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Published as a conference paper at ICLR 2020
MOGRIFIER LSTM
Gábor Melis y , Tomáš Kociskýˇ y , Phil Blunsom yz
{melisgl,tkocisky,pblunsom}@google.com
y DeepMind, London, UK
z University of Oxford
ABSTRACT
arXiv:1909.01792v2 [cs.CL] 29 Jan 2020 Many advances in Natural Language Processing have been based upon more expres-
sive models for how inputs interact with the context in which they occur. Recurrent
networks, which have enjoyed a modicum of success, still lack the generalization
and systematicity ultimately required for modelling language. In this work, we
propose an extension to the venerable Long Short-Term Memory in the form of
mutual gating of the current input and the previous output. This mechanism affords
the modelling of a richer space of interactions between inputs and their context.
Equivalently, our model can be viewed as making the transition function given
by the LSTM context-dependent. Experiments demonstrate markedly improved
generalization on language modelling in the range of 3–4 perplexity points on Penn
Treebank and Wikitext-2, and 0.01–0.05 bpc on four character-based datasets. We
establish a new state of the art on all datasets with the exception of Enwik8, where
we close a large gap between the LSTM and Transformer models.
1 I NTRODUCTION
The domination of Natural Language Processing by neural models is hampered only by their limited
ability to generalize and questionable sample complexity (Belinkov and Bisk 2017; Jia and Liang
2017; Iyyer et al. 2018; Moosavi and Strube 2017; Agrawal et al. 2016), their poor grasp of grammar
(Linzen et al. 2016; Kuncoro et al. 2018), and their inability to chunk input sequences into meaningful
units (Wang et al. 2017). While direct attacks on the latter are possible, in this paper, we take a
language-agnostic approach to improving Recurrent Neural Networks (RNN, Rumelhart et al. (1988)),
which brought about many advances in tasks such as language modelling, semantic parsing, machine
translation, with no shortage of non-NLP applications either (Bakker 2002; Mayer et al. 2008). Many
neural models are built from RNNs including the sequence-to-sequence family (Sutskever et al. 2014)
and its attention-based branch (Bahdanau et al. 2014). Thus, innovations in RNN architecture tend to
have a trickle-down effect from language modelling, where evaluation is often the easiest and data
the most readily available, to many other tasks, a trend greatly strengthened by ULMFiT (Howard
and Ruder 2018), ELMo (Peters et al. 2018) and BERT (Devlin et al. 2018), which promote language
models from architectural blueprints to pretrained building blocks.
To improve the generalization ability of language models, we propose an extension to the LSTM
(Hochreiter and Schmidhuber 1997), where the LSTM’s inputxis gated conditioned on the output of
the previous stephprev . Next, the gated input is used in a similar manner to gate the output of the
previous time step. After a couple of rounds of this mutual gating, the last updatedxandhprev are
fed to an LSTM. By introducing these additional of gating operations, in one sense, our model joins
the long list of recurrent architectures with gating structures of varying complexity which followed
the invention of Elman Networks (Elman 1990). Examples include the LSTM, the GRU (Chung et al.
2015), and even designs by Neural Architecture Search (Zoph and Le 2016).
Intuitively, in the lowermost layer, the first gating step scales the input embedding (itself a representa-
tion of theaveragecontext in which the token occurs) depending on theactualcontext, resulting in a
contextualized representation of the input. While intuitive, as Section4 shows, this interpretation
cannot account for all the observed phenomena.
In a more encompassing view, our model can be seen as enriching the mostly additive dynamics of
recurrent transitions placing it in the company of the Input Switched Affine Network (Foerster et al.
1 Published as a conference paper at ICLR 2020
h0 ⦁ h2 ⦁ h4
LSTM
x-1 ⦁ x1 ⦁ x3 ⦁ x5
Figure 1: Mogrifier with 5 rounds of updates. The previous stateh0 =hprev is transformed linearly (dashed
arrows), fed through a sigmoid and gatesx1 =xin an elementwise manner producingx1 . Conversely, the
linearly transformedx1 gatesh0 and producesh2 . After a number of repetitions of this mutual gating cycle, the
last values ofh andx sequences are fed to an LSTM cell. Theprevsubscript ofhis omitted to reduce clutter.
