Pushing the bounds of dropout
Authors: Gábor Melis
Charles Blundell
Tomáš Kočiský
Karl Moritz Hermann
Chris Dyer
Phil Blunsom
Venue:  (2020)
Keywords: dropout
language

Abstract: We push on the boundaries of our knowledge about dropout by showing theoretically that dropout training can be understood as performing MAP estimation concurrently for an entire family of conditional models whose objectives are themselves lower bounded by the original dropout objective. This discovery allows us to pick any model from this family after training, which leads to a substantial improvement on regularisation-heavy language modelling. The family includes models that compute a power mean over the sampled dropout masks, and their less stochastic subvariants with tighter and higher lower bounds than the fully stochastic dropout objective. The deterministic subvariant's bound is equal to its objective, and the highest amongst these models. It also exhibits the best model fit in our experiments. Together, these results suggest that the predominant view of deterministic dropout as a good approximation to MC averaging is misleading. Rather, deterministic dropout is the best available approximation to the true objective.

Under review as a conference paper at ICLR 2020
PUSHING THE BOUNDS OF DROPOUT
Anonymous authors Paper under double-blind review
ABSTRACT
We push on the boundaries of our knowledge about dropout by showing theo- retically that dropout training can be understood as performing MAP estimation concurrently for an entire family of conditional models whose objectives are them- selves lower bounded by the usual dropout objective. This discovery allows us to pick any model from this family after training, which leads to substantial improve- ment on regularisation-hen heavy language modelling. The family includes models that compute a power mean over the sampled dropout masks, and their less stochastic subvariants with tighter and higher lower bounds than the fully stochastic dropout objective. The deterministic subvariant's bound equal to its objective, and the highest amongst these models. It also exhibits the best model fit in experiments. Together, these results suggest that the predominant view of deterministic dropout good approximation to MC averaging is misleading. Rather, deterministic dropout is the best available approximation to the true objective.
INTRODUCTION
The regularisation technique known dropout underpins numerous state-of-the-art results in deep learning (Hinton et al. 2012; Srivastava et al. 2014), and its application has received much attention in the form of optimisation (Wang& & Manning 2013) and attempts explaining improving its approximation properties (Baldi & Sadowski 2013; Zolna et al. 2017: Ma et al. 2016). The dominant perspective today views dropout as either an implicit ensemble method (Warde-Farley et al. 2013) or averaging over an approximate Bayesian posterior (Gal & Ghahramani 201 Regardless of which view we take, dropout training is carried out the same way, by minimising the expectation of the loss over randomly sampled dropout masks. However, at test time these views naturally lead to different algorithms: the Bayesian approach computes an arithmetic average as marginalises out the weight uncertainty, while the ensemble approach typically uses the geometric average due to its close relationship to the loss. Collectively they are called MC dropout and neither is clearly better than the other (Warde-Farle et al. 2013). A third way make predictions is to "turn dropout off", that is, propagate expected values through the network in a single, deterministic pass. This deterministic (also known standard) dropout is considered to be an excellent approximation to MC dropout.
This situation is unsatisfactory as it does not provide theoretical grounding for dropout, without which the choice of dropout variant remains arbitrary. In this paper, we provide such theoretical foundations. First, we prove the dropout objective to be a common lower bound on the objectives of a family of infinitely many models. This family includes models corresponding to the three aforementioned methods of evaluation: the arithmetic averaging, the geometric averaging, and the deterministic. Thus by maximising the dropout objective we get single set of parameters and many models that all have the same parameters but differ in how they make predictions. This allows us to train once and perform model selection at validation time by evaluating the different methods of making predictions corresponding to individual models in the family. Second, we turn the conventional perspective on its head by showing that while dropout training performs stochastic regularisation, the trained model best viewed as deterministic, not stochastic model with deterministic approximation.
This paper is structured as follows. In $2, we revisit variational dropout (Gal & Ghahramani 2016a) and demonstrate that, despite common perception, sharing of masks is not necessary, neither in theory nor in practice. Then, by recasting dropout in a simple conditional form, we highlight the counterintuitive role played by the variational posterior. $3 contains construct a family of conditional models whose MAP objectives are all lower bounded by the usual dropout objective, and identify a member of this family as best in terms of model fit. In $4, select
main contributions. Here

---

Under review as a conference paper at ICLR 2020
the best of this family in terms of generalisation improve language modelling. Finally, creating cheap approximation to the bias of this model allows us to get better results from model tuning.
VARIATIONAL DROPOUT
Since its original publication (Hinton et al. 2012), dropout had been considered a stochastic regularisa- tion method, implemented tweak to the loss function. That was until Gal & Ghahramani (2016a) grounded dropout in much-needed theory. Their subsequent work (Gal & Ghahramani 2016b) fo- cused on RNNs, showing that if dropout masks shared between time steps, the objective for their proposed variational model is the same as the commonly used dropout objective with an &2 penalty. Their method became known variational dropout, not be confused with Kingma al. (2015), and is used in state-of-the-art sequential models (Merity et al. 2017; Melis et al. 2017). Before we move on more general formulation revisit it to better understand its critical features.
First, we recall the derivation of variational dropout. Consider an RNN that takes input æ, maps it to output y and trained on a of N data points paired sets X, Y. variational lower bound on the log likelihood is obtained as follows:
In [yp(Y In ((Y(X,))p(w)
(
g(w)
E
[n KL(g(W)lp(w))
wogg(w)
((w) (w) KL(g(w))|p(w))
((w) In wdd KL(g(W)] p(w)),
where p(y|a, w) is defined by the RNN with weights w. Variational Bayesian methods then maximise this lower bound with respect to the variational distribution q(w). For variational dropout, ((w) takes the form of mixture of two gaussians with small variances: one with zero mean that represents the dropped out rows of weights, and another with mean O:
q(w+) |0 1 + (1 (),(
In the above, r is the index of a row of a weight matrix. Dropping whole rows of weights equivalent to the more familiar view of dropout over units. The prior over the weights is
gaussian:
p(w) = (w=(0, o
The loss is defined based on Eq. 1. The integrals are approximated using single sample W ~ q(w), and the KL term is approximated with weight decay on O:
N
L L=
Σ In Imp(u - In (())
(2)
This loss coincides with the usual dropout objective (Srivastava et al. 2014). Although using the same dropout masks for all time steps considered defining characteristic of variational dropout, we note in passing that the theory for the non-shared masks case is very similar and there is little between them in practice with LSTMs (see Appendix A). With this we conclude the recap of variational dropout, and describe our contributions in the rest of the paper.
2.1 DROPOUT AS A CONDITIONAL MODEL
There are questions, though, surrounding the suitability of variational inference in the above formula- tion which we will briefly cover in this section. Then, to avoid getting bogged down in these issues and ease interpretation, we construct a straightforward conditional model and lower bound its MAP objective in the same form the variational objective.
In variational inference the idea to approximate the intractable and complicated posterior with simple, parameterised distribution q. Crucially, this approximation affects inferences and
2

