Minimum Description Length Recurrent Neural Networks

Nur Lan1,2, Michal Geyer2, Emmanuel Chemla1,3∗, Roni Katzir2∗
1Ecole Normale Sup´erieure, Francia
2Tel Aviv University, Israel
3EHESS, PSL University, CNRS
{nlan,chemla}@ens.fr
michalgeyer@mail.tau.ac.il
rkatzir@tauex.tau.ac.il

Abstracto

We train neural networks to optimize a Min-
imum Description Length score, eso es, a
balance between the complexity of the net-
work and its accuracy at a task. We show
that networks optimizing this objective func-
tion master tasks involving memory challenges
and go beyond context-free languages. Estos
languages such as anbn,
learners master
anbncn, anb2n, anbmcn+m, and they perform
addition. Además, they often do so with
100% exactitud. The networks are small, y
their inner workings are transparent. We thus
provide formal proofs that their perfect accu-
racy holds not only on a given test set, pero
for any input sequence. To our knowledge,
no other connectionist model has been shown
to capture the underlying grammars for these
languages in full generality.

Introducción

A successful learning system is one that makes
appropriate generalizations. Por ejemplo, después
seeing the sequence 1,0,1,0,1 we might suspect
that the next element will be 0. If we then see
0, we might be even more confident that the
next input element will be 1. Artificial neural
networks have shown impressive results across a
wide range of domains, including linguistic data,
computer vision, and many more. They excel
at generalizing when large training corpora and
computing resources are available, but they face
serious challenges that become particularly clear
when generalizing from small input sequences like
the one presented above.

Primero, they tend to overfit the learning data. A
avoid this, they require external measures to con-
trol their own tendency for memorization (semejante
as regularization) as well as very large training

∗Both authors contributed equally to this work.

785

corpus. Además, standard regularization tech-
niques fall short in many cases, as we show below.
Segundo, even when successful, they tend to
produce non-categorical results. Eso es, they out-
put very high probabilities to target responses,
but never 100%. Adequate, human-like general-
ización, on the other hand involves having both
a probabilistic guess (which neural networks can
hacer) y, at least in some cases, a clear statement
of a categorical best guess (which neural networks
cannot do).

Tercero, these networks are often very big, y eso
is generally very hard to inspect a given network
and determine what it is that it actually knows
(though see Lakretz et al., 2019, for a recent
successful attempt to probe this knowledge in the
context of linguistics).

Some of the challenges above arise from the
reliance of common connectionist approaches
on backpropagation as a training method, cual
keeps the neural architecture itself constant
throughout the search. The chosen architecture
must therefore be large enough to capture the
given task, and it is natural to overshoot in terms of
tamaño. Además, it must allow for differentiable
operations to be applied, which prevents certain
categorical patterns from even being expressible.
en este documento, we propose to investigate a train-
ing method that differs from common approaches
in that its goal is to optimize a Minimum Descrip-
tion Length objective function (MDL; Rissanen,
1978). This amounts to minimizing error as usual,
while at the same time trying to minimize the size
of the network (a similar pressure to a Bayesian
size prior). Como resultado, the objective function of-
fers a guide to determining the size of the network
(a guide that error minimization alone does not
provide), which means that the architecture itself
can evolve during learning and typically can de-
crease in size. One potential side effect is that

Transacciones de la Asociación de Lingüística Computacional, volumen. 10, páginas. 785–799, 2022. https://doi.org/10.1162/tacl a 00489
Editor de acciones: Daniel Gildea. Lote de envío: 12/2021; Lote de revisión: 3/2022; Publicado 8/2022.
C(cid:3) 2022 Asociación de Lingüística Computacional. Distribuido bajo CC-BY 4.0 licencia.

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optimization cannot be done through backprop-
agation alone. We here use a genetic algorithm
to search through the very large search space of
neural networks of varying sizes.

We find that MDL-optimized networks reach
adequate generalizations from very small corpora,
and they avoid overfitting. The MDL-optimized
networks are all small and transparent; En realidad, nosotros
provide proofs of accuracy that amount to infinite
and exhaustive test sets. They can also provide
deterministic outputs when relevant (expressing
pure 100% confidence). We illustrate this across
a range of tasks involving counters, stacks, y
simple functions such as addition.

2 Previous Work

Our primary concern in this paper is the objective
función. The idea of applying a simplicity crite-
rion to artificial neural networks dates back at least
to Hinton and Van Camp (1993), who minimize
the encoding length of a network’s weights along-
side its error, and to Zhang and M¨uhlenbein (1993,
1995), who use a simplicity metric that is essen-
tially the same as the MDL metric that we use in the
present work (and describe below). Schmidhuber
(1997) presents an algorithm for discovering net-
works that optimize a simplicity metric that is
closely related to MDL. Simplicity criteria have
been used in a range of works on neural networks,
including recent contributions (p.ej., Ahmadizar
et al., 2015; Gaier and Ha, 2019). Outside of
neural networks, MDL—and the closely related
Bayesian approach to induction—have been used
in a wide range of models of linguistic phenomena,
in which one is often required to generalize from
very limited data (see Horning, 1969; Berwick,
1982; Stolcke, 1994; Gr¨unwald, 1996; and de
Marcken, 1996, among others; and see Rasin and
Katzir, 2016, and Rasin et al., 2021, for recent
proposals to learn full phonological grammars
using MDL within two different representational
frameworks). In the domain of program induction,
Yang and Piantadosi (2022) have recently used a
Bayesian learner equipped with a simplicity prior
to learn formal languages similar to the ones we
present below.

