new character image, and to complete the parsing task, we identify the maximum a posteriori parse k max; Tk. reporting the corresponding stroke configuration. Fig. shows a visualization of the GNS predicted parses for 27 different raw images drawn from 3 unique character classes, plotted alongside ground- truth human parses (how the images were actually drawn) along with predicted parses from the BPL model. Compared to BPI,, GNS parses possess a few unique desirable qualities. The first character class has an obvious segmentation to the human eye- evidenced by the consistency of human parses in all examples- and the GNS model replicates this consistency across all 9 predicted parses. In contrast, BPL predicts scemingly-unlikely parses for 2 of the examples shown. The second character is more complex, and it was drawn in different styles by different human subjects. The GNS model, which is trained on data from subjects with different styles, captures the uncertainty in this character by predicting a variety of unique parses. BPL, on the other hand, produces a single, ubiquitous segmentation across all examples. In Appendix Fig. ATT we provide a larger set of parses from the GNS model for a diverse range of Omniglot characters.
Generating new exemplars. Given just one training image of a novel character concept, GNS produces new exemplars of the concept by sampling from the approximate conditional P(r 8(2)
1() of Eq. g In Fig. 4a we show new exemplars produced by GNS for a handful of target images, plotted next to human productions (more examples in Appendix Fig. (12)). In the majority of cases, samples from the model demonstrate that it has successfully captured the causal structure and invariance of the target class. In contrast, deep generative models applied to the same task miss meaningful compositional and causal structure, producing new examples that are easily discriminated 43 21 (see Appendix Fig. A13). In some cases, such as the third column of Fig. 4a samples from GNS exhibit sloppy stroke junctions and connections. Compared to BPL, which uses engineered symbolic relations to enforce rigid constraints at stroke junetions, GNS misses some of these structural elements. Nevertheless, new examples from GNS appear strong enough to pass for human in many cases, which we would like test in future work with visual Turing tests.
from human productions
Concept learning experiments can he reproduced using pre-trained generative model and source code: https://githuul www.hhuw..cmmmmeeeomeeeeeeeeeeee www.offeeee.....o...oooo...o.eee MMMaa/GGSS-MMe1111 in
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Target
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(a) Generating new exemplars
(b) Generating new concepts
Figure 4: Generation tasks. (a) GNS produced 9 new exemplars for each of 5 target images, plotted here next human and BPL productions. (b) A grid 36 new character concepts sampled unconditionally from GNS, shown next BPL. samples.
Generating new concepts (unconstrained), In addition to generating new exemplars of a particular concept, GNS can generate new character concepts altogether, unconditioned on training images. Whereas the BPL model uses a complicated procedure for unconditional generation that involves preliminary inference step and a supplemental nonparametrie model, GNS generates new concepts by sampling directly from the type prior P(t). Moreover, the resulting GNS productions capture more of the structure found in real characters than either the raw BPL. prior (Fig 1
4h) or the supplemental nonparametric BPL prior [29]. In Fig. 4b we show a grid of 36 character concepts sampled from our generative model at reduced temperature setting 7 = 0.5 (Appendix (A.)). The model produces characters in multiple distinct styles, with some having more angular, line-based structure and others relying on complex curves. In Appendix Fig. A14 we show larger set of characters sampled from grid alongside a corresponding grid of "nearest neighbor"
GNS, plotted in a paglloiccally-oggniine training examples. In many cases, samples from the model have a distinct style and are visually dissimilar from their nearest Omniglot neighbor.
Marginal image likelihoods. As final evaluation, we computed likelihoods of held-out character images by
Table 3: Test log likelihood bounds.
marginalizing over the latent type and token variables of GNS to estimate P())= = P(t. 0, 1)0 We hy- pothesized that causal generative model of character concepts would yield better test likelihoods compared to deep generative models trained directly image pixels. As detailed in Appendix [ under the minimal assump- tion that our K posterior parses represent sharply peaked modes of the joint density, we can obtain an approximate lower bound on the marginal P(T) by using Laplace's method to estimate the integral around each mode and summing the resulting integrals. (LL) bounds obtained from GNS for a random subset of 1000 evaluation images, compared against test LL bounds from both the SG 43 and the VHE (1 models. Our GNS model performs stronger than each alternative, reporting the best overall logllikelihood per dimension.