---

Under review as a conference paper at ICLR 2020
predictions. If we are serious about it being approximation to the posterior and want to reduce its distortion of the model p. then 9 can be made more flexible. But making q more flexible in variational dropout potentially ruin the regularisation effect. So the particular choice of q plays important, active role: it effectively performs posterior regularisation and acts as an integral part of the model.
Coming from another angle, Osband (2016) makes the point that in variational dropout the posterior over weights does not concentrate with more data, unlike for example in Graves (2011) 1), which is unexpected behaviour from Bayesian model. This conundrum caused by encoding dropout with fixed rate mixture of fixed variance components in which also necessitates expensive tuning of the dropout rate. Gal al. (2017) proposes a way address these shortcomings.
Resolving these issues is not the focus of the present paper, so we construct conditional model with the same lower bound. Suppose we want do MAP estimation for the model parameters (the means of the distribution of weights, θ): arg maxe p(O Y). Consider conditional model p(Y|X, b) as a crippled generative model with p(a;) constant, 2; and è independent. Place normal prior on the means O and otherwise make the weights w conditional on θ the same way they were in the variational posterior g(w):
(()) = NN@e0, of
pw (0) = pu(w,110 10, 2 (1 p)W ,r, Or. plu, w w/r, Q) = p(y|æ, w)
(3)
The log posterior of this model has similar lower bound to the variational objective (Eq. I):
N
In Inp(o|X,y) Y) >
Σ p(w(O) In p(us lo, wdd + In p(O) - CMAP
(4)
See Appendix C for detailed derivation. Dropping the normalisation constant CMAP that doesn't 't depend on θ, and approximating the above integrals with a single sample, the loss corresponding to the MAP objective becomes:
N
LMAP =
Σ p(p a.) In p(e)
(5)
With this loss being identical to that of variational dropout (Eq 2) and the usual dropout objective, in the following we focus on MAP estimation for the conditional model to sidestep any questions about whether variational inference makes sense in this case and move to investigating the properties of this bound.
THE DROPOUT FAMILY OF MODELS
Having developed a conditional model for dropout that leads to the same objective as variational dropout, we now derive family of models whose objectives are all lower bounded by the usual dropout objective. We draw inspiration from the different evaluation methods employed for dropout:
Deterministic dropout propagates the expectation of each unit through the network in single pass. This is very efficient and is viewed as a good approximation to the next option.
MC dropout mimicks the training procedure, and averages the predicted probabilities over randomly sampled dropout masks. With one forward pass per sample, this can be rather expensive. There is some ambiguity as what kind of averaging shall be applied: oftentimes the geometric average (GMC) is used, because of its close relationship to the loss, but use of the arithmetic average (AMC) is also widespread.
Our goal in this section is to demonstrate the consequences of optimising a lower bound instead of the true objective. While is easy to argue in general that objectives of more than one model may share any given lower bound, for dropout particularly simple explicit construction exists for such family of models. As we will show, this allows for post-training model selection based on validation results given trained set of parameters, while in the absence of validation results to guide model selection, inspection of the tightness the lower bound indicates the deterministic model as the most reasonable choice from the family.
3

---

Under review as a conference paper at ICLR 2020
3.1 GEOMETRIC MODEL
First, we investigate whether the geometric or the arithmetic mean is the correct choice for making predictions in the context of classification. Recall the predictive term of the MAP loss in Eq.5: 5: pups (M, w.). Notice how with SGD and multiple epochs, for each data point several dropout masks are encountered, and the approximating quantity is the geometric mean of the predicted probabilities pluila, @y) over the masked weights. For this reason, the posterior predictive distri- bution p(y 1=" X, Y) often computed as the renormalised geometric mean. This is in apparent conflict with the conditional model that prescribes the arithmetic mean (integrating w out of Eq. 3). However, we can define another model where the conditional distribution is directly defined to be renormalised geometric mean
p(yl +) = exp( (()-(())) ltpppp/llllppppppa..l Inplylz, Z θ)
Z(x, +)
E exp
E amp((a1)
Imp(c/ar,)) w))
(6)
with a slight abuse of notation, due to using the symbol in plyla, Q) although plylæ, Q) o Ew (z))))((ylz..))) w). It can be shown that the arithmetic model's (Eq. 3) lower bound (Eq. 4) is a lower bound for this renormalised geometric model (Eq. 6). as well. See Appendix D for the derivation. The answer to the question whether we should use GMC AMC that it depends: they correspond to different models, but the dropout objective is lower bound on the objectives of both models. So one can freely choose between GMC and AMC evaluation time, doing model selection retrospectively after training.
3.2 THE POWER MEAN MODEL FAMILY
Having two models to choose from, it is natural to ask whether these are just instantiations of larger class of models. We propose the power mean family of models to extend the set of models to continuum between the geometric and arithmetic models described in $3.1 and $2.1, respectively, and show that they have the same lower bound. The power mean is defined
1/a
Ma (Z
In
For a 1 we arrive at the arithmetic mean while the natural extension to a 0 the geometric mean as it is the limit of M,, at -> 0, which can be proven with I. L''Hôpital's 's rule. Similarly to the construction of the geometric model, we define the power mean model by directly conditioning on O:
plylar θ) \Barp(ule) P(ylæ,
7x, O) Σ
E
p(c|,, w)
(7)
7(x, Q)
where (xx, +) is at most 1 if (-oo, 1] because Ma is monotonically increasing in and Z is 1 for 1. Here we provide concise derivation of a lower bound on the log posterior (the full derivation can be found in Appendix E):
In (np(O)X,
Σ
In
(ant))))-nn((aar ()) In In(Z(x;, 01) -lnp(() CMAP
In A p(uiizr, (n) CMAP
(8)
E Inplu/z, ~0 + Inp(() - CMAP
(9)
Σ p(w|O) Infllee,w) pm)( CMAP
The first inequality above follows from Z(z, +) < for all θ, while the second is application of Jensen's 's rule assuming a > 0. We arrived at the same lower bound on the objective as we had for the geometric (Eq. 6) and arithmetic models (Eq. 3), thus defining the power mean family with parameter a e [o, 1] of models from which we can choose at evaluation time. For a > 1, the normalising constant Z would be greater than and this would not be lower bound in general.
4