2017) with a separate transition matrix for each possible input, and the Multiplicative RNN (Sutskever
et al. 2011), which factorizes the three-way tensor of stacked transition matrices. Also following
this line of research are the Multiplicative Integration LSTM (Wu et al. 2016) and – closest to our
model in the literature – the Multiplicative LSTM (Krause et al. 2016). The results in Section3.4
demonstrate the utility of our approach, which consistently improves on the LSTM and establishes a
new state of the art on all but the largest dataset, Enwik8, where we match similarly sized transformer
models.
2 M ODEL
To allow for ease of subsequent extension, we present the standard LSTM update (Sak et al. 2014)
with input and state of sizemandnrespectively as the following function:
LSTM:Rm Rn Rn !Rn Rn
LSTM(x;cprev ;hprev ) = (c;h):
The updated statecand the outputhare computed as follows:
f=(Wfx x+Wfh hprev +bf )
i=(Wix x+Wih hprev +bi )
j= tanh(Wjx x+Wjh hprev +bj )
o=(Wox x+Woh hprev +bo )
c=fcprev +ij
h=otanh(c);
whereis the logistic sigmoid function,is the elementwise product,W andb are weight
matrices and biases.
While the LSTM is typically presented as a solution to the vanishing gradients problem, its gatei
can also be interpreted as scaling the rows of weight matricesWj (ignoring the non-linearity in
j). In this sense, the LSTM nudges Elman Networks towards context-dependent transitions and
the extreme case of Input Switched Affine Networks. If we took another, larger step towards that
extreme, we could end up with Hypernetworks (Ha et al. 2016). Here, instead, we take a more
cautious step, and equip the LSTM with gates that scale thecolumnsof all its weight matricesW
in a context-dependent manner. The scaling of the matricesWx (those that transform the cell input)
makes the input embeddings dependent on the cell state, while the scaling ofWh does the reverse.
The Mogrifier 1 LSTM is an LSTM where two inputsxandhprev modulate one another in
an alternating fashion before the usual LSTM computation takes place (see Fig.1). That is,
Mogrify(x;cprev ;hprev ) = LSTM(x" ;cprev ;h" )where the modulated inputsx" andh" are prev prev
defined as the highest indexedxi andhi , respectively, from the interleaved sequences prev
xi = 2(Qi hi1 )xi2 ; for odd i2[1:::r] (1) prev
1 It’s like a transmogrifier 2 without the magic: it can only shrink or expand objects.
2 Transmogrify (verb, 1650s): to completely alter the form of something in a surprising or magical manner.
2 Published as a conference paper at ICLR 2020
hi = 2(Ri xi1 )hi2 ; for even i2[1:::r] (2) prev prev
withx1 =xandh0 =h 2N, is a hyperparameter;r= 0 prev prev . The number of “rounds”,r
recovers the LSTM. Multiplication with the constant2ensures that randomly initializedQi ;Ri
matrices result in transformations close to identity. To reduce the number of additional model
parameters, we typically factorize theQi ;Ri matrices as products of low-rank matrices:Qi =
Qi Qi withQi 2Rmn ;Qi 2Rmk ;Qi 2Rkn , wherek < min(m;n)is the rank. left right left right
3 E XPERIMENTS
3.1 T HE CASE FOR SMALL -S CALE
Before describing the details of the data, the experimental setup and the results, we take a short detour
to motivate work on smaller-scale datasets. A recurring theme in the history of sequence models is
that the problem of model design is intermingled with optimizability and scalability. Elman Networks
are notoriously difficult to optimize, a property that ultimately gave birth to the idea of the LSTM,
but also to more recent models such as the Unitary Evolution RNN (Arjovsky et al. 2016) and fixes
like gradient clipping (Pascanu et al. 2013). Still, it is far from clear – if we could optimize these
models well – how different their biases would turn out to be. The non-separability of model and
optimization is fairly evident in these cases.
Scalability, on the other hand, is often optimized for indirectly. Given the limited ability of current
models to generalize, we often compensate by throwing more data at the problem. To fit a larger
dataset, model size must be increased. Thus the best performing models are evaluated based on their
scalability 3 . Today, scaling up still yields tangible gains on down-stream tasks, and language mod-
elling data is abundant. However, we believe that simply scaling up will not solve the generalization
problem and better models will be needed. Our hope is that by choosing small enough datasets, so
that model size is no longer the limiting factor, we get a number of practical advantages:
?Generalization ability will be more clearly reflected in evaluations even without domain adaptation.