Model
Im. Size
LL
LL/dim
VHE
28x28
-61.2
-0.0496
SG
52x52
-134.1
-0.0781
GNS
105x105
-383.2
-0.0348
Table we report average log-likelihood
6 DISCUSSION
We introduced generative neuro-symbolic (GNS) model for learning flexible, task-general representations of character concepts. We demonstrated GNS on the Omniglot challenge, showing that it performs variety of inductive tasks in ways difficult to distinguish from human behavior.
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Some evaluations were still qualitative, and future work will further quantify these results using Visual Turing Tests [29].
Whereas many machine learning algorithms emphasize breadth of data domains, isolating just single task across datasets, we have focused our efforts in this paper on a single domain, emphasizing depth of the representation learned. Human concept learning is distinguished for having both a breadth and depth of applications [37 31,, and ultimately, we would like to capture both of these unique qualities. We see our character model as belonging a broader class of generative neuro-symbolic (GNS) models for capturing the data generation process. We have designed our model based on general principles of visual concepts-nammll that concepts are composed of reusable parts and locations and we describe how it generalizes to 3D object concepts in Appendix E As in the human mind, machine learning practitioners have far more prior knowledge about some domains vs. others. Handwritten characters is domain with strong priors mind and body. For concepts like these with more explicit causal knowledge, is beneficial to include priors about how causal generative factors translate into observations, endowed to character model through its symbolic rendering engine. For other types of concepts where these processes are less clear, it may be appropriate use more generic neural networks that generate concepts and parts directly as raw stimuli, using less symbolic machinery and prior knowledge. We anticipate that GNS can flexibly model concepts in both types of domains, although further experiments are needed to demonstrate this.
33 23
implemented directly in the human
Our current token model for character concepts is much too simple, and we acknowledge a few important shortcomings. First, as shown in Appendix Fig. [AI0] there are a number of scenarios in which the parses from a training character cannot adequately refit to a new example of the same character without a token model that allows for changes to discrete variables. By incorporating this allowance in future work, we hope to capture more knowledge in this domain and further improve performance. Furthermore, although our vision for GNS is to represent both concepts and background knowledge with neuro-symbolic components, the current token-level model uses only simple parametric distributions. In future work, we hope incorporate token-level models that use neural network sub-routines, as in the type-level model presented here.
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A
GENERATIVE MODEL
Тype
Token
Image
M
Figure A5: The GNS hierarchical generative model.
The full hierarchical generative model of GNS is depicted in Fig. A5 token ((m), and image ((m) factors as
The joint density for type 1/,
P(v. @(m) [(m) ((g(m) (((()))))))((m))) g(m)).
The type &i parameterizes a motor program for generating character tokens ((m²), unique exemplars of the concept. Both v and ((mm) are expressed as causal drawing parameters. An image / [(m) is obtained from token ((m) by rendering the drawing parameters and sampling binary pixel values.
A.I TRAINING ON CAUSAL DRAWING DATA
original stroke
minimal spline
Figure A6: Spline representation. Raw strokes (left) converted into minimal splines (right) using least-squares optimization. Crosses (left) indicate pen locations and red dots (right) indicate spline control points.
To learn the parameters of P(t) and P(O(m) v)). we fit our models to the human drawing data from the Omniglot background set. In this drawing data, a character is represented as a variable-length sequence of strokes, and each stroke is a variable-length sequence of pen locations (21,.., 27), with 2, E IR? (Fig. A6 left). Before training our model on background drawings, we convert each stroke into minimal spline representation using least-squares optimization (Fig. A6] right), borrowing the B-spline tools from [29]. The number of spline control points depends on the stroke complexity and is determined by residual threshold. Furthermore, on the trajectory length. These processing steps help suppress noise and emphasize signal in the drawings. Our generative models trained to produce character drawings, where each drawing is represented as an ordered set of splines (strokes). The number of strokes, and the number spline coordinates per stroke, allowed vary in the model.