---

Under review as a conference paper at ICLR 2020
3.3 TIGHTNESS OF THE LOWER BOUND
To better understand the quality of fit for models in the power mean family we examine the tightness of their lower bounds. There are two steps involving inequalities in the derivation of the bound: one where the normalisation constant Z is dropped (Eq. 8) and another where the logarithm is moved inside the expectation (Eq. 9). We show that the gaps introduced by these steps can be made arbitrarily small by reducing the variance of ply|2, w) with respect to w.
Notice that the Jensen gap with the logarithm function is scale invariant:
In (ELL) - En = In(EE) - Em((L)
Intuitively, this suggests that var( var(L.)/C (L. L) is closely related to the size of the gap. Indeed, Maddison et al. (2017) show that if the first inverse moment of 7. is finite, then
var(L) 2)
In(E (L))
O(VE[(L- L)])
(10)
Here we go a bit further and show that if there is a positive lower and upper bound on L. then there non-trivial lower and upper bounds its Jensen gap and these are bounds are multiplicative in var(I), Let L be a random variable such that PlL E (a, b)) = where -00 & a < b & co. Furthermore, let (I) be a convex function. Jensen's inequality states that [L)( > ((((T)). Liao & Berg (2017) show that the Jensen gap [[p( (PB[4] can be bounded from below and above:
(h):k() (a,b), b)} vLr(L) < [[[[[]] <p(B[[]) & sup((((;s)) LE (a, var(L)
k(1; (0() ((()
('(m)
14)²
7-p
where h(l; μi) does not depend on the distribution of L, only on its expected value μ and on the function yp. Substituting L = p(c|z,,w) w) (a random variable on [0, due to the randomness of the dropout masks) and (dl) = - In()), we can see that the gap introduced by Eq. 9 can be made smaller by decreasing the variance of the predictions while maintaining the expected value of L (i.e. the expected probability), assuming that there is a positive lower and upper bound on them (so that the supremum is finite and the infimum is positive, respectively). A similar argument based shows that approximately monotonically approaches as the variance decreases, so the gap of Eq. 8 can also be reduced.
on Ee Ma (E& p(c|2, ())
Suppose we pick a base model from the power mean family and have continuum of subvariants with gradually reduced variance in their predictions but the same expectation. Clearly, for each of them we can derive lower bound the same way as we did for the power mean family. And as we showed above, the lower bounds will tend to increase the variance of the predictions decreases (see Fig. la). They do not strictly increase, only tend to, due to how the Jensen gap is bounded from above and below and also due the O term of Eq. 10. Nonetheless, bound is forced into increasingly tighter ranges with strictly monotonically increasing bounds around it, thus we can always reduce the variance such that there is no overlap between the ranges and we get a guaranteed improvement on the lower bound. This effect reaches its apex at the deterministic model whose lower bound is both exact and higher than any other model's. s. Fig. 1b illustrates that regardless of the choice of base model, reducing the prediction variance will eventually transform it into the same deterministic model.
approach determinism the lower
3.4 THE EXTENDED POWER MEAN FAMILY: CONTROLLING THE TIGHTNESS OF THE BOUND
Intuitively, in the absence of other sources of stochasticity the dropout rate controls the variance of the predictions and it is low, the lower bound can be pretty snug. However, there are two problems.
First, decreasing the dropout rate does not necessarily keep the expectation of the predictions the same. We offer no solution to this bias issue, but refer the reader to previous studies of dropout's approximation properties such as (Baldi & Sadowski 2013) and our subsequent empirical results.
Second, reducing the dropout rate would trade off generalisation for tighter bounds. But doing so only at evaluation time leaves the training time regularisation effect intact, and can be seen as picking another model whose lower bound tends to be higher than that of the base model. Having thus extended the dropout family further,
now tweak both a and dropout rates at evaluation time.
5

---

Under review as a conference paper at ICLR 2020
Geometric
tightness
tightness
Determin
Base model
Arithmetic
variance
variance
(a) Lower and upper bounds on the lower bound of the model objective as a function of prediction variance for any model in the power mean family.
(b) With reduced variance, all members of the power mean family (brown shading) converge to the determin- istic model while their lower bounds tighten.
Figure I:
Tightness of lower bounds vs evaluation time prediction variance in the extended dropout family.
Table 1:
PTB training XEs with various dropout rate multipliers between deterministic and GMC. Observe the monotonic improvement in training fit when reducing the dropout evaluation only.
x0.0
x0.1
x0.2
x0.3
x0.4
x0.5
x0.6
x0.7
x0.8
x0.9
x 1.0
2.731
2.738
2.746
2.755
2.766
2.777
2.791
2.807
2.826
2.849
2.878
Depending on the severity of the introduced bias compared to the benefits of having tighter lower bound, the optimal variance may lie anywhere between the deterministic and the base model. We show experimentally that across number of datasets the benefits of tighter bounds matter more, and observe monotonic improvement in model fit as evaluation time dropout rates are decreased all the way to full determinism. The experiment the dropout rate was multiplied by À E [0, 1]. As Table shows, the model fit as measured by cross entropy (XE) the training improves monotonically when reducing A. Results other datasets and with other power mean models are very similar. We call the union of the reduced dropout rate subvariants of all power mean family models the extended dropout family paramaterised by a, A.
conducted follows. On an already trained model,
Based on our experimental results, we can say that dropout training optimises deterministic model subject to regularisation constraints, and deterministic evaluation, widely believed to approximate MC evaluation, is the closest match to the true objective at our disposal. It is not that dropout evaluation has a deterministic approximation: dropout trains a deterministic model first and foremost and continuum of stochastic ones to various extents.
In summary, we described dropout training as optimising a common lower bound for family of models. Since this lower bound is the same for all models in the family, them at evaluation time. However, the tightness of the bound varies, which affects model fit. Having trained model with dropout, the best fit is achieved by the deterministic model with no dropout. This result isolates the regularisation effects from the biases of the lower bound and the dropout family.
nominate any of
4 APPLYING DROPOUT
We investigate how members of the extended dropout model family perform in terms generalisation. We follow the experimental setup of Melis et al. (2017) and base our work on their best performing model variant for each dataset. Unless explicitly stated, no retraining was performed and their model weights were reused. In the experiments with the tuning objective, we follow their experimental setup, using Google Vizier (Golovin et al. 2017), a black-box hyperparameter tuner based on batched Gaussian Process Bandits.
See Table for results of image classification on MNIST, character based language modelling on Enwik8, word based language modelling PTB and Wikitext-2. On MNIST, deterministic dropout is the best in terms of cross entropy. which matches our theoretical predictions. contrast, on language modelling arithmetic averaging produces the best results, which necessitates further analysis.