Turning to the optimization algorithm that we
use to search for the MDL-optimal network, nuestro
work connects with the literature on using evolu-
tionary programming to evolve neural networks.
Early work that uses genetic algorithms for various

aspects of neural network optimization includes
Miller et al. (1989), Montana and Davis (1989),
Whitley et al. (1990), and Zhang and M¨uhlenbein
(1993, 1995). These works focus on feed-forward
architectures, but Angeline et al. (1994) present
an evolutionary algorithm that discovers recurrent
neural networks and test it on a range of sequen-
tial tasks that are very relevant to the goals of
the current paper. Evolutionary programming for
neural networks remains an active area of research
(see Schmidhuber, 2015, and Gaier and Ha, 2019,
among others, for relevant references).

Our paper connects also with the literature
on using recurrent neural networks for grammar
induction and on the interpretation of such net-
works in terms of symbolic knowledge (a menudo
formal-language theoretic objects). These chal-
lenges were already taken up by early work on
recurrent neural networks (see Giles et al., 1990
and Elman, 1990, among others), and they re-
main the focus of recent work (ver, p.ej., Wang
et al., 2018; Weiss et al., 2018a). Jacobsson (2005)
and Wang et al. (2018) provide discussion and
further references.

3 Learner

3.1 Objective: Minimum Description Length
Consider a hypothesis space G of possible gram-
Marte, and a corpus of input data D. In our case,
G is the set of all possible network architectures
expressible using our representations, and D is
a set of input sequences. For a given G ∈ G
we may consider the ways in which we can en-
code the data D given G. The MDL principle
(Rissanen, 1978), a computable approximation
of Kolmogorov Complexity (Solomonoff, 1964;
Kolmogorov, 1965; Chaitin, 1966), aims at the
G that minimizes |GRAMO| + |D : GRAMO|, dónde |GRAMO| es
the size of G and |D : GRAMO| is the length of the
shortest encoding of D given G (with both com-
ponents typically measured in bits). Minimizing
|GRAMO| favors small, general grammars that often fit
the data poorly. Minimizing |D : GRAMO| favors large,
overly specific grammars that overfit the data.
By minimizing the sum, MDL aims at an inter-
mediate level of generalization: reasonably small
grammars that fit the data reasonably well.

The term |D : GRAMO| corresponds to the surprisal of
the data D according to the probability distribution
defined by G (es decir., the negative log of the probabil-
ity assigned to targets by the network). The term

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3.2.4 Units
The encoding of a unit includes its activation
función, the number of its outgoing connections,
the encoding of each of its outgoing connections,
and its bias weight, if any.

3.2.5 Activation Functions
Possible activation functions are: the linear acti-
vación (identity), ReLU, sigmoid, square, también
as the floor function and the unit step function
(returns 0 for inputs ≤ 0 y 1 de lo contrario). A
build an intuitive measure of simplicity into the
model’s choice of activation functions, we add a
cost to each function, encoded as a unary string:
The linear activation has no cost; square costs 2
bits; ReLU, sigmoid, and floor cost 4 bits; y el
unit step function costs 8 bits.

3.2.6 Connections and Weights
A connection’s encoding includes its target unit
number (each connection is specified within the
description of its source unit, hence the source
needs not be encoded), its weight, and its type:
forward (0) or recurrent (1).

To simplify the representation of weights in
classical neural networks and to make it easier to
mutate them in the genetic algorithm described
abajo, we represent weights as signed fractions
± n
d , which are serialized into bits by concatenating
the codes for the sign (1 para +, 0 for −), el
numerador, and the denominator. Por ejemplo, el
weight wij = + 2

5 would be encoded as:

1(cid:2)(cid:3)(cid:4)(cid:5)
+

(cid:2)

mi(2) = 10…10
(cid:5)
(cid:3)(cid:4)
(cid:2)
2
(cid:3)(cid:4)
wij

mi(5) = 1110…11
(cid:5)
(cid:3)(cid:4)
(cid:2)
5
(cid:5)

3.3 Search Algorithm

Our interest in this paper is the MDL objective
function and not the training method. Sin embargo,
identifying the MDL-optimal network is hard: El
space of possible networks is much too big for an
exhaustive search, even in very simple cases. Nosotros
therefore need to combine the objective function
with a suitable search procedure. We chose to use
a genetic algorithm (Georgia; Holanda, 1975), cual
frees us from the constraints coming from back-
propagation and is able to optimize the network
structure itself rather than just the weights of a
fixed architecture. For simplicity and to highlight
the utility of the MDL metric as a standalone ob-
jective, we use a vanilla implementation of GA,
summarized in Algorithm 1.

Cifra 1: Example network, encoded in Fig. 2.

|GRAMO| depends on an encoding scheme for mapping
networks onto binary strings, described below.

3.2 Our Networks and Their Encoding

The MDL learner explores a space of directed
graphs, made of an arbitrary number of units and
weighted connections between them. We describe
the actual search space explored in the experi-
ments below, by explaining how these networks
are uniquely encoded to produce an encoding
length |GRAMO|.

3.2.1 Ejemplo

We will consider networks such as the one rep-
resented and encoded in Fig. 2. It consists of two
input units (yellow units 1 y 2) and one output
unit with a sigmoid activation (blue unit 3). El
network has one forward connection (from unit 1
a 3) and one recurrent connection (unit 2 a 3),
represented by a dashed arrow. Recurrent connec-
tions feed a unit with the value of another unit
at the previous time step, and thus allow for the
development of memory across the different time
steps of the sequential tasks we are interested in.
Aquí, unit 3 is fed with input 2 from the previous
step. The connection weights are w1,3 = 0.5 y
w2,3 = 2. Unidad 3 is also fed a bias term b3 = 1
represented by a sourceless arrow.

We will now explain how such a network is

represented to measure its encoding size |GRAMO|.

3.2.2 Preliminary: Encoding Numbers

To ensure unique readability of a network from
its string representation we use the prefix-free
encoding for integers from Li and Vit´anyi (2008):

mi(norte) = 11111…1111
(cid:2)
(cid:5)
(cid:3)(cid:4)
Unary enc. de (cid:5)log2n(cid:6)

0(cid:2)(cid:3)(cid:4)(cid:5)
Separator

10101…00110
(cid:2)
(cid:5)
(cid:3)(cid:4)
Binary enc. of n

3.2.3 Encoding a Network

The encoding of a network is the concatenation of
(i) its total number of units, y (ii) the ordered
concatenation of the encoding of each of its units.