removed small strokes using threshold
A.2
TYPE PRIOR
The type prior P(b) represents character as a sequence of strokes, with each stroke decomposed into starting location yi E R.2 conveying the first spline control point, and a stroke trajectory J; (A,,... AN). conveying deltas between spline control points. It generates character types one stroke time, using symbolic rendering procedure called frender as intermediate processing step after forming each stroke, An image canvas is used as a memory state to convey information sampled
about previous strokes. At each step i, the next stroke's 's starting location and trajectory with procedure Generater art. In this procedure, the current image canvas C is first read by the location model (Fig. ? convolutional neural network (CNN) that processes the image and returns
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probability distribution for starting location y::
~ p(y: C).
The starting location yi is then passed along with the image canvas to the stroke model, a Long Short-Term Memory (LSTM) architecture with a CNN-based image attention mechanism. The stroke model samples the next stroke trajectory 2; sequentially one offset time, selectively attending to different parts of the image canvas at each sample step and combining this information with the context of y::
2; p(#; | ys: C).
After Generat returns, the stroke parameters Vi, are rendered to produce an updated can- vas C = Jemder (i, Jys C). The new canvas is then fed the termination model, a CNN architecture that samples a binary termination indicator ey:
& ~ p(e; C).
Both location model and stroke model follow technique from T6 who proposed networks with mixture outputs to model handwriting data. Parameters {ir l.K L:K o I:KK _plkk output by our network specify Gaussian mixture model (GMM) with K components (Fig. 2 colored ellipsoids), where x* E (0, 1) is the mixture weight of the hth component, μe E IR² its means, of E R2 its standard deviations, and pf (-1,1) correlation. In our location model, a single GMM describes the distribution p(v: C). In our stroke model, the LSTM outputs one GMM at each timestep, describing p( ( Yä, C). The termination model CNN has no mixture outputs; predicts a single Bernoulli probability to sample binary variable 14 When sampling from the model test time, we use temperature parameter proposed by Ha & Eck [20 (see (20] Eq. 8)) to control the entropy of the mixture density outputs.
neural
A.3 TOKEN MODEL
procedure GENERATETOKENIO
{n. Min 2 t V
Unpack type-level variables
for i = K do 1, (m) P(y (m)
Sample token-level location Sample token-level part Sample affine warp transformation
x(()) P(x (xa))
(m)) P(A(m))
(m))
Tae Alm))
return
Return concept token
Figure A7: Token model sampling procedure.
Character types s are used to parameterize the procedure GenerateTokern()), a probabilistic program representation of token model P(0(m)) ) The psuedo-code of this sampling procedure is
provided in Fig. A7 The location model Py (m) y:) and part model P(a. (x;) are each zero- mean Gaussians, with standard deviations fit the background drawings following the procedure of
Lake et al. (29] (see SM 2.3.3). The location model adds noise to the start of each stroke, and the part model adds isotropic noise the 2d cooridnates of each spline control point in a stroke. In the affine warp Am)) RR1, the first dimensions control global re-scaling of spline coordinates, and the second two control global translation of the center mass. The distribution is
(m)()) = (11, 1,0, 0). EA),
with the parameter ΣA similarly fit from background drawings (see SM 2.3.4 in 29
B APPROXIMATE POSTERIOR
To obtain parses {v), 1. for approximate posterior (Eq. 3) given image I, follow the high-level strategy of Lake et al. (2), using fast bottom-up search followed by discrete selection and continuous optimization. The algorithm proceeds by the following steps.
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Step 1: Propose range of candidate parses with fast bottom-up up methods. The bottom-up algorithm extracts an undirected skeleton graph from the character image and uses random walks on the graph to propose range of candidate parses. There are typically about 10-100 proposal parses, depending on character complexity (Fig. A8
Input
X
X
X
X
X
R
Figure A8: The initial "base" parses proposed for an image with skeleton extraction random walks.