---

Under review as a conference paper at ICLR 2020
Table Validation XEs datasets varying the power and the dropout rate mulitiplier A. Deterministic dropout is not the best evaluation method for the language modelling datasets due to simple smoothing effect.
Geometric (ar
0)
Power = 0.5
Arithmetic (a
1)
Dataset
DET
x0.8
x0.9
x1.0
x0.8
x0.9
x1.0
x0.8
x0.9
x 1.0
MNIST
0.070
0.087
0.087
0.088
0.92
0.93
0.93
0.100
0.100
0.100
Enwik8
0.886
0.879
0.878
0.881
0.877
0.877
0.877
0.875
0.875
0.875
PTB 4.110
4.090
4.090
4.093
4.072
4.070
4.073
4.061
4.064
4.080
Wikitext-2
4.236
4.229
4.231
4.235
4.025
4.026
4.208
4.203
4.212
4.228
Table 3: PTB training and validation XEs for AMC at À {0, 0.8, 1) per word frequency. Note how DET dominates AMC on the training set, AMC better for rare words the validation set.
training
validation
frequency
targets
DET
x0.8
AMC
DET
x0.8
AMC
25000
13580
1.40
1.50
1.56
1.58
1.64
1.68
5000
26658
1.65
1.75
1.81
1.93
1.98
2.02
500<
44702
2.19
2.30
2.36
2.58
2.63
2.66
<500
29058
4.07
4.19
4.29
6.49
6.39
6.39
<100
14222
4.24
4.38
4.49
7.81
7.64
7.61
<20
5008
4.00
4.19
4.33
9.20
9.01
8.97
We suspected that the particularly severe form of class imbalance exhibited by the power-law word distribution (Zipf 1935) might play a role. To verify this, we contrasted training and validation XEs on PTB for words grouped by frequency (see Table 3). On the training set, the gap between deterministic dropout and AMC is wider for low frequency words. On the validation set, AMC is worse for frequent words but better for rare words. The x.88 dropout multiplier just finds reasonable compromise.
4.1 SOFTMAX TEMPERATURE
The observed effect is consistent with smoothing, thus posit that the reason MNIST results worse with AMC is that the marginal distributions of labels in the training and test set are identical by construction and further smoothing is unnecessary. On the other hand, PTB and Wikitext-2 benefit from AMC's smoothing because the penalty for underestimating low probabilities is harsh, hence the large improvement on rare words. The character based Enwik8 dataset lies somewhere in between: the training and test distributions are better matched and there are no very low probability characters.
To the hypothesis that AMC's advantage lies in smoothing, we tested how performing smoothing by other means affects the results. In this experiment, on a trained model the temperature of the final
Table Validation and test perplexities on PTB and Wikitext-2 with various evaluation strategics and default or optimal validation softmax temperatures. Our bascline results correspond DET at temperature Note that AMC does not benefit from setting the optimal softmax temperature ("opt"), while DET is improved by it almost to the point of matching AMC which supports the smoothing hypothesis.
Geometric (a = 0)
Power & = 0.5
Arithmetic (a = 1)
Dataset
Temp
DET
x0.8
x0.9
x 1..0
x0.8
x0.9
x0.8
x0.9
x1.0
Validation
WT-2
69.1
68.6
68.8
69.1
67.0
67.2
67.2
66.9
67.5
68.6
opt
67.4
67.5
67.7
68.0
67.0
67.1
67.2
66.9
67.4
68.1
60.9
59.6
59.7
59.7
58.1
$7.9
58.0
$7.3
57.5
58.5
PTB
opt
57.5
57.5
57.9
58.3
57.1
57.3
57.8
57.1
57.5
58.4
65.9
65.3
65.4
65.6
63.8
63.9
64.2
63.7
64.5
65.5
WT-2
Test
opt
64.5
64.7
64.8
64.9
63.8
63.8
64.2
63.7
64.2
64.9
58.6
57.3
57.4
57.4
56.0
55.8
55.9
55.3
55.5
56.5
PTB
opt
56.0
56.0
56.
56.5
55.7
55.7
56.0
55.3
55.5
56.3