?Turnaround time in experiments will be reduced, and the freed up computational budget can be
put to good use by controlling for nuisance factors.
?The transient effects of changing hardware performance characteristics are somewhat lessened.
Thus, we develop, analyse and evaluate models primarily on small datasets. Evaluation on larger
datasets is included to learn more about the models’ scaling behaviour and because of its relevance
for applications, but it is to be understood that these evaluations come with much larger error bars
and provide more limited guidance for further research on better models.
3.2 D ATASETS
We compare models on both word and character-level language modelling datasets. The two word-
level datasets we picked are the Penn Treebank (PTB) corpus by Marcus et al. (1993) with prepro-
cessing from Mikolov et al. (2010) and Wikitext-2 by Merity et al. (2016), which is about twice
the size of PTB with a larger vocabulary and lighter preprocessing. These datasets are definitely
on the small side, but – andbecauseof this – they are suitable for exploring different model biases.
Their main shortcoming is the small vocabulary size, only in the tens of thousands, which makes
them inappropriate for exploring the behaviour of the long tail. For that, open vocabulary language
modelling and byte pair encoding (Sennrich et al. 2015) would be an obvious choice. Still, our
primary goal here is the comparison of the LSTM and Mogrifier architectures, thus we instead opt
for character-based language modelling tasks, where vocabulary size is not an issue, the long tail
is not truncated, and there are no additional hyperparameters as in byte pair encoding that make
fair comparison harder. The first character-based corpus is Enwik8 from the Hutter Prize dataset
(Hutter 2012). Following common practice, we use the first 90 million characters for training and
the remaining 10 million evenly split between validation and test. The character-level task on the
3 Note that the focus on scalability isnota problem per se. Indeed the unsupervised pretraining methods
(Peters et al. 2018; Devlin et al. 2018) take great advantage of this approach.
3 Published as a conference paper at ICLR 2020
Table 1: Word-level perplexities of near state-of-the-art models, ourLSTMbaseline and theMogrifieron PTB
and Wikitext-2. Models with Mixture of Softmaxes (Yang et al. 2017) are denoted withMoS, depth N withdN.
MCstands for Monte-Carlo dropout evaluation. Previous state-of-the-art results in italics. Note the comfortable
margin of 2.8–4.3 perplexity points the Mogrifier enjoys over the LSTM.
No Dyneval Dyneval
Val. Test Val. Test
FRAGE(d3, MoS15) (Gong et al. 2018) 22M 54.1 52.4 47.4 46.5
AWD-LSTM (d3, MoS15) (Yang et al. 2017) 22M 56.5 54.4 48.3 47.7
Transformer-XL (Dai et al. 2019) 24M 56.7 54.5
PTB EN LSTM(d2) 24M 55.8 54.6 48.9 48.4
Mogrifier(d2) 24M 52.1 51.0 45.1 45.0
LSTM(d2, MC) 24M 55.5 54.1 48.6 48.4
Mogrifier(d2, MC) 24M 51.4 50.1 44.9 44.8
FRAGE(d3, MoS15) (Gong et al. 2018) 35M 60.3 58.0 40.8 39.1
AWD-LSTM (d3, MoS15) (Yang et al. 2017) 35M 63.9 61.2 42.4 40.7
WT2 EN LSTM(d2, MoS2) 35M 62.6 60.1 43.2 41.5
Mogrifier(d2, MoS2) 35M 58.7 56.6 40.6 39.0
LSTM(d2, MoS2, MC) 35M 61.9 59.4 43.2 41.4
Mogrifier(d2, MoS2, MC) 35M 57.3 55.1 40.2 38.6
Mikolov preprocessed PTB corpus (Merity et al. 2018) is unique in that it has the disadvantages of
closed vocabulary without the advantages of word-level modelling, but we include it for comparison
to previous work. The final character-level dataset is the Multilingual Wikipedia Corpus (MWC,
Kawakami et al. (2017)), from which we focus on the English and Finnish language subdatasets in
the single text, large setting.