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Cifra 2: Binary encoding of the network in Fig. 1.

Algoritmo 1 Genetic algorithm

function TOURNAMENTSELECTION(pop):
T ← t random networks from pop
winner ← argminM DL(t )
loser ← argmaxM DL(t )
return winner, loser

end function
population ← ∅
mientras |población| < N do: (cid:2) Population initialization generate a random network net add net to population end while generation ← 0 while generation < Gen do: for N steps do: parent, loser ← (cid:2) Evolution loop TOURNAMENTSELECTION(population) oﬀspring ← mutate(parent) eval(oﬀspring) remove loser from population add oﬀspring to population (cid:2) MDL score end for generation ← generation + 1 end while return argminM DL(population) The algorithm is initialized by creating a pop- ulation of N random neural networks. A network is initialized by randomizing the following pa- rameters: activation functions, the set of forward and recurrent connections, and the weights of each connection. Networks start with no hid- den units. In order to avoid an initial population that contains mostly degenerate (specifically, dis- connected) networks, output units are forced to have at least one incoming connection from an input unit. The algorithm is run for Gen generations, where each generation involves N steps of selection fol- lowed by mutation. During selection, networks compete for survival on to the next generation based on their fitness, namely, their MDL score, where lower MDL is better. A selected network is then mutated using one of the following oper- ations: add/remove a unit; add/remove a forward or recurrent connection; add/remove a bias; mu- tate a weight or bias by changing its numerator or denominator, or flipping its sign; and change an activation function. These mutations make it possible to grow networks and prune them when necessary, and to potentially reach any architec- ture that can be expressed using our building blocks. The mutation implementations are based on Stanley and Miikkulainen (2002).1 On top of the basic GA we use the Island Model (Gordon and Whitley, 1993; Adamidis, 1994; Cantu-Paz, 1998) which divides a larger population into ‘islands’ of equal size N , each run- ning its own GA as described above, periodically exchanging a fixed number of individuals through a ‘migration’ step. This compartmentalization serves to mitigate against premature convergence which often occurs in large populations. The simulation ends when all islands complete Gen generations, and the best network from all islands is taken as the solution. 4 Experiments ran tasks based on several classical We formal-language learning challenges. We use both deterministic and probabilistic expectations to test the ability of a learner to work on proba- bilistic and symbolic-like predictions. In addition to showing that the MDL learner performs well on test sets, we provide proofs that it performs well on the whole infinite language under consideration. 1A mutation step can potentially produce a network that contains loops in its non-recurrent connections, most com- monly after a new connection is added. In the feed-forward phase, we detect loop-closing connections (using depth-first search) and ignore them. This avoids circular feeding, and at the same time creates a smoother search space, in which architectures freely evolve, even through intermediate defec- tive networks. Stagnant loop connections that don’t end up evolving into beneficial structures are eventually selected out due to the |G| term. 788 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . e d u / t a c l / l a r t i c e - p d f / d o i / . 1 0 1 1 6 2 / t l a c _ a _ 0 0 4 8 9 2 0 3 7 1 3 0 / / t l a c _ a _ 0 0 4 8 9 p d . f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 4.1 General Setup and Comparison RNNs All simulations reported in this paper used the following hyper-parameters: 250 islands, each with population size 500 (total 125,000 networks), 25,000 generations, tournament size 2, migration size 2, and a migration interval of 30 minutes or 1,000 generations (earliest of the two). The num- ber of generations was chosen empirically to allow enough time for convergence. Each task was run three times with different random seeds.2 To compare the performance of the MDL- optimized recurrent neural networks (MDLRNNs) with classical models, we trained standard RNNs on the same tasks, varying their architecture— GRU (Cho et al., 2014), LSTM (Hochreiter and Schmidhuber, 1997), and Elman networks (Elman, 1990)—as well as the size of their hidden state vectors (2, 4, 32, 128), weight regularization method (L1/L2/none), and the regularization con- stant in case regularization was applied (λ = 1.0/0.1/0.01), totaling 84 RNN configurations. Each configuration was run three times with dif- ferent random seeds. These RNNs were trained with a cross-entropy loss, which corresponds to the |D : G| term divided by the number of char- acters in the data.3 Table 1 summarizes the results for both MDLRNNs and classical RNNs for all the tasks that will be described below. For each task, the representative network for each model (out of all configurations and random seeds) was cho- sen based on performance on the test set, using MDL scores for MDLRNNs and cross-entropy for RNNs. It should be noted that model selection based on test performance is at odds with the premise of MDL: By balancing generalization and data fit during training, MDL automatically delivers a model that generalizes well beyond the train- ing data; MDL also does away with the post-hoc, trial-end-error selection of hyper-parameters and in classical regularization techniques inherent 2All experimental material and the source code for the model are available at https://github.com/taucompling /mdlrnn. 