---

Under review as a conference paper at ICLR 2020
softmax was optimised on the validation set and the model was applied with the optimal temperature to the validation and test sets. Our experimental results in Table 4 support the hypotheses that AMC smooths the predicted distribution increasing the temperature improves DET and GMC considerably but not AMC. In fact, the optimal temperature for AMC with = was slightly lower than 1, which corresponds to sharpening, not smoothing.
Tuning the evaluation time softmax temperature is similar to label smoothing (Pereyra et al. 2017), the main difference being that our method does not affect training. While this convenient, for tuning model hyperparameters, ideally we would determine the optimal evaluation parameters a, and the temperature for the calculation of the validation score for each set of hyperparameters tried, but this would be prohibitively expensive. Since deterministic evaluation coupled with the optimal temperature very close to the best performing AMC model, it serves as good proxy for the ideal tuning objective. The optimal temperature can be approximately determinined using a linear search on subset of the validation data which is orders of magnitude faster than MC dropout. In our experiments, hyperparameter tuning with validation scores computed at the optimal softmax temperature did improve results, albeit very slightly (about half a perplexity point). Thus we can conclude that deterministic dropout is already reasonable proxy for which to optimise.
4.2 RESULTS
We have improved the best test result of Melis et al. (2017) from 58.3 to 55.7 on PTB, and from 65.9 to 63.7 on Wikitext-2 using their model weights, only tuning the evaluation parameters a, and the softmax temperature on the validation set. By retuning the hyperparameters of the PTB model with optimal temperature deterministic evaluation, we improved to 55.3 on PTB. For lack of resources, we did not retune for Wikitext-2. For comparison, the state of the art in language modelling without resorting to dynamic evaluation or a continuous cache pointer is Mixture of Softmaxes (Yang et al. 2017) with 54.44 and 61.45 on PTB and Wikitext-2, respectively. At present, it is unclear whether the benefits of their approach and ours combine.
In summary, looked at how different models and evaluation methods rank in terms of generalisation. Across a number of tasks and datasets the ranking differed from what was observed on the training set. We found that AMC smooths the distribution of the prediction probabilities and we achieved similar effect without resorting to expensive sampling simply by adjusting the temperature of the final softmax. Finally, we brought the tuning objective more in line with the improved evaluation by automatically determining the optimal softmax temperature when evaluating on the validation set which further improved results.
IMPLICATIONS
The construction of conditional model family with a common lower bound on their objectives is applicable to other latent variable models with similar structure and inference method. This lower bound admits ambiguity as to what model is being fit to the data, which in turn allows for picking any such model evaluation time. However, the tightness of the bound and the quality of the fit varies. For dropout, the deterministic model has the best fit even though the training objective is highly stochastic, but this result hinges on the approximation properties of deterministic dropout and will not carry over to other probabilistic models in general. In particular, standard VAEs (Kingma & Welling 2013) with their lower bound being very similar in construction to Eq. cannot quite collapse to deterministic model else they suffer an infinite KL penalty. Still, the lower bound being looser on the tails of q is related to problem of underestimating posterior uncertainty (Turner & Sahani 2011), 1).
In related works, expectation- -inear dropout (Ma et al. 2016) and fraternal dropout (Zolna et al. 2017) both try to reduce the "inference gap"" the mismatch between the training objective and deterministic evaluation. The gains reported in those works might be explained by reducing the bias of deterministic evaluation and also by encouraging small variance in the predictions and thus getting tighter bounds. Another recent work, activation regularisation (Merity et al. 2017), could be thought of as a mechanism to reduce the variance of predictions to a similar effect. In the context of language modelling, the connection between noise and smoothing established by Xie et al. (2017). Our improved understanding further emphasises that connection, and at the same time challenges the way we think about dropout.
8

---

Under review as a conference paper at ICLR 2020
REFERENCES
Pierre Baldi and Peter J Sadowski. Understanding dropout. In Advances in neural information processing systems, pp. 2814-2822, 2013.
Justin Bayer, Christian Osendorfer, Daniela Korhammer, Nutan Chen, Sebastian Urban, and Patrick van der Smagt. On fast dropout and its applicability to recurrent networks. arXiv preprint arXiv: 131 1,0701, 2013.
Yarin Gal and Zoubin Ghahramani. Dropout as a bayesian approximation: Representing model uncertainty deep learning. In international conference on machine learning, pp. 1050-1059,
201
Yarin Gal and Zoubin Ghahramani. A theoretically grounded application of dropout in recurrent neural networks. In Advances in Newral Information Processing Systems, pp. 1019-1027, 2016b.
Yarin Gal, Jiri Hron, and Alex Kendall. Concrete dropout. In Advances in Neural Information Processing Systems, pp. 3584-3593, 2017.
Daniel Golovin, Benjamin Solnik, Subhodeep Moitra, Greg Kochanski, John Karro, and D Sculley. Google vizier: service for black-box optimization. In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1487-1495. ACM, 2017.
Alex Graves. Practical variational inference for neural networks. In Advances in Neural Information Processing Systems, pp. 2348-2356, 2011.
Geoffrey E Hinton, Nitish Srivastava, Alex Krizhevsky, Ilya Sutskever, and Ruslan R Salakhutdinov. Improving neural networks by preventing co-adaptation of feature detectors. arXiv preprint 1rXiv:1207.0580, 2012.
Diederik P Kingma and Max Welling. Auto-encoding variational bayes. arXiv:1312.6114 2013.
arXiv preprint
Diederik P Kingma, Tim Salimans, and Max Welling, Variational dropout and the local reparameteri- zation trick. In Advances in Neural Information Processing Systems, pp. 2575-2583, 2015.
JG Liao and Arthur Berg. Sharpening jensen's inequality. The American Statistician, 2017.
Xuezhe Ma. Yingkai Gao, Zhiting Hu, Yaoliang Yu, Yuntian Deng, and Eduard H. Hovy. Dropout with expectation-linear regularization. CoRR, htss/1609.00011 2016. URL http://arxiv. /109. 08017.
Chris J Maddison, John Lawson, George Tucker, Nicolas Heess, Mohammad Norouzi, Andriy Mnih, Amaud Doucet, and Yee Teh. Filtering variational objectives. In Advances in Neural Information Processing Systems, pp.
2017.
Gábor Melis, Chris Dyer, and Phil Blunsom. On the state of the art of evaluation in neural language models. arXiv preprint 1rXiv:1707.05589 2017.
Stephen Merity, Nitish Shirish Keskar, and Richard Socher. Regularizing and optimizing Istm language models. arXiv preprint arXiv: 700.002188 2017.
Ian Osband. Risk versus uncertainty in deep learning: Bayes, bootstrap and the dangers of dropout. In NIPS Bayesian Deep Learning Workshop, 2016.
Marius Pachitariu and Maneesh Sahani. Regularization and nonlincarities for neural language models: when they needed? arXiv preprint arXiv: 1301.5650, 2013.
Gabriel Pereyra, George Tucker, Jan Chorowski, Eukasz Kaiser, and Geoffrey Hinton. Regularizing neural networks by penalizing confident output distributions. arXiv preprint arXiv: /701. 1765486588 2017.
Stanislau Semeniuta, Aliaksei Severyn, and Erhardt Barth. Recurrent dropout without memory loss.
arXiv preprint arXiv: 1603.000 18, 2016.