Step 2: Select stroke order and stroke directions for each parse using exhaustive search with the type
prior P()). Random search is used for complex parses with large configuration spaces.
Step 3: Score each of the proposal parses using type prior P(v) and select the top K parses. We use K- following previous work (29).
Step 4: Separate each parse into type and token 0 and optimize the continuous type- and token-level parameters with gradient descent to maximize the full joint density P(*), 0, T) of Eq. [
Step 5: Compute weights Tı:K for each parse by computing मे P(th,, 02, 7) and normalizing
Th /K/ Ek 7k-
C INFERENCE FOR GENERATING NEW EXEMPLARS
When generating new exemplars, we are given a single image 7(1) of novel class and asked to generate new instances (2) (overloading the parenthesis notation from classification). To perform this task with GNS, we first sample from approximate posterior Q(v,, 0 I(1) to obtain parse {vs 0} (see Eq. 3). and then re- re-sample token parameters 0 from token model P(((2) v)). Due to high-dimensional images, mass in the approximate posterior often concentrates on the single best parse. To model the diversity seen in different human parses, apply a temperature parameter to the log of unnormalized parse weights log( = log(TT) (Tx))/ before normalization, selecting T = 8 for our experiments. With updated weights u. our sampling distribution is written as
(T(2),0(2) ((0) x((((2 P(B()) (b))
D MARGINAL IMAGE LIKELIHOODS
Let 1 U 0 be stand-in for the joint of type- and token-level random variables in our GNS generative model. The latent z includes both continuous and discrete variables: the number strokes and the number of control points per stroke dıx are discrete, and all remaining variables are continuous. Decomposing into its discrete variables zp E Ωt and continuous variables 2c E lc. the marginal density for image is written as
P(T) = Σ
((T.2,,z))0cc
(9)
EDEND
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For any subset Up QD of the discrete domain, the following inequality holds:
Σ /
P()
p(1,,n,=c)0s0
(10)
EpEAp
3 gives us K parses that represent promising modes {ED, =c hisk of the joint density P(I, zD. &C) for an image I, and by setting Ωi = {aD b to be the set of K unique discrete configurations from our parses, we can compute the lower bound of Eq. by computing the integral P(T, =D, =c) )0cc each of these
Our approximate posterior (Eq.
At each ZDR E I=D the log-density function f(c) = log P(T, EDR, =c) has a gradient- maximum at 2Ck. the continuous configuration of the corresponding posterior parse. These maxima were identified by our gradient-bar we assume that these maxima are sharply peaked, then we can use Laplace's method to estimate the integral f P(I, 2DR, zc) Jōzc at each ZDK- Laplace's ' method uses Taylor expansion to approximate the integral of ef(2) for a eiffereiffeet function f around a maximum Tog as
continuous optimizer during parse selection (Appendix (B).
f ( dr (aso)
(2T)
(11)
- H
where R& and H2))) is the Hessian matrix of evaluated at zo. Our log-density function ((c) fully differentiable w.r.t continuous parameters cc, therefore can compute H(2c) = 0 f0 with case. Our approximate lower bound on P(1) is therefore written as the sum of Laplace approximations K parses:
Σ P(I, (2T)
P(I) 2
Σ
PCL,
(12)
(=cw))
E APPLYING GNS TO 3D OBJECT CONCEPTS
The GNS modeling framework is designed to capture inductive biases for concept learning that generalize across different types of visual concepts. We are actively working applying GNS as a task-general generative model of 3D object concepts, such as chairs and vehicles, again with neuro-yymbolie generative process for parts and relations. Here, briefly review the path forward for training GNS models of object concepts.
procedure GeNcHAISTONNE
GENGRATEPARTIC) p/r (C)
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If then
GENERATE/NAT
breek
Terminata sample
i- [tc return
Returm concept token
GENERATEEPARTCC)
instance model pix
category model PdA
et k)
generate token
Figure A9: A GNS model for the concept of a chair.