3All RNNs were trained using the Adam optimizer (Kingma and Ba, 2015) with learning rate 0.001, β1 = 0.9, and β2 = 0.999. The networks were trained by feed- ing the full batch of training data for 1,000 epochs. The cross-entropy loss for RNNs is calculated using the natural logarithm, converted in Table 1 to base 2 for comparison with MDL scores. models. In other words, MDL models can just as well be selected based on training perfor- mance. This is in contrast to standard RNNs, for which the training-best model is often the one that simply overfits the data the most. We show that even when given post-hoc advantage, RNNs still underperform.4,5 4.2 General Measures We will report on two measures: (i) Accuracy, which we explain below based on each task’s unique properties; and (ii) Cross-entropy, av- eraged by character for better comparability across set sizes, and compared with a baseline value calculated from the probability distribu- tion underlying each task. Most originally, on some occasions we report on these measures on whole, infinite languages. This is possible be- cause, by design, MDL-optimized networks are just large enough for their task, allowing us to fully understand their inner workings. 4.3 Experiment I: Counters and Functions We test our model’s ability to learn formal languages that require the use of one or mul- tiple counters: anbn, anbncn, anbncndn. These languages can be recognized using unbounded counting mechanisms keeping track of the number n of a’s and balance the other characters accord- ingly. We also test our model’s ability to learn languages that not only encode integers as above, but also operate over them: anbmcn+m (addition) and anb2n (multiplication by two). We did not test general multiplication through the language anbmcnm for practical reasons, namely, that the length of the sequences quickly explodes. 4When models are selected based on training performance (and then evaluated on the test sets), MDLRNNs outperform standard RNNs in all tasks in terms of cross-entropy and accu- racy. We make the full training-based comparison available as part of the experimental material. 5Training-based selection yields different MDLRNN win- ners for three out of the seven relevant tasks when trained on the smaller data sets, and for two tasks when trained on the larger sets. However, only one of these cases, for anbncn with the larger training set, results in a drop from 100% accuracy when selected by test to a suboptimum (97.6%), while other models remain at the same accuracy levels. MDL optimization is thus not immune to overfitting, which could occur for example due to accidental bad sampling. However, as shown by our results, MDL training produces models that generalize well across data sets. 789 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . e d u / t a c l / l a r t i c e - p d f / d o i / . 1 0 1 1 6 2 / t l a c _ a _ 0 0 4 8 9 2 0 3 7 1 3 0 / / t l a c _ a _ 0 0 4 8 9 p d . f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 optimal MDLRNN Training set size 100 500 100 500 100 500 100 500 100 500 100 500 20,000 100 400 anbn anbncn anbncndn anb2n anbmcn+m Dyck-1 Dyck-2 Addition MDLRNN 29.4 25.8 49.3 17.2 65.3 13.5 17.2 17.2 39.8 26.8 110.7 88.7 Test cross-entropy (×10−2) RNN 53.2 51.0 62.6 55.4 68.1 63.6 38.0 34.7 47.6 45.1 94.5 93.0 1.19 75.8 72.1 25.8 25.8 17.2 17.2 12.9 12.9 17.2 17.2 26.9 26.9 88.2 88.2 1.18 0.0 0.0 1.19 0.0 0.0 Test accuracy (%) RNN 99.8 99.8 99.8 99.8 99.8 99.8 99.9 99.9 98.9 98.9 10.9 10.8 89.0 74.9 79.4 100.0 100.0 96.5 100.0 68.6 99.9 100.0 100.0 98.9 100.0 69.9 100.0 99.3 100.0 100.0 Best RNN Type Elman Elman Elman Elman GRU GRU Elman GRU Elman + L1 Elman Elman LSTM GRU Elman Elman Size 2 2 4 4 4 4 4 4 128 128 4 4 128 4 4 MDLRNN proof Th. 4.1 Th. 4.2 Th. 4.3 Th. 4.4 Th. 4.5 Th. 4.6 Table 1: Performance of the networks found by the MDL model compared with classical RNNs for the tasks in this paper. Test accuracy indicates deterministic accuracy, the accuracy restricted to deterministic steps; Dyck-n tasks have no deterministic steps, hence here we report categorical accuracy, defined as the fraction of steps where a network assigns a probability lower than (cid:4) = 0.005 to each of the illegal symbols. When available, the last column refers to an infinite accuracy theorem for MDL networks: describing their behavior not only for a finite test set but over the relevant, infinite language. 4.3.1 Language Modeling The learner is fed with the elements from a se- quence, one input after the next, and at each time step its task is to output a probability distribu- tion for the next character. Following Gers and Schmidhuber (2001), each string starts with the symbol #, and the same symbol serves as the tar- get prediction for the last character in each string. If the vocabulary contains n letters, the inputs and outputs are one-hot encoded over n input units (in yellow in the figures), and the outputs are given in n units (in blue). To interpret these n outputs as a probability distribution we zero negative values and normalize the rest to sum to 1. In case of a degenerate network that outputs all 0’s, the probabilities are set to the uniform value 1/n. 4.3.2 Setup Each task was run with data set sizes of 100 and 500. The training sets were generated by sampling positive values for n (and m, when relevant) from a geometric distribution with p = .3. The maximal values K observed for n and m in our batches of size 100 and 500 were 14 and 22, respectively. We test the resulting networks on all unseen se- quences for n in [K +1, K +1001]. For anbmcn+m we test on n and m in [K + 1, K + 51], that is, the subsequent 2,500 unseen pairs. Only parts of the sequences that belong to the formal languages presented here can be pre- dicted deterministically, for example, for anbn, the deterministic parts are the first a (assuming n > 0), all b’s except the first one, y el
end-of-sequence symbol. For each of the tasks
in this section, entonces, we report a metric of de-
terministic accuracy, calculated as the number
of matches between the output symbol predicted
with maximum probability and the ground truth,
relative to the number of steps in the data that can
be predicted deterministically.