---

Under review as a conference paper at ICLR 2020
Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: A simple way to prevent neural networks from overfitting. The Journal of Machine Learning Research, 15(1):1929-199) 2014.
Richard E Turner and Maneesh Sahani. Two problems with variational expectation maximisation for time-series models. Bayesian Time series models, 1(3.1):3-1, 2011. 2011
Sida Wang and Christopher Manning. Fast dropout training. In international conference on machine learning, pp. 118 118-126, 2013.
David Warde-Farley, lan J Goodfellow, Aaron Courville, and Yoshua Bengio. An empirical analysis of dropout in piecewise linear networks. arXiv preprint arXiv: 1312.6197, 2013.
Ziang Xie, Sida I Wang, Jiwei Li, Daniel Lévy, Aiming Nie, Dan Jurafsky, and Andrew Ng. Data noising as smoothing in neural network language models. arXiv preprint arXiv:/703.02573. 2017.
Zhilin Yang, Zihang Dai, Ruslan Salakhutdinov, and William W Cohen. Breaking the softmax bottleneck: high-rank rnn language model. arXiv preprint arXiv: 2017.
Wojciech Zaremba, Ilya Sutskever, and Oriol Vinyals. Recurrent neural network regularization. arXiv preprint 1409 2329, 2014.
George Kingsley Zipf. The psycho-biology of language. Houghton, Mifflin, 1935.
Konrad Zolna, Devansh Arpit, Dendi Suhubdy, and Yoshua Bengio. Fraternal dropout. arXiv preprint arXiv:1711.0006,, 2017.
10

---

Under review as a conference paper at ICLR 2020
APPENDIX A VARIATIONAL DROPOUT WITH NON-SHARED MASKS
If q and p are redefined for the non-shared setting be products of identical and independent, per time step factors, neither term of the variational objective requires rethinking: the MC approximation still works since q easy to sample from, while the KL term becomes a sum of componentwise KL divergences and can still be implemented as weight decay. Consequently, both shared and non-shared masks fit into the variational framework. For detailed derivation see Appendix B.
In related works, Pachitariu & Sahani (2013) in their investigation of regularisation of standard RNN based language models dismiss applying dropout to recurrent connections "to avoid introducing instabilities into the recurrent part of the LMs". Bayer et al. (2013) echo this claim about RNNs, which is then cited by Zaremba et al. (2014), but their work is based on LSTMs not standard RNNs. Finally, Gal & Ghahramani (2016b) cite all of the above but also work with LSTMs. Their results indicate a large, about 15 perplexity point advantage to shared mask dropout for language modelling on the Penn Treebank (PTB) corpus (see Fig. 2 in their paper).
Our experimental results obtained with careful and extensive hyperparameter tuning, listed in Table indicate only a small difference between the two which is in agreement with the empirical study of Semeniuta et al. (2016).
Table 5: Validation and test set perplexities on PTB with shared (S) or non-shared (NS) dropout masks for a small, layer and large, 4 layer LSTM with 10 and 24 million weights, respectively. Non-shared masks perform nearly well shared masks and as have seen neither "more variational" than the other.
dataset
10M
24M
NS
NS
validation
59.4
60.2
57.5
58.3
test
57.5
58.6
56.0
56.9
In any case, non-shared masks, in addition to being variational, are also surprisingly competitive with shared masks for LSTMs (we make no claims about standard RNNs). We also tested whether embedding dropout (in which dropout is applied to entire vectors in the input embedding lookup table) proposed by Gal & Ghahramani (2016b) improves results, and find that embedding dropout does not offer any improvement on top of input dropout.
APPENDIX B DERIVATION OF VARIATIONAL DROPOUT WITH NON-SHARED MASKS
In this section, we formulate naive (i.e. non-shared mask) dropout in the variational setting. In contrast to the shared mask case, where was single set of weights, here wil (or w, for short) has a set of weights for each time step that differ in their dropout masks. The variational posterior g(w) and the prior p(w) are both products of identical distributions over time:
T
II ('(w")
II
(1- (
t=1
P(
II p' (w")
An unbiased approximation to the integrals in Eq. 1 is based on a single, easy to obtain sample is g(w):
I g(w) In phyly, dww In (()la,
Showing that the KL term can still be approximated with weight decay with non-shared masks is not much more involved. Both distributions are products of densities over independent random variables,
11

---

Under review as a conference paper at ICLR 2020
so the componentwise KL divergencies sum. In particular:
(T(((W))ll(w) = - [I
d'ww)) (
T In (ww))
Hr (ww))) de
In of ('w') dws pl(ut)
((()))
In ((w²t) dw p((wt)
(w+t)
((w)) (c(w() In {(w')]
p((wt] du
Σ II iliigt
[f ('(w) de) pl(w)
= T - KL(g (w))((())))
We partitioned the variables into two mutually exclusive sets w' and its complement w and split the multiple integral using Fubini's theorem (or, equivalently, using the expectation of independent random variables rule). After the split, the first integral is trivially 1 and the second has no dependence on T.
What we end up with is sum of identical KL terms of the same distributions as in the shared mask case, so the full KI. can be approximated with weight decay.
APPENDIX C DERIVATION OF THE MAP LOWER BOUND FOR THE ARITHMETIC MODEL
We can rewrite the posterior as:
,OIX, p(X,))
X p(x, (10)p(0)
= / | x)) X)
/ (XX11x))))())))0000
Moving to the log domain and using Jensen's 's inequality allows us to construct lower bound that is sum of per data point terms (i.e. something that can be conveniently optimised):
Inp(O)X,,)) = In f ((((xx.))(())))((()) - -MMAP
N II (i In p(e) CMAP
=In [nm((w p(wl))
i=1 N
> / p(w|O) In II
p(u. w)w + In p(0) -CMAP
In In Wiw + Inp(@) CMAP
12