3.3 S ETUP
We tune hyperparameters following the experimental setup of Melis et al. (2018) using a black-box
hyperparameter tuner based on batched Gaussian Process Bandits (Golovin et al. 2017). For the
LSTM, the tuned hyperparameters are the same:input_embedding_ratio,learning_rate,l2_penalty,
input_dropout,inter_layer_dropout,state_dropout,output_dropout. For the Mogrifier, the number
of roundsrand the rankkof the low-rank approximation is also tuned (allowing for full rank, too).
For word-level tasks, BPTT (Werbos et al. 1990) window size is set to 70 and batch size to 64. For
character-level tasks, BPTT window size is set to 150 and batch size to 128 except for Enwik8 where
the window size is 500. Input and output embeddings are tied for word-level tasks following Inan
et al. (2016) and Press and Wolf (2016). Optimization is performed with Adam (Kingma and Ba
2014) with1 = 0, a setting that resembles RMSProp without momentum. Gradients are clipped
(Pascanu et al. 2013) to norm 10. We switch to averaging weights similarly to Merity et al. (2017)
after a certain number of checkpoints with no improvement in validation cross-entropy or at 80% of
the training time at the latest. We found no benefit to using two-step finetuning.
Model evaluation is performed with the standard, deterministic dropout approximation or Monte-
Carlo averaging (Gal and Ghahramani 2016) where explicitly noted (MC). In standard dropout
evaluation, dropout is turned off while in MC dropout predictions are averaged over randomly
sampled dropout masks (200 in our experiments). Optimal softmax temperature is determined on
the validation set, and in the MC case dropout rates are scaled (Melis et al. 2018). Finally, we report
results with and without dynamic evaluation (Krause et al. 2017). Hyperparameters for dynamic
evaluation are tuned using the same method (see AppendixA for details).
We make the code and the tuner output available at https://github.com/deepmind/lamb.
3.4 R ESULTS
Table1 lists our results on word-level datasets. On the PTB and Wikitext-2 datasets, the Mogrifier
has lower perplexity than the LSTM by 3–4 perplexity points regardless of whether or not dynamic
evaluation (Krause et al. 2017) and Monte-Carlo averaging are used. On both datasets, the state of
the art is held by the AWD LSTM (Merity et al. 2017) extended with Mixture of Softmaxes (Yang
4 Published as a conference paper at ICLR 2020
Table 2: Bits per character on character-based datasets of near state-of-the-art models, ourLSTMbaseline
and theMogrifier. Previous state-of-the-art results in italics. Depth N is denoted withdN. MC stands for
Monte-Carlo dropout evaluation. Once again the Mogrifier strictly dominates the LSTM and sets a new state of
the art on all but the Enwik8 dataset where with dynamic evaluation it closes the gap to the Transformer-XL of
similar size (yKrause et al. (2019),zBen Krause, personal communications, May 17, 2019). On most datasets,
model size was set large enough for underfitting not to be an issue. This was very much not the case with Enwik8,
so we grouped models of similar sizes together for ease of comparison. Unfortunately, a couple of dynamic
evaluation test runs diverged (NaN) on the test set and some were just too expensive to run (Enwik8, MC).