Everyday 3D object concepts have more within-class variability than handwritten characters; for example, different chair tokens vary in the number arm rests, legs, etc. much as different handwrit- ten character types vary in their parts, although both domains characterized by similar structural
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and statistical considerations. To address these differences within GNS, shift up one level in the generative hierarchy and design a token-level model for generating new exemplars of an individual concept (chairs, cars, etc.) that mirrors our type-level model for characters. The architecture and sampling procedure of GNS for 3D object tokens is given in Fig. A9 As with characters, our object model produces samples one part
time, using 3D canvas C' in place of the previous 2D canvas. The procedure Generate eaneratePaae Part consists of two neural network components: 1) a discriminitively- trained category model p(k C)) that predicts the category label of the next part given the current canvas (leg, arm, back, etc.), and 2) a generative instance model p(z k, C)) that is trained with variational autoencoder objective to sample an instance of the next part æ given the current canvas and predicted part category label. Objects and object parts are represented as 3D voxel grids, and all neural modules, including the category model and the encoder/decoder of the instance model, are parameterized by 3D convolutional neural networks. A function Jupdate is used to update the current canvas with the most recent part by summing the voxel grids. A GNS model for a particular concept trained examples of the 3D voxel grids with semantic part labels.
F EXPERIMENTS: SUPPLEMENTAL FIGURES
F.1 ONE-SHOT CLASSIFICATION
In Fig. A1O we show a collection GNS fits from different classification trials, including 2 trials that were misclassified (a misrepresentative proportion).
F.2 PARSING
In Fig. All we show a collection of predicted parses from GNS for 100 different target images.
F.3 GENERATING NEW EXEMPLARS
Fig. A122shows new exemplars produced by GNS for 12 different target images, and Fig. A13 shows new exemplars produced by two alternative neural models from prior work Omniglot.
F,4 GENERATING NEW CONCEPTS (UNCONSTRAINED)
In Fig. A14 we show grid of 100 new character concepts produced by GNS, plotted alongside corresponding grid of "nearest neighbor" training examples.
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Figure A 14; Generating new concepts (unconstrained). 100 new concepts sampled unconditionally from GNS are shown in topologically organized grid alongside corresponding grid of "nearest neighbor" training examples. To identify nearest neighbors, we used cosine distance in the last hidden layer of CNN classifier metric perceptual similarity. CNN was trained to classify characters from Omniglot background set, 964-way classification task.
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Reviews:
[{"review_id": "saiX3ANqSEl", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "This paper introduces generative neuro-symbolic modelling, advertised as a probabilistic programming framework in which the distributions are modelled by neural networks. This is a very exciting idea, and well past its time. However, the work discussed here is limited to modelling the drawn characters in the Omniglot dataset, rather than a general framework. It is clear there is not a usable tool that would allow for practical construction of a wide range of probabilistic programs, from the fact that the paper sticks with a simple model for the Omniglot problem, rather than doing a lot of manipulations on the model, and, more importantly, the fact that the inference is specific to the Omniglot model, and is presumably not very fast, given the description. Thus, while the sales pitch to the paper is very exciting, compared to that, the content itself, which is really just a model of one problem, is a bit disappointing. That was also a property of the earlier Lake et al paper - I never understood what the \"framework\" was there either. That is not to dismiss this paper. The insight for this problem is that, rather than relying on a database of character primitives, like the previous Bayesian model of the problem, you can sequentially generate characters (types)  non-independently using the expected cast of characters (convolutions and LSTMs - although there are also splines involved, so it's not exactly typical CNN either). The evaluations are not all that impressive, except for the unconditional character generation shown at the beginning, which is clearly outstripping the BPL model in terms of human-like ness. In Table 3, I don't understand why different image sizes are used, and why BPL is not compared. The paper is very clear.