4.3.3 Resultados

The performance of the resulting networks is pre-
sented in Table 1. In Figures 3–6, we show the
networks that were found and their typical behav-
ior on language sequences. Thanks to their low
number of units and connections, somos capaces de
provide simple walkthroughs of how each network
operates. We report the following measures:

Deterministic accuracy: Perfect for almost all
tareas, both with small and large training sets.
The MDL learner achieves perfect accuracy for the

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tasks anbn and anb2n, both with small and large
training sets. The learner also achieves perfect
accuracy for anbncn and anbmcn+m with a larger
training set, and in fact the networks found there
would be better optima also for the respective
smaller training sets, therefore showing that the
suboptimal results for the small training sets are
only due to a limitation of the search, y eso
perfect accuracy should in principle be reachable
there too with a more robust search.

The only task for which MDLRNNs did not
reach 100% accuracy is anbncndn. Since the other
tasks show that our representations make it pos-
sible to evolve counters, we attribute this failure
to the search component, assuming a larger pop-
ulation or more generations are needed, bastante
than lack of expressive power; networks for this
task require more units for inputs and outputs,
which enlarge the number of possible mutations
the search can apply at each step.

Cross-entropy: Near perfect. For all tasks but
anbncndn, the MDLRNN per-character average
cross-entropy is also almost perfect with respect
to the optimal cross-entropy calculated from the
underlying probability distribution.

RNNs: No perfect generalization. Among the
competing models, no standard RNN reached
100% deterministic accuracy on the test sets,
and all RNNs reached suboptimal cross-entropy
puntuaciones, indicating that they failed to induce the
grammars and probability distributions underly-
ing the tasks. In terms of architecture size, el
best-performing RNNs are often those with fewer
units, while L1 and L2 regularizations do not yield
winning models except for one task.

for the whole,

Transparency supports formal proofs that
results are perfect
infinite
tasks but anbncndn then,
idioma. Para todos
deterministic accuracy and cross-entropy are per-
fect/excellent on training and test sets. Because the
MDL networks are small and transparent, podemos
go beyond these results and demonstrate formally
that the task is performed perfectly on the entire
infinite underlying language. To our knowledge,
such results have never been provided for any
classic neural network in these tasks or any other.

Teorema 4.1. The anbn network represented in
Higo. 3 outputs the correct probability for each

Cifra 3: The network found by the MDL learner for
the anbn task, for a training set with data set size 500.
See Theorem 4.1 for a description of how this network
accepts any anbn sequence and why it rejects any
other sequence.

anbn

Unidad 6

Initial #

kth a

kth b,
k < n nth b 0 k n−k 0 Unit 4 P (a) 7/3 ∼ 1 7/3 ∼ .7 −2/3 0 −2/3 0 Unit 5 P (b) 0 ∼ 0 1 ∼ .3 1 ∼ 1 0 0 Unit 3 P (#) σ(−15) ∼ 0 σ(−15) ∼ 0 σ(−15) ∼ 0 σ(−15) 1 Table 2: Unit values (columns) during each phase of a valid anbn sequence (rows). The second line for output units, given in bold, indicates the final normalized probability. character, for each sequence in the anbn lan- guage, with a margin of error below 10−6. Proof. Table 2 traces the value of each unit at each step in a legal sequence for the relevant network. When normalizing the outputs to obtain probabili- ties, the values obtained are the exact ground-truth values, up to the contribution of σ(−15) to that normalization (sigmoid is abbreviated as σ), which is negligible compared to all other positive val- ≈ 3.10−7, ues (the largest deviance is observed during the b’s). The network not only accepts valid anbn sequences, but also rejects other sequences, visible by the zero probability it assigns to irrelevant outputs at each phase in Table 2. σ(−15) 1+σ(−15) More informally, the network uses a single hid- den unit (6) as a counter, recognizable from the recurrent loop onto itself. The counter is incre- mented by 1 for each a (+2 from unit 1, –1 from the bias), and then decremented by 1 for each b (signaled by a lack of a, which leaves only the –1 bias as input to the counter). 791 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . e d u / t a c l / l a r t i c e - p d f / d o i / . 1 0 1 1 6 2 / t l a c _ a _ 0 0 4 8 9 2 0 3 7 1 3 0 / / t l a c _ a _ 0 0 4 8 9 p d . f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Figure 4: The network found for the anbncn task for the larger training set. See Theorem 4.2 for a description of how this network accepts only sequences of the language anbncn. anbncn Unit 8 Unit 9 Initial # 1 − 1 3 kth a kth b k < n nth b kth c k < n nth c 1 − k − k+1 3 k+1−n − k+n+1 3 1 − 2n+1 3 1+k 2k−2n−1 3 1+n − 1 3 Unit 4 Unit 5 Unit 6 Unit 7 P (#) P (a) P (b) P (c) − 1 0 1 σ(−15) 3 0 ∼ 0 0 ∼ 1 − k+1 7 1 σ(−15) 3 3 ∼ 0 ∼ .7 ∼ .3 ∼ 0 0 1 σ(−15) 0 ∼ 1 ∼ 0 0 0 σ(−15) 1 0 0 0 − 2n+1 3 − k+n+1 3 0 2(k+1−n) 3 0 2 3 ∼ 1 0 0 0 0 σ(−15) 1 0 0 0 σ(−15) 0 ∼ 0 Table 3: Unit values (columns) during each phase of a valid anbncn sequence (rows). Theorem 4.2. The network represented in Fig. 4 outputs the correct probability for each character, for each sequence in the anbncn language, with a margin of error below 10−6. Proof. The proof is again obtained by tracing the values each unit holds at each phase of a valid sequence in the language, see Table 3. The network uses two hidden units that serve as counters for the number of a’s (unit 8) and c’s (unit 9). Each occurrence of a simultaneously feeds the output unit for a (5) and the a counter (8) connected to the b output (6), using weights to create the correct probability distribution between a’s and b’s. Once a’s stop, P (a) flatlines, and the a counter (8) starts decreasing until n b’s are seen. Another counting system has evolved in unit 9 which counts the number of a’s and b’s (signaled by lack of c’s), and then decreases for each c, 792 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . e d u / t a c l / l a r t i c e - p d f / d o i / . 1 0 1 1 6 2 / t l a c _ a _ 0 0 4 8 9 2 0 3 7 1 3 0 / / t l a c _ a _ 0 0 4 8 9 p d . f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Figure 5: The network found for the anb2n task for the larger training set. See Theorem 4.3 for a description of how this network accepts only sequences of the language anb2n. finally triggering the end-of-sequence output #. Note how the model avoids holding a third counter for the number of b’s, by reusing the a counter. This makes it possible to disconnect the b input unit (2), which minimizes encoding length. Theorem 4.3. The anb2n network represented in Fig. 5 outputs the correct probability for each character, for each sequence in the anb2n lan- guage, with a margin of error below 10−6. Proof. The network is similar to the one found for anbn (Fig. 3). The proof that this network is accurate is also similar (Theorem 4.1), the only dif- ference being that the hidden unit is incremented with 2 instead of 1 for each a input. Theorem 4.4. The network represented in Fig. 6 outputs the correct probability for each character, for each sequence in the anbmcn+m language, with a margin of error below .2 (and below 10−4 for deterministic steps, i.e., probabilities 0 or 1).6 Proof. In Table 4 we trace the values of each unit during feeding of a valid sequence in anbmcn+m. We do not represent the internal memory unit 8, its value is the seventh of that of unit 4. Here, a network with a single counter (unit 8) has evolved which recognizes the language with 100% accuracy. While one would think that this task requires at least two counters—for n and m—the pressure for parsimony leads the model to evolve a more compact solution: Since the num- ber of c’s is always n + m, and no other symbols 6For this task, the average test cross-entropy per character of the network trained on the larger data set goes slightly below the optimum (see Table 1); this can happen, for example, if the model picks up on unintentional regularities in the training set that are also present in the test set. the next closing element must be a parenthesis or a bracket (and similarly for any Dyck-n language for n > 1; Suzgun et al., 2019). We ask here
whether MDL-optimized networks can evolve not
only counters but also stacks.