No Dyneval Dyneval
Val. Test Val. Test
TrellisNetworks (Bai et al. 2018) 13.4M 1.159
AWD-LSTM (d3) (Merity et al. 2017) 13.8M 1.175
PTB EN LSTM(d2) 24M 1.163 1.143 1.116 1.103
Mogrifier(d2) 24M 1.149 1.131 1.098 1.088
LSTM(d2, MC) 24M 1.159 1.139 1.115 1.101
Mogrifier(d2, MC) 24M 1.137 1.120 1.094 1.083
HCLM withCache (Kawakami et al. 2017) 8M 1.591 1.538
LSTM (d1) (Kawakami et al. 2017) 8M 1.793 1.736
MWC EN LSTM(d2) 24M 1.353 1.338 1.239 1.225
Mogrifier(d2) 24M 1.319 1.305 1.202 1.188
LSTM(d2, MC) 24M 1.346 1.332 1.238 NaN
Mogrifier(d2, MC) 24M 1.312 1.298 1.200 1.187
HCLM withCache (Kawakami et al. 2017) 8M 1.754 1.711
LSTM (d1) (Kawakami et al. 2017) 8M 1.943 1.913
MWC LSTM(d2) 24M 1.382 1.367 1.249 1.237
FI Mogrifier(d2) 24M 1.338 1.326 1.202 1.191
LSTM(d2, MC) 24M 1.377 1.361 1.247 1.234
Mogrifier(d2, MC) 24M 1.327 1.313 1.198 NaN
Transformer-XL (d24) (Dai et al. 2019) 277M 0.993 0.940y
Transformer-XL (d18) (Dai et al. 2019) 88M 1.03
LSTM(d4) 96M 1.145 1.155 1.041 1.020
Mogrifier(d4) 96M 1.110 1.122 1.009 0.988
LSTM(d4, MC) 96M 1.139 1.147
Enwik8 Mogrifier(d4, MC) 96M 1.104 1.116
EN Transformer-XL (d12) (Dai et al. 2019) 41M 1.06 1.01z
AWD-LSTM (d3) (Merity et al. 2017) 47M 1.232
mLSTM (d1) (Krause et al. 2016) 46M 1.24 1.08
LSTM(d4) 48M 1.182 1.195 1.073 1.051
Mogrifier(d4) 48M 1.135 1.146 1.035 1.012
LSTM(d4, MC) 48M 1.176 1.188
Mogrifier(d4, MC) 48M 1.130 1.140
et al. 2017) and FRAGE (Gong et al. 2018). The Mogrifier improves the state of the art without either
of these methods on PTB, and without FRAGE on Wikitext-2.
Table2 lists the character-level modelling results. On all datasets, our baseline LSTM results are much
better than those previously reported for LSTMs, highlighting the issue of scalability and experimental
controls. In some cases, these unexpectedly large gaps may be down to lack of hyperparameter tuning
as in the case of Merity et al. (2017), or in others, to using a BPTT window size (50) that is too small
for character-level modelling (Melis et al. 2017) in order to fit the model into memory. The Mogrifier
further improves on these baselines by a considerable margin. Even the smallest improvement of
0.012 bpc on the highly idiosyncratic, character-based, Mikolov preprocessed PTB task is equivalent
to gaining about 3 perplexity points on word-level PTB. MWC, which was built for open-vocabulary
language modelling, is a much better smaller-scale character-level dataset. On the English and the
Finnish corpora in MWC, the Mogrifier enjoys a gap of 0.033-0.046 bpc. Finally, on the Enwik8
dataset, the gap is 0.029-0.039 bpc in favour of the Mogrifier.
5 Published as a conference paper at ICLR 2020
h0 ⦁ h2 ⦁ h4
LSTM
x-1 ⦁ x1 ⦁ x3 ⦁ x5
Figure 2: “No-zigzag” Mogrifier for the ablation study. Gating is always based on the original inputs.
57 Table 3: PTB ablation study validation
perplexities with 24M parameters.
56
55 Mogrifier 54.1
Full rankQi ;P i 54.6
54 No zigzag 55.00 1 2 3 4 5 6 LSTM 57.5
mLSTM 57.8 Figure 3: Perplexity vs the roundsrin the PTB ablation study.
Of particular note is the comparison to Transformer-XL (Dai et al. 2019), a state-of-the-art model
on larger datasets such as Wikitext-103 and Enwik8. On PTB, without dynamic evaluation, the
Transformer-XL is on par with our LSTM baseline which puts it about 3.5 perplexity points behind
the Mogrifier. On Enwik8, also without dynamic evaluation, the Transformer-XL has a large, 0.09 bpc
advantage at similar parameter budgets, but with dynamic evaluation this gap disappears. However,
we did not test the Transformer-XL ourselves, so fair comparison is not possible due to differing
experimental setups and the rather sparse result matrix for the Transformer-XL.
4 A NALYSIS
4.1 A BLATION STUDY
The Mogrifier consistently outperformed the LSTM in our experiments. The optimal settings were
similar across all datasets, withr2 f5;6gandk2[40:::90](see AppendixB for a discussion of
hyperparameter sensitivity). In this section, we explore the effect of these hyperparameters and show
that the proposed model is not unnecessarily complicated. To save computation, we tune all models
using a shortened schedule with only 145 epochs instead of 964 and a truncated BPTT window
size of 35 on the word-level PTB dataset, and evaluate using the standard, deterministic dropout
approximation with a tuned softmax temperature.