\n\nIn short, I really like the idea in this paper, and I think it is worth building on. The section in the appendix detailing the future work on 3D object modelling sounds very promising. However, the result itself is not all that significant beyond the idea; the evaluations are not particularly strong, except for the generation of novel characters, which would benefit from a formal evaluation, and perhaps a transfer to a less artificial problem.", "extra_scores": {}, "extra_text": {}}, {"review_id": "pZyaaZAMGQD", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 7, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "### Summary\n\nIn this paper, the authors seek to combine the advantages of symbolic, compositional models and neural approaches, in particular by using \"probabilistic programs with neural network subroutines\" which should reflect a causal generative process. This is certainly an interesting area to explore. But I'm not convinced that the approach they propose (GNS) lives up to their stated goals.\n\nThe concrete model they describe (a generative model of Omniglot characters) seems to be essentially a recurrent neural controller whose outputs are connected to a differentiable stroke-rendering output function, and it seems to me to have very little \"symbolic\" structure. There is also a hierarchy of \"type\" and \"token\" levels (similar to BPL) but this hierarchy makes strong independence assumptions and seems too low-level to capture the true generative process. The authors claim this is a framework for learning task-general representations, but I struggle to see what this framework consists of; their discussion of GNS in section 3 seems highly tailored to the Omniglot task.\n\nThe paper is well written and the model itself is described in detail. The experimental results on four Omniglot tasks are strong and quite thorough, and the GNS model does seem to outperform the BPL model and other baselines. The authors also acknowledge some of the shortcomings of their model and describe some interesting possible future extensions.\n\nI lean toward accepting the paper, since it is an interesting approach to the Omniglot tasks and does have some solid contributions. But, at least in it's current state, I feel like the paper falls short of fulfilling its own promises.\n\n### Detailed comments\n\nThe introduction and motivation sections are very well written, and put forth a vision of Generative Neuro-Symbolic Modeling (GNS) as a way to combine the advantages of symbolic approaches with learned neural ones. The authors describe how symbolic models can generalize well and behave compositionally, but only when their strong assumptions correctly match the data-generating process; on the other hand, neural models can flexibly learn from data but can struggle to capture compositionality and may not generalize as well. GNS is described as a way to unify these strengths and obtain flexible models that also generalize.\n\nThe high-level goals of this approach seem to overlap to some degree with the HOUDINI framework [1], in that both are trying to combine the advantages of neural and symbolic approaches for generalizing to new tasks. There are significant differences, of course, but it is likely worth a brief discussion in the related work section.\n\nThe generative model proposed in section 3 seems strange to me in a couple of ways. Like BPL, it is separated into a \"type\" level, which is meant to capture character concepts, and a \"token\" level, which is meant to capture individual drawings of each character. In BPL, the \"type\" level values are structured objects that combine sequences of subparts and link them with discrete relationships, and the \"token\" level values are collections of strokes that are then rendered. In GNS, however, both the \"type\" and \"token\" levels are collections of rendered strokes, and the authors note that the type prior alone already specifies an \"auto-regressive density function that can evaluate exact likelihoods of character drawings\". The \"token\" model simply perturbs the strokes from the \"type\" level with simple 2D Gaussians, independently for each control point in the stroke, and then applies a global affine transformation. The authors claim that this is a causal model of character drawings, but it doesn't match my intuition about what the causal generative process of those drawings should be. In particular, it seems to me that the character concept needs to include the kinds of relational structure captured by BPL, at least implicitly. The representation used by GNS seems too low-level to adequately capture a character concept, and the strong assumption of independent perturbations leads to the loss of certain structural elements (as the authors note in section 5 under \"Generating new exemplars\").\n\nGiven the strong claims made initially, I was really hoping for a more powerful neural token model combined with a more abstract type model. Perhaps this could look something like Sketch-RNN [2] or DRAW, with a \"type\" prior over an abstract latent space and then a \"token\" decoder that renders strokes (similar to the proposed GNS type model). It seems to me that that would still having many of the same advantages but would also have the ability to generate more flexible output “tokens” than the GNS model (for instance, it might learn to connect strokes to previous strokes even when those previous strokes are perturbed by noise). The VAE framework might also provide a nice way of generalizing the very-domain-specific bottom-up parse proposal distribution described in the appendix.