4.4.1 Setup

The setup is that of a language modeling task, as in
Experiment I. For Dyck-1, the training sequences
were generated from a PCFG with probability
pag = .3 of opening a new bracket, with data set
sizes 100 y 500. The test sets contained 50,000
sequences generated from the same grammar that
were not seen during training.

For Dyck-2, a fully operational stack is needed
in order to recognize the language. We thus first
make sure that such a network exists in the search
espacio. We do this by manually designing a network
that implements a fully operational stack. Usamos
this network as a baseline for comparison with the
results of the MDL simulations.

The stack network and a detailed description of
its mechanism are given in Fig. 8. We add two
additional building blocks in order to implement
this mechanism: the modulo 3 activation function
used in the ‘pop’ implementation, and a second
type of unit that applies multiplication to its inputs,
in order to create gates such as the ones used in
LSTM networks. Because the inclusion of these
new representations enlarges the search space, y
because the baseline network is larger in terms
of number of units than the networks found in
previous tasks (23 vs. 7-10), we double the genetic
algorithm’s overall population size (500 islands
vs. 250), allowing more hypotheses to be explored.
We also enlarge the training set to 20,000 muestras,
which allows networks with costlier |GRAMO| terms to
evolve. Here again the training sequences were
generated from a PCFG with probability p = .3
for opening a new bracket or parenthesis, y
tested on 50,000 novel sequences generated from
the same grammar.

Dyck sequences don’t have any sub-parts that
can be predicted deterministically (one can always
open a new bracket), which makes deterministic
accuracy reported above irrelevant. Nosotros informamos
instead a metric we call categorical accuracy,
defined as the fraction of steps where the network
predicts probability p ≥ (cid:4) for symbols that could
appear at the next step, y P < (cid:4) for irrelevant symbols. For example, for Dyck-2, when the up- coming closing character is a bracket (i.e., the last Figure 6: The network found for the anbmcn+m task for the larger training set. See Theorem 4.4 for a description of how this network accepts only sequences of the language anbmcn+m. anbmcn+m Initial # kth a kth b kth c k < m + n (m + n)th c unit 5 P (a) 312 ∼ .996 112 unit 7 unit 6 P (c) P (b) σ(−4) 0 ∼ 0 0 σ(−4) 72 ∼ 0 ∼ .71 ∼ .29 σ(−4) 0 .04 0 ∼ .69 ∼ .31 σ(−4) 0 0 1 0 0 σ(−4) 0 0 ∼ 0 0 0 7 unit 4 P (#) 7 2 ∼ .004 2 (1 − k) 0 2 (1−n−k) 7 0 2 (1−n−m+k) 0 7 2 ∼ 1 7 Table 4: Unit values during each phase of a valid anbmcn+m sequence. appear between the first and last #, the net- work uses the signal of lack of a’s and b’s as an indication of positive occurrences of c’s. This might raise a suspicion that the network recog- nizes out-of-language sequences such as balanced yet unordered strings, e.g., abbaaccccc. In prac- tice, however, the network imposes a strict order: a receives a positive probability only after # or a; b only after a or b; and c receives a signifi- cant proportion of the probability mass only as a last resort. 4.4 Experiment II: Dyck-1 vs. Dyck-2 In previous tasks, we showed the capability of MDLRNNs to evolve counters. A counter is also what is needed to recognize the Dyck-1 language of well-matched parentheses sequences. In the Dyck-2 language, however, there are two pairs of opening and closing characters, such as parenthe- ses and brackets. Counters are not sufficient then, and a stack is needed to additionally track whether 793 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . e d u / t a c l / l a r t i c e - p d f / d o i / . 1 0 1 1 6 2 / t l a c _ a _ 0 0 4 8 9 2 0 3 7 1 3 0 / / t l a c _ a _ 0 0 4 8 9 p d . f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Dyck-1 Unit 6 o > 0

1 − o

o = 0

Unidad 4
PAG ([)

1/2

∼ 3

7+2oh
1/2
1/3

Unidad 5
Unidad 3
PAG (]) PAG (#)
1 − o
0

f loor( 2+oh
3 )
∼ 4+2o
7+2oh
3 ) = 0
0

f loor( 2

1
2/3

Cifra 7: The network found by the MDL learner
for the Dyck-1 task for the larger training set. Ver
Teorema 4.5 for a description of how it accepts only
valid Dyck-1 sequences.

seen opening character was a bracket), the network
should assign probability 0 to the closing parenthe-
hermana; and similarly for the end-of-sequence symbol
as long as a sequence is unbalanced. Porque
classical RNNs cannot assign categorical 0 proba-
bilities to outputs due to their reliance on softmax
capas, we use (cid:4) = 0.005 as a categorical margin.