Fig.3 shows that the number of roundsrgreatly influences the results. Second, we found the low-rank
factorization ofQi andRi to help a bit, but the full-rank variant is close behind which is what we
observed on other datasets, as well. Finally, to verify that the alternating gating scheme is not overly
complicated, we conditionallnewly introduced gates on the original inputsxandhprev (see Fig.2).
That is, instead of Eq.1 and Eq.2 the no-zigzag updates are
xi = 2(Qi hprev )xi2 for odd i2[1:::r];
hi = 2(Ri x)hi2 for even i2[1:::r]:prev prev
In our experiments, the no-zigzag variant underperformed the baseline Mogrifier by a small but
significant margin, and was on par with ther= 2model in Fig.3 suggesting that the Mogrifier’s
iterative refinement scheme does more than simply widen the range of possible gating values ofx
andhprev to(0;2dr=2e )and(0;2br=2c ), respectively.
4.2 C OMPARISON TO THE M LSTM
The Multiplicative LSTM (Krause et al. 2016), or mLSTM for short, is closest to our model in
the literature. It is defined asmLSTM(x;cprev ;hprev ) = LSTM(x;cprev ;hm ), wherehm =prev prev
6 Published as a conference paper at ICLR 2020
LSTM 4 LSTM 1:5 Mogrifier Mogrifier
1 2
0:5
0 0
50 100 150 200 50 100 150 200
(a) 10M model parameters with vocabulary size 1k. (b) 24M model parameters with vocabulary size 10k.
Figure 4: Cross-entropy vs sequence length in the reverse copy task with i.i.d. tokens. Lower is better. The
Mogrifier is better than the LSTM even in this synthetic task with no resemblance to natural language.
(Wmx x)(Wmh hprev ). In this formulation, the differences are readily apparent. First, the mLSTM
allows for multiplicative interaction betweenxandhprev , but it only overrideshprev , while in the
Mogrifier the interaction is two-way, which – as the ablation study showed – is important. Second,
the mLSTM can change not only the magnitude but also the sign of values inhprev , something with
which we experimented in the Mogrifier, but could not get to work. Furthermore, in the definition of
hm , the unsquashed linearities and their elementwise product make the mLSTM more sensitive to prev initialization and unstable during optimization.
On the Enwik8 dataset, we greatly improved on the published results of the mLSTM (Krause et al.
2016). In fact, even our LSTM baseline outperformed the mLSTM by 0.03 bpc. We also conducted
experiments on PTB based on our reimplementation of the mLSTM following the same methodology
as the ablation study and found that the mLSTM did not improve on the LSTM (see Table3).
Krause et al. (2016) posit and verify the recovery hypothesis which says that having just suffered
a large loss, the loss on the next time step will be smaller on average for the mLSTM than for the
LSTM. This was found not to be the case for the Mogrifier. Neither did we observe a significant
change in the gap between the LSTM and the Mogrifier in the tied and untied embeddings settings,
which would be expected if recovery was affected byxandhprev being in different domains.
4.3 T HE REVERSE COPY TASK
Our original motivation for the Mogrifier was to allow the context to amplify salient and attenuate
nuisance features in the input embeddings. We conduct a simple experiment to support this point
of view. Consider the reverse copy task where the network reads an input sequence of tokens and
a marker token after which it has to repeat the input in reverse order. In this simple sequence-to-
sequence learning (Sutskever et al. 2014) setup, the reversal is intended to avoid the minimal time lag
problem (Hochreiter and Schmidhuber 1997), which is not our focus here.
The experimental setup is as follows. For the training set, we generate500000examples by uniformly
sampling a given number of tokens from a vocabulary of size1000. The validation and test sets
are constructed similarly, and contain10000examples. The model consists of an independent,
unidirectional encoder and a decoder, whose total number of parameters is10million. The decoder
is initialized from the last state of the encoder. Since overfitting is not an issue here, no dropout is
necessary, and we only tune the learning rate, the l2 penalty, and the embedding size for the LSTM.
For the Mogrifier, the number of roundsrand the rankkof the low-rank approximation are also
tuned.