\n\nMore broadly, I get the sense from the introduction that this generative model of sketches is intended to be an instance of a more general framework of some kind (although this isn't particularly obvious, since the sketch-specific model is frequently referred to as simply \"GNS\" in the paper). But it is not at all clear to me what this framework actually is. On the most general side, is any model that composes neural submodules according to domain knowledge an instance of the framework? This is the impression that I got from the introduction, but it would seem to include a large number of preexisting models; for instance, the DRAW model specifies a structured probabilistic generation process of images. Or perhaps it requires combining neural and domain-specific primitives together into differentiable processes? This would include various papers on differentiable rendering (such as [3] and [4]) or audio synthesis (such as [5]). On the other hand, the conclusion suggests that the GNS framework is only applicable for visual tasks, which is a considerably smaller scope than the introduction suggested. (The authors also include some brief discussion of a GNS model of 3d structures in the appendix, but it seems very different from the model for characters, and I'm not sure what the common thread is.) The paper could be much improved by being more explicit about what the claimed general concept-learning framework actually consists of.\n\n### Questions and suggestions\n\nI was a bit surprised that neural architectures are referred to as \"nonparametric\". I suppose this is because vastly overparameterized networks behave like nonparametric models, despite having parameters? It might be worth clarifying the meaning here to avoid confusion.\n\nIs equation (4) doing maximum a-posteriori (MAP) inference over the token-level parameters? It would be useful to go into a bit more detail on what this approximation is and why approximating in this way is reasonable.\n\nIn section 5 under \"Parsing\", it wasn't clear to me why having a variety of parses is bad for the first character but desirable for the second one. Is it because that matches what humans do?\n\nMinor issue: I noticed that the authors appear to have used the ICLR 2020 style files instead of the ICLR 2021 style files.\n\n### References\n\n[1] Valkov, Lazar, et al. \"Houdini: Lifelong learning as program synthesis.\" Advances in Neural Information Processing Systems. 2018.\n\n[2] Ha, David, and Douglas Eck. \"A neural representation of sketch drawings.\" arXiv preprint arXiv:1704.03477 (2017).\n\n[3] Li, Tzu-Mao, et al. \"Differentiable monte carlo ray tracing through edge sampling.\" ACM Transactions on Graphics (TOG) 37.6 (2018): 1-11.\n\n[4] Thies, Justus, Michael Zollhöfer, and Matthias Nießner. \"Deferred neural rendering: Image synthesis using neural textures.\" ACM Transactions on Graphics (TOG) 38.4 (2019): 1-12.\n\n[5] Engel, Jesse, et al. \"DDSP: Differentiable digital signal processing.\" arXiv preprint arXiv:2001.04643 (2020).\n\n----\nAfter reading the author responses and updated manuscript, I have raised my score from 6 to 7.", "extra_scores": {}, "extra_text": {}}, {"review_id": "jx-vuiM9BnX", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "Summary: This paper presents a generative neuro-symbolic model for learning the task-general representations. The model is inspired by the BPL approach, with less manually-defined prior or rules but more learning-based ingredients. I generally appreciate the ideas and improvements it made. However, compared with BPL, the claims of task-general representation, and neural-symbolic modeling does not match the actual contributions of the work.\n\nPros\n1. The paper is well-written with its motivations,  methods, and corresponding experimental results. It shows a promising direction of learning a representation and model for multiple tasks, \\ie, the Omniglot challenge.\n\n2. The formulation of the proposed GNS and the corresponding method of inference provides a well-established and holistic solution for the visual simple concepts learning and inference.\n\nCons\n1. The claimants of human-level visual understanding, and task-general representations are similar to the BPL methods, which is well-known enough. I think the author should put more effort into comparing the BPL and demonstrates the actual improvement compared with it. \n\n2. The current framework is still under the Bayesian statistical modeling, I don't think it provides a task-general representation since it cannot be transferred to more complex domains and perceptual tasks. Moreover, the symbolic representations and symbolic module in the proposed method are quite trivial. The proposed method does not provide deeper insights into the neural symbolic representations.\n\n3. The causal structures in the model are captures in an autoregressive way, which is similar to lots of existing modules.  I don't see any actual causal problems are solved in the framework.\n\nMy main concern is that it put too much space in telling the stories that BPL has already proposed and demonstrated, but not the detailed comparisons with BPL and how the technical contributions and novelty of this specific model can benefit the overall human-level understanding of visual concepts. I would like to raise my scores if the concern is properly addressed.