4.4.2 Resultados
Full performance details are given in Table 1.

For the Dyck-1 language, the networks for the
small and large training sets reach average test
cross-entropy of 1.11 y 0.89, respectivamente, com-
pared to an optimal 0.88. This result is in line
with those of Experiment I, where we have shown
that our representations are capable of evolv-
ing counters, which are sufficient for recognizing
Dyck-1 as well. An Elman RNN reaches a better
cross-entropy score, but worse categorical accu-
racy, for the smaller training set, while MDLRNN
wins with the larger set, reaching a score close to
the optimum and 100% categorical accuracy.

Teorema 4.5. When brackets are well balanced,
the Dyck-1 network in Fig. 7 correctly predicts
that no closing bracket can follow by assign-
ing it probability 0. En cambio, when brackets
are unbalanced, it assigns probability 0 to the
end-of-sequence symbol.

Prueba. Call o the number of open brackets in
a prefix. Throughout a Dyck-1 sequence, unit 6
holds the value 1 − o: it holds the value 1 after the
initial ‘#’; entonces +1 is added for each ‘[', and −1
for each ‘]'. The output probabilities in the cases of
balanced and unbalanced sequences are then given
en mesa 5. The theorem follows from the fact that
PAG (#) = 0 if o > 0, y P (]) = 0 if o = 0. En

794

Mesa 5: Unit values and output probabilities
during the two possible phases of a Dyck-1
secuencia: (i) the number of open brackets
o is positive, o (ii) all brackets are well
balanced (o = 0).

the target language, we note that opening brackets
have a constant probability of P ([) = .3, while in
the found network this probability decreases with
oh (visible in unit 4’s output probability, Mesa 5).
This makes a potential difference for high values
of o, cual, sin embargo, are very rare (o decreases
with probability .7 at all time steps).

For Dyck-2, the MDL model fails to reach the
architecture of the baseline manual network, o
another architecture with a similar cross-entropy
puntaje, reaching a network which has a worse MDL
score than the baseline (148,497 vs. 147,804).
Respectivamente, MDLRNN reaches a non-perfect
99.27% categorical accuracy, compared to 89.01%
for RNNs, which reflects both models’ failure to
correctly balance certain sequences. Both models
tie at 1.19 cross-entropy, close to the optimal 1.18.
Since we confirmed that the baseline archi-
tecture exists in the search space, Concluimos
that reaching a fully operational stack network is
hindered by the non-exhaustive search procedure,
rather than by the MDL metric. This may be solv-
able by tweaking the hyper-parameters or putting
more computational resources into the search. Él
could be, sin embargo, that this difficulty is due to a
more interesting property of the task at hand. Tiene
been claimed that evolutionary algorithms tend to
struggle with so-called ‘deceptive’ optimization
problems—tasks for which series of intermediate
good solutions don’t necessarily lead to a global
optimum (see overview in Lehman and Stanley,
2011). For the stack network, it could be the case
that a stack is only operational in its full form, y
that intermediate networks deceive and lead the
search to local minima, like the one found in the
current simulation.

A recent line of work has addressed the need
for stacks by manually designing stack-like mech-
anisms using continuous representations, y

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Cifra 8: A manually designed network implementing a fully operational stack, which recognizes the Dyck-2
idioma. The network uses an additional type of unit, which calculates the product of its inputs instead of
summing them, making it possible to create gate units similar to those of LSTM networks (gray striped units in
the figure). The stack’s memory is implemented as an integer, stored here in unit 13; the integer is shifted to the
left or right in base 3, making it possible to store the value 2 for a parenthesis and 1 for a bracket, visible in their
respective input weights. Unidad 12 is the ‘push’ gate, which opens when a non-zero value flows from the opening
bracket or parenthesis inputs. Unidad 16 is the ‘pop’ gate, opened by a non-zero input from a closing symbol. El
recurrent connection from memory unit 13 to unit 11 performs the base-3 left shift by multiplying the memory by
3. For ‘pop’, a right shift is applied by dividing the memory by 3. To extract the value of the topmost element,
modulo 3 is applied. The bias for unit 22 handles outputting the probability p of opening a new bracket/parenthesis.

integrating them manually into standard archi-
tectures (Graves et al., 2014; Joulin and Mikolov,
2015; Suzgun et al., 2019, among others). En efecto,
when they are explicitly augmented with manu-
ally designed continuous stack emulators, neural
networks seem to be able to capture nonregular
grammars such as the one underlying the Dyck-2
idioma. Similarmente, we could allow our search
to add stacks in one evolution step. This could
overcome the risk of a deceptive search target
mentioned above. If successful, we can expect
this approach to come with all the benefits of
the MDL approach: The winning network would
remain small and transparent, and it would even-
tually contain a memory stack only if this is
intrinsically needed for the task.

4.5 Experiment III: General Addition

In the previous experiments, we saw that
MDL-optimized networks are capable of repre-
senting integers and add them in what amounts to
unary representation (see anbmcn+m language).
Aquí, we show that addition can be performed
when the numbers and outputs are represented in
a different format. Específicamente, consideramos el
familiar task of adding two integers in binary rep-
resentation when the numbers are fed bit-by-bit

in parallel, starting from the least significant bit.
While this problem has good approximate solu-
tions in terms of standard RNNs,7 we will show
that our model provides an exact solution. As far
as we are aware, this has not been shown before.

4.5.1 Setup

In this setting, we diverge from a language mod-
eling task. The network here is fed at each time
step i with a tuple of binary digits, representing
the digits ni and mi of two binary numbers n and
metro, starting from the least significant bit. The two
input units are assigned the values ni and mi. El
output is interpreted as the predicted probability
eso (norte + metro)i = 1, eso es, eso 1 is the ith digit
in the sum (norte + metro). Output values are capped to
make them probabilities: values at or below 0 son
interpreted as probability 0, values at or above 1
are interpreted as probability 1.