We compare the case where both the encoder and decoder are LSTMs to where both are Mogrifiers.
Fig.4a shows that, for sequences of length 50 and 100, both models can solve the task perfectly. At
higher lengths though, the Mogrifier has a considerable advantage. Examining the best hyperparameter
settings found, the embedding/hidden sizes for the LSTM and Mogrifier are 498/787 vs 41/1054 at
150 steps, and 493/790 vs 181/961 at 200 steps. Clearly, the Mogrifier was able to work with a much
smaller embedding size than the LSTM, which is in line with our expectations for a model with a
more flexible interaction between the input and recurrent state. We also conducted experiments with
a larger model and vocabulary size, and found the effect even more pronounced (see Fig.4b).
7 Published as a conference paper at ICLR 2020
4.4 W HAT THE MOGRIFIER IS NOT
The results on the reverse copy task support our hypothesis that input embeddings are enriched by
the Mogrifier architecture, but that cannot be the full explanation as the results of the ablation study
indicate. In the following, we consider a number of hypotheses about where the advantage of the
Mogrifier lies and the experiments that provide evidenceagainstthem.
E Hypothesis: the benefit is in scalingxandhprev .We verified that data dependency is a crucial
feature by adding a learnable scaling factor to the LSTM inputs. We observed no improvement.
Also, at extremely low-rank (less than 5) settings where the amount of information in its gating is
small, the Mogrifier loses its advantage.
E Hypothesis: the benefit is in making optimization easier.We performed experiments with different
optimizers (SGD, RMSProp), with intra-layer batch normalization and layer normalization on
the LSTM gates. While we cannot rule out an effect on optimization difficulty, in all of these
experiments the gap between the LSTM and the Mogrifier was the same.
E Hypothesis: exact tying of embeddings is too constraining, the benefit is in making this rela-
tionship less strict.Experiments conducted with untied embeddings and character-based models
demonstrate improvements of similar magnitude.
E Hypothesis: the benefit is in the low-rank factorization ofQi ;Ri implicitly imposing structure on
the LSTM weight matrices.We observed that the full-rank Mogrifier also performed better than
the plain LSTM. We conducted additional experiments where the LSTM’s gate matrices were
factorized and observed no improvement.
E Hypothesis: the benefit comes from better performance on rare words.The observed advantage
on character-based modelling is harder to explain based on frequency. Also, in the reverse copy
experiments, a large number of tokens were sampled uniformly, so there were no rare words at all.
E Hypothesis: the benefit is specific to the English language.This is directly contradicted by the
Finnish MWC and the reverse copy experiments.
E Hypothesis: the benefit is in handling long-range dependencies better.Experiments in the episodic
setting (i.e. sentence-level language modelling) exhibited the same gap as the non-episodic ones.
E Hypothesis: the scaling up of inputs saturates the downstream LSTM gates.The idea here is that
saturated gates may make states more stable over time. We observed the opposite: the means
of the standard LSTM gates in the Mogrifier were very close between the two models, but their
variance was smaller in the Mogrifier.
5 C ONCLUSIONS AND FUTURE WORK
We presented the Mogrifier LSTM, an extension to the LSTM, with state-of-the-art results on
several language modelling tasks. Our original motivation for this work was that the context-free
representation of input tokens may be a bottleneck in language models and by conditioning the
input embedding on the recurrent state some benefit was indeed derived. While it may be part of the
explanation, this interpretation clearly does not account for the improvements brought by conditioning
the recurrent state on the input and especially the applicability to character-level datasets. Positioning
our work on the Multiplicative RNN line of research offers a more compelling perspective.
To give more credence to this interpretation, in the analysis we highlighted a number of possible
alternative explanations, and ruled them all out to varying degrees. In particular, the connection
to the mLSTM is weaker than expected as the Mogrifier does not exhibit improved recovery (see
Section4.2), and on PTB the mLSTM works only as well as the LSTM. At the same time, the
evidence against easier optimization is weak, and the Mogrifier establishing some kind of sharing
between otherwise independent LSTM weight matrices is a distinct possibility.
Finally, note that as shown by Fig.1 and Eq.1-2, the Mogrifier is a series of preprocessing steps
composed with the LSTM function, but other architectures, such as Mogrifier GRU or Mogrifier
Elman Network are possible. We also leave investigations into other forms of parameterization of
context-dependent transitions for future work.