\n\nI raised my scores from 4 to 6 after the author updated their draft and answered my questions. I appreciate their efforts in addressing my concerns and improve the paper. The current version is good enough to be accepted and it also compares with other approaches thoroughly. However, I still feel the symbolic module is too simple in this work and does not distinguish it from other works. ", "extra_scores": {}, "extra_text": {}}, {"review_id": "Zb_oViDpM-y", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 7, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "Motivated by few-shot learning challenges such as Omniglot, the authors propose a human-like model for learning how to draw visual concepts that consists of three components: (1) a location model for picking the starting point of the next stroke given the current \"canvas\", (2) a stroke model that continues an existing stroke, (3) a termination model that decides when to stop drawing.  The three components of this model are trained on example glyphs that are heuristically parsed into strokes.  The authors evaluate performance of this method, called GNS (Generative Neuro-Symbolic) versus more classical program learning approaches based on pre-existing libraries of subroutines and more generic deep neural network architectures.  This approach (1) represents an advance over methods based on pre-defined subroutines in the sense that primitives such as \"draw a stroke intersecting an existing stroke\" are learned rather than supplied by the programmer and (2) performs much better at few-shot learning of glyphs than generic approaches.\n\nQuality: The idea behind the method makes sense.  The authors evaluate the model on four different tasks, and compare against reasonable baselines.  They also generalize beyond glyphs to learning 3D objects, although results are not shown.  The authors are careful to describe details such preprocessing of the input into splines and candidate parses, although details of the actual architecture do not seem to be given.  The weakest part of the approach is the image model.  It seems to work well enough for performance purposes, but the image distribution involving randomly transforming parts does not seem intuitively plausible as a model of human-like image generation.  As the authors note, this results in their approach underperforming against BPL.  The approach would probably work better if the authors had included stroke variability as a part of the core model.\n\nClarity:  I had to read the paper a few times to understand the approach.  It would have helped if the authors could better explain the architecture of the various model components.  Besides this issue of vagueness on technical details, the paper is well-written.\n\nOriginality:  To my knowledge the use of a separate location and LSTM stroke model for glyphs is novel.  The previous work section is adequate for contextualizing the paper.\n\nSignificance:  The paper represents an important approach to mimicking human concept learning.  (I base this on my own intuition; the authors could perhaps make a stronger case for this by including relevant references from the cognitive science literature.) \nChaining together multiple different neural networks to create a new type of ML algorithm has been a promising source of innovation in deep learning, e.g. AlphaGo combining a policy and evaluation network, GANs combining a generator and discriminator.  This paper is yet another example via combining a location and stroke model for learning glyphs.  On the other hand, the authors may have oversold the significance of the work by claiming that this model represents an example of fundamentally new approach in harmonizing artificial neural networks and symbolic program learning.  I am not convinced that this combination of location+stroke model is doing hierarchical concept learning in the same way that humans do.  If this is not what the authors are claiming, they should clarify this in the main paper.\n\nPros:\n * novel and plausible model for human-like learning\n * achieves reasonable performance on a variety of one-shot learning tasks\n * qualitatively exhibits human-like performance at free glyph generation (in my opinion)\n\nCons:\n * the type -> token part of the model seems underdeveloped\n * paper is vague on the architectures used for each of the model components (location, stroke, termination)\n * paper makes a claim that the model is doing symbolic modeling, but the term \"symbolic modeling\" is not well-defined and there is insufficient analysis showing that the model learns something analogous to existing methods such as BPL\n", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper was reviewed by four experts in the field. Based on the reviewers' feedback, the decision is to recommend the paper for acceptance to ICLR 2021. The reviewers did raise some valuable concerns that should be addressed in the final camera-ready version of the paper. The authors are encouraged to make the necessary changes and include the missing references.
Decision: Accept (Poster)