The model was trained on two corpus sizes:
one that contained all pairs of integers up to
K = 10 (total 100 muestras), and a larger set of
all pairs up to K = 20 (total 400). El resultado-
ing networks were then tested on the set of all

7An example implementation that reportedly works up
to a certain number of bits: https://github.com
/mineshmathew/pyTorch_RNN_Examples.

795

Cifra 9: The network found by the MDL learner
for the binary addition task, trained on all 400 pares
of numbers up to 20. This network is correct for all
numbers (Teorema 4.6).

pairs of integers n, m ∈ [k + 1, k + 251], es decir.,
62,500 pairs not seen during training. Desde el
task is fully deterministic, we report a standard
accuracy score.

4.5.2 Resultados
MDLRNNs reached 100% accuracy on both test
conjuntos, and an optimal cross-entropy score of zero.
Higo. 9 shows the MDLRNN result for the larger
training set. It provably does perfect addition, con
perfect confidence, for all pairs of integers:

Teorema 4.6. For the net in Fig. 9, La salida
unit at time step i is the ith digit of the sum of
las entradas.

Prueba. Call c3
i−1 the value of unit 3 en el paso del tiempo
i − 1; this value is the carry-over for the next
time step, feeding unit 4 through their recurrent
connection at time step i. This can be proven in
two steps. (1) At the first time step i = 1 el
carry-over going into unit 4 es 0, since recurrent
inputs are 0 by default at the first time step. (2) Por
induction, c4
i is the sum of the relevant carry-over
(c3
i−1) and the two input digits at time i. El
combination of the 1/2 multiplication and floor
operation extracts a correct carry-over value from
that sum and stores it in unit 3. From there, we see
that c2
i holds the correct binary digit: the sum of
current inputs and carry-over (from c4
i ), minus the
part to be carried over next (from −2 × c3
i ).

De nuevo, the task is learned perfectly and in a
readable fashion. As a side remark, the network
obtained here can also naturally be extended to
perform addition of more than 2 numbers, simply
by adding the necessary inputs for the additional
digits and connecting them to cell 4. To our
knowledge no other RNN has been proven to hold
a carry-over in memory for an unbounded number
of digits, eso es, to perform general addition of
any arbitrary pair of numbers. The best competing

Size

CE (×10−2)
units
prueba
train
158.5 158.5
0
37.3 ∞ 126
6
37.6
55.3
37.5 ∞
6
1
38.1

25.8

conn
0
299
23
33
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Mesa 6: Cross-entropy and number of units
and connections on the anbn task using differ-
ent objective functions; MDL yields ground-truth
optimal CE for both training and test.

classical RNNs trained here were never able to
reach more than 79.4% accuracy on the test sets,
indicating that they learned a non-general way to
do addition.

4.6 Objective Function Probe

In order to further probe the value of the MDL ob-
jective function—and to isolate the effects of the
objective function, which is our main focus, de
those of the training method and the activation
functions—we ran four additional simulations us-
ing variations of MDL while keeping the setting
without change. The variants of the objective func-
tion that we tested are: (i) |GRAMO| solo, a saber, solo
the description length of the network is minimized;
(ii) |D : GRAMO| solo, eso es, the model only optimizes
training data fit, similarly to a cross-entropy loss
in traditional models; (iii)-(iv) replacing |GRAMO| con
traditional L1 and L2 regularization terms.

The different objective functions were tested
on the anbn task using the same hyper-parameters
given in Section 4.1. Mesa 6 summarizes the
performance for each resulting network. As ex-
pectado, cuando |GRAMO| alone is minimized, the result
is a degenerate network with no hidden units or
connections. En cambio, |D : GRAMO|-only training
results in a network growing large and picking up
on accidental regularities in the training set. El
overfitting leads to below-optimal cross-entropy
on the training set. Test cross-entropy is infinite
because the model assigns a categorical zero prob-
ability to some possible targets. Both L1 and L2
regularizations indirectly constrain the encoding
length of the resulting networks and have the
advantage of being compatible with backpropa-
gation search. Sin embargo, these constraints are not
as effective as pure MDL in avoiding overfitting

796

(cross-entropy is below optimal on the training set
and above on the test set).

39:1–13. https://doi.org/10.1016/j
.engappai.2014.11.003

5 Conclusión

Classical RNNs optimized for accuracy can
partially recognize nonregular
languages and
generalize beyond the data up to a certain n
(Gers and Schmidhuber, 2001; Weiss et al.,
2018b). However large this n may be, the failure
of these networks to fully generalize to arbitrary
values of n reveals that they fail to lock in on the
correct grammars that underlie these tasks.

We found that an MDL-optimized learner ar-
rives at networks that are reliably close to the
true distribution with small training corpora, para
classically challenging tasks. In several cases,
the networks achieved perfect scores. Beyond the
usual evaluation in terms of performance on test
conjuntos, the networks lent themselves to direct in-
spection and showed an explicit statement of the
pattern that generated the corpus.

Expresiones de gratitud

We wish to thank Matan Abudy, Moysh Bar-Lev,
Artyom Barmazel, Marco Baroni, Adi Behar-Medrano,
Maxime Caut´e, Rahma Chaabouni, Emmanuel
Dupoux, Nicolas Gu´erin, Jean-R´emy King, Yair
Lakretz, Tal Linzen, A¨el Quelennec, Ezer Rasin,
Mathias Sabl´e-Meyer, and Benjamin Spector;
the audiences at CNRS/ENS Paris, Facebook AI
París, NeuroSpin, Tel Aviv University, and ZAS
Berlina; and Nitzan Ron for creating the figures
in this paper. We also thank the action editors at
TACL and three anonymous reviewers for their
helpful comments.

This work was granted access to the HPC/AI
resources of IDRIS under the allocation 2021-
A0100312378 made by GENCI.

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