8 Published as a conference paper at ICLR 2020
ACKNOWLEDGMENTS
We would like to thank Ben Krause for the Transformer-XL dynamic evaluation results, Laura
Rimell, Aida Nematzadeh, Angeliki Lazaridou, Karl Moritz Hermann, Daniel Fried for helping with
experiments, Chris Dyer, Sebastian Ruder and Jack Rae for their valuable feedback.
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APPENDIX A H YPERPARAMETER TUNING RANGES
In all experiments, we tuned hyperparameters using Google Vizier (Golovin et al. 2017). The tuning
ranges are listed in Table4. Obviously,mogrifier_roundsandmogrifier_rankare tuned only for the
Mogrifier. Ifinput_embedding_ratio>1, then the input/output embedding sizes and the hidden
sizes are set to equal and the linear projection from the cell output into the output embeddings space
is omitted. Similarly,mogrifier_rank60is taken to mean full rankQ ,R without factorization.
Since Enwik8 is a much larger dataset, we don’t tuneinput_embedding_ratioand specify tighter
tuning ranges for dropout based on preliminary experiments (see Table5).
Dynamic evaluation hyperparameters were tuned according to Table6. The highest possible value
formax_time_steps, the BPTT window size, was 20 for word, and 50 for character-level tasks. The
batch size for estimating the mean squared gradients over the training data was set to 1024, gradient
clipping was turned off, and the l2 penalty was set to zero.
Table 4: Hyperparameter tuning ranges for all tasks except Enwik8.
Low High Spacing
learning_rate 0.001 0.004 log
input_embedding_ratio 0.0 2.0
l2_penalty 5e-6 1e-3 log
input_dropout 0.0 0.9
inter_layer_dropout 0.0 0.95
state_dropout 0.0 0.8
output_dropout 0.0 0.95
mogrifier_rounds (r) 0 6
mogrifier_rank (k) -20 100
Table 5: Hyperparameter tuning ranges for Enwik8.
Low High Spacing
learning_rate 0.001 0.004 log
l2_penalty 5e-6 1e-3 log
input_dropout 0.0 0.2
inter_layer_dropout 0.0 0.2
state_dropout 0.0 0.25
output_dropout 0.0 0.25
mogrifier_rounds (r) 0 6
mogrifier_rank (k) -20 100
Table 6: Hyperparameter tuning ranges for dynamic evaluation.
Low High Spacing
max_time_steps 1 20/50
dyneval_learning_rate 1e-6 1e-3 log
dyneval_decay_rate 1e-6 1e-2 log
dyneval_epsilon 1e-8 1e-2 log
12 Published as a conference paper at ICLR 2020
APPENDIX B H YPERPARAMETER SENSITIVITY
The parallel coordinate plots in Fig.5 and 6, give a rough idea about hyperparameter sensitivity. The
red lines correspond to hyperparameter combinations closest to the best solution found. To find the
closest combinations, we restricted the range for each hyperparameter separately to about 15% of its
entire tuning range.
For both the LSTM and the Mogrifier, the results are at most 1.2 perplexity points off the best result,
so our results are somewhat insensitive to jitter in the hyperparameters. Still, in this setup, grid search
would require orders of magnitude more trials to find comparable solutions.
On the other hand, the tuner does take advantage of the stochasticity of training, and repeated runs
with the same parameters may be give slightly worse results. To gauge the extent of this effect, on
PTB we estimated the standard deviation in reruns of the LSTM with the best hyperparameters to be
about 0.2 perplexity points, but the mean was about 0.7 perplexity points off the result produced with
the weights saved in best tuning run.
Figure 5: Average per-word validation cross-entropies for hyperparameter combinations in the neighbourhood of
the best solution for a 2-layer LSTM with 24M weights on the Penn Treebank dataset.
Figure 6: Average per-word validation cross-entropies for hyperparameter combinations in the neighbour-
hood of the best solution for a 2-layer Mogrifier LSTM with 24M weights on the Penn Treebank dataset.
feature_mask_rankandfeature_mask_roundsare aliases formogrifier_rankandmogrifier_rounds
.
13