---

Under review as a conference paper at ICLR 2020
APPENDIX D DERIVATION OF THE MAP LOWER BOUND FOR THE GEOMETRIC
MODEL
From Eq. 6 recall that:
p(y Q) exp (Banp(w)e) 0) Inp(y/a, @))
[ 6)
The normalisation constant Z is most due to the geometrie mean being bounded from above by the arithmetic mean on per class c basis:
7 0)
Σ
exp(
E Inp(cla, 2))
Dmpp( (+) ())
Σ p(ck2, ()) =
(mp(w(e) ()
Since this conditional model, we can rewrite the posterior as:
p(O)x.Y)) ((X)p)))p(O)
p(X,Y)
x ((0))))))(()) = PY|X, ()p(( PY||,, ())(()
p(X(0) is dropped in the last step it constant. Moving to the log domain once again:
Inp(O)X,Y) = lny(() 9) + Imp(e) CMAP
In p(ul,,,,)) In OMAP
E Imp(ys/=;, - In(2(x;, 61) Imp(()) CMAP
@rup(w(0)
E w
Inplu.z,,4) + Imp(()) -MAP
= Σ / p(w|0) In ww + In ((e)
CMAP
where the lower bound arises due Vi: ((z +) < 1.
APPENDIX E DERIVATION OF THE MAP LOWER BOUND FOR THE POWER MEAN FAMILY
In $3.2 we proved that Vi: Z(x;, θ) < Starting from p()]X,Y) x pY O)p(O) just like in the geometric case, we derive a lower bound in the log domain:
In (() p +) + p(e)
CMAP
In p(y pr, θ) + In (e) CMAP
N
In
p(yiles, 4) - In(2 0))
+ +lp((e) - -MAP
@nmp(=/e)
13

---

Under review as a conference paper at ICLR 2020
In
p(lile,, + Inp(() -CMAP
@mppwwee
In
E
p())) lmp -MAP
(nmp(w/e)
inp(yl2z,,)) + Inp(e) -MAP
a (nplleee)
Inp(vrle, + Inp( CMAP
@mp(()))
p(w|O) Inp(ul + In (()) -MAP
14


Reviews:
[{"review_id": "rJlsryRpFB", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "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": "Authors consider dropout as MAP for conditional model and consider different types of averaging to obtain predictive distribution p(y|x, theta) during inference stage. The paper proposes power mean family and shows that well-known types of MC averaging (arithmetic and geometric) are particular cases of proposed family. Authors show that power mean family objective is lower-bounded by the original dropout lower-bound. Therefore it is consistent to use original dropout on the training stage and do any kind of averaging from power mean family during inference stage.\n\nConcerns:\n1) In general, paper is hard to follow and main motivation of the accomplished work is not clearly stated in the paper.\n2) One of the most confusing things about this paper is the analysis of the lower bound tightness. The authors state that the quality of fit for models in the power mean family depends on the Jensen gap ln(E[L]) - E[ln(L)] where L = p(y|x, w). This gap reflects the difference between the training objective and the objective which is used at the evaluation stage. From this perspective the expectation for the second term E[ln(L)] (which corresponds to the training objective and is the same in all settings) is fixed because we change the dropout scheme only in the inference. Therefore, the Eq. 10 for the gap from this paper is misleading. From this equality the authors derive that reducing the variance leads to decreasing the gap. However, from the Bayes inference it is known that the zero gap between the training objective and the one at evaluation will be reached when we use expectation with respect to the true posterior distribution on the model weights given training data. The authors however claim that the gap is zero when deterministic dropout (that is weight scaling rule) is used. This would be true if we used the same deterministic objective during training stage (that would correspond to no dropout). But we use expectation wrt non-degenerate noise distribution during training and at evaluation stage we take expectation wrt degenerate distribution (i.e. apply deterministic mode). Since the distributions are different we cannot conclude that the gap is zero. Moreover it can be even negative. Hence the statement about zero gap and justification of deterministic mode seems to be wrong. \n3) Experiments results are also hard to follow. In Tables 2, 3, 4 the differences in metric values are insignificant and there are no error bars. From such empirical results it is difficult to draw any conclusions. \n\nOverall, the motivation of the paper is not clear, the analysis for the lower bound tightness and following conclusions are misguided and experiments results are unconvincing. Therefore, I would suggest rejecting the current version.", "extra_scores": {"experience_assessment": "I have published in this field for several years.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}, {"review_id": "SyluRQZf9r", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "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:\nThe paper aims to develop a more principled framework for choosing between different inference procedures in neural network models employing dropout as a stochastic regularizer. In particular, they posit a family of conditional models and show that the learning objectives of all these models are lower bounded by the usual dropout training objective with L2 regularization on weights. They proceed to show empirically that the deterministic inference procedure (multiplying the node's output by the droput rate) achieves the tightest lower bound. From this observation the authors conclude that deterministic inference should be seen as the best available approximation to the true dropout objective rather than an approximation to Monte Carlo averaging. \n\nStrengths: The paper builds on recent works viewing dropout as a Bayesian approximation to the predictive posterior distribution. Introducing a conditional model and showing that the dropout objective is akin to MAP estimation of the parameters of this model is interesting. The result that dropout simultaneously optimizes a lower bound to an entire family of conditional distributions is novel.\n\nWeaknesses:\nThe writing is often not clear, ambiguous or misleading and needs improvement. For instance: \nIn Section 2.1, comments on weaknesses of variational dropout seems out of place. It should either be ommitted or shifted to the previous section where variational dropout is introduced. \nIn Section 2.1, \"Consider a conditional model p(Y |X, Θ) as a crippled generative model with p(X) constant, X and Θ independent.\" Assuming p(X), the input features, to be constant is a very strong assumption. The variational lower bound is still true if p(X) is assumed arbitrary independent of Θ.\nIn Section 3.3, last paragraph, \"Suppose we pick a base model from the power mean family and have a continuum of subvariants with gradually reduced variance in their predictions but the same expectation.\" It is not clear what the authors mean by continuum of subvariants? Is it the dropout rate?\nSeveral statements are made without any citations or explanations. \nIn Section 3. \"While it is easy to argue in general that objectives of more than one model may share any given lower bound\" How? More explanation needed.\nIn Section 3.1, \"Notice how with SGD and multiple epochs, for each data point several dropout masks are encountered, and the approximating quantity is the geometric mean of the predicted probabilities\". Citation needed.\nIn Section 3.2, \"because M_α is monotonically increasing in α\". Proof (in appendix) or citation needed. \nIn Section 5, \"The construction of a conditional model family with a common lower bound on their objectives is applicable to other latent variable models with similar structure and inference method.\" What general structure and inference method are the authors referring to?\n\nTechnical Concerns:\nSection 3.3, last paragraph. The entire paragraph is extremely convoluted. It is not clear how one achieves deterministic dropout inference by reducing variance of the prediction y keeping its expectation constant. \nSection 3.3, \"A similar argument based ... shows that Z monotonically increases... \" How? The authors should deliberate more on this statement since, the Z term is also important in the difference between the true objective of each conditional model and the droput training objective.\nIn Section 5, \"The gains reported in those works might be explained by reducing the bias of deterministic evaluation and also by encouraging small variance in the predictions and thus getting tighter bounds.\" Isn't this contradictory to Section 3.3, where from Eq. 10 and Eq. 11? If the bias is reduced and variance increases, according to Eq. 10, the lower bound would become looser.\nHow is Fig. 1a and 1b computed?\nConclusions drawn from experiments not convincing.\nSection 3.4, last line. \"Having trained a model with dropout, the best ﬁt is achieved by the deterministic model with no dropout. This result isolates the regularisation effects from the biases of the lower bound and the dropout family.\" How does this isolate the regularization effects? What biases of the lower bound? Do you mean the difference between the model's true objective and the dropout training objective?\nTable 4 indicates that not only AMC, also power with alpha=0.5 is better than deterministic in some cases. Did the authors try other values of alphas? Deterministic seems to be good just for MNIST. This strongly refutes the most important claim of this paper, written in the abstract, \"Together, these results suggest that the predominant view of deterministic dropout as a good approximation to MC averaging is misleading. Rather, deterministic dropout is the best available approximation to the true objective.\" \nChanging the dropout multiplier and adjusting the softmax temperature of the network output layer, to achieve comparable performance to AMC on several datasets seems to support the existing hypothesis that deterministic dropout is a good approximation to MC average, and not the other way around!\n\nSummary: \nThe paper introduces some interesting ideas about dropout but suffers from bad writing and presentation of results. One of the most important claims made in this paper, \"dropout trains a deterministic model ﬁrst and foremost and a continuum of stochastic ones to various extents\", is not well-motivated theoretically in Section 3.3. Consequently, this seems to be purely an empirical observation, which is contradicted by further experiments on linguistic datasets. Several conclusions made from experiments seem adhoc.\n", "extra_scores": {"experience_assessment": "I have read many papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I carefully checked the derivations and theory."}, "extra_text": {}}, {"review_id": "BkenstNatH", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "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 proposes a new understanding of dropout on top of variational dropout, which shows that training with dropout equals to maximizing an empirical variational lower bound on the log-likelihood. This paper shows that the log posterior have the same lower bound when the inference model p(y|x) is defined by different methods, i.e., the arithmetic mean of predictions with different dropout masks, the geometric mean, and a power-mean family as an interpolation between these two cases. This indicates that with the same training objective, different inference methods have different gaps to the posterior lower bound. Intuitively, a smaller gap might lead to better performance. The paper then uses an existing result from Liao & Berg (2017) to show that the gap can be bounded by the variance of prediction probability. With empirical observations, the paper gives an unrigorous conclusion that the deterministic inference with dropout rate 0 achieves the smallest gap. However, this does not hold theoretically due to the extra bias on expectation. The optimality of deterministic inference does not hold empirically due to class imbalance or discrepancy between training and test sets. The paper then proposes two practical solutions for better inference: 1) tuning dropout rate, softmax temperature, and the power mean parameter; and 2) deterministic inference with tuned softmax temperature. By using the first inference solution, the performance on PTB and Wikitext2 LM can be improved by 2-3 on perplexity but is still slightly worse than the SOTA achieved by the mixture of softmaxes.\n\nThe idea of analyzing the gap to variational posterior lower bound for different dropout inference model is interesting. The derivations are correct. The organization is not perfect and readers might find it hard to follow here and there, but the main idea is understandable. Experiments show that the suggested tuning of inference hyperparameters can bring improvements to LM tasks, which is convincing. However, there are still major gaps between the theoretical analysis, the conclusion and the empirical solution (please see the detailed comments). Such gaps make the main contribution questionable and make it as a pure empirical paper on its value.\n\nDetailed comments:\n\n1) Reducing the variance of output prediction can reduce the gap on variational posterior, but how does the gap relate to the generalization error? The current paper only indicates that a small gap gives more consistency between the true objective and the optimized objective defined on the training set: they can be still far away from the expected posterior over data distribution. Hence, it is hard to directly relate \"reducing the gap\" and \"improve the test-set performance\".\n\n2) As the author mentioned in Section 3.4, reducing the dropout rate causes a bias issue on the expectation. So it is not clear whether deterministic inference with zero dropout rate can achieve the smallest gap or not. In this way, the conclusion is only supported by the empirical observations but not the presented theoretical analysis.\n\n3) One main contribution of this paper is the power-mean family of dropout. However, only one member (alpha=0.5) from the family has been evaluated in the experiments, and it does not achieve the best performance in most experiments. So this contribution seems not practically useful according to the empirical result.\n\n4) It is not clear how the prediction variance is reduced gradually in order to generate the results in Figure 1(b). I guess reducing dropout rate is not the correct way to do so since it causes the bias issue and the tightness will be influenced.", "extra_scores": {"experience_assessment": "I have published one or two papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly.", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I carefully checked the derivations and theory."}, "extra_text": {}}]

Author rebuttal:

Meta-review: The reviewers have uniformly had significant reservations for the paper. Given that the authors did not even try to address them, this suggests the paper should be rejected.
Decision: Reject