instruction:
In this task, you will be presented with a context from an academic paper and a question separated with a 
. You have to answer the question based on the context.
question:
In this work, we introduce a new logical inference engine called MonaLog, which is based on natural logic and work on monotonicity stemming from vanBenthemEssays86. Since our logic operates over surface forms, it is straightforward to hybridize our models. We investigate using MonaLog in combination with the language model BERT BIBREF20, including for compositional data augmentation, i.e, re-generating entailed versions of examples in our training sets.  We perform two experiments to test MonaLog. We first use MonaLog to solve the problems in a commonly used natural language inference dataset, SICK BIBREF1, comparing our results with previous systems. Second, we test the quality of the data generated by MonaLog. To do this, we generate more training data (sentence pairs) from the SICK training data using our system, and performe fine-tuning on BERT BIBREF20, a language model based on the transformer architecture BIBREF23, with the expanded dataset.  
 Question: How do they combine MonaLog with BERT?
answer:
They use Monalog for data-augmentation to fine-tune BERT on this task


question:
Training and testing are done in alternating steps: In each epoch, for training, we first present to an LSTM network 1000 samples in a given language, which are generated according to a certain discrete probability distribution supported on a closed finite interval. We then freeze all the weights in our model, exhaustively enumerate all the sequences in the language by their lengths, and determine the first $k$ shortest sequences whose outputs the model produces inaccurately.  experimented with 1, 2, 3, and 36 hidden units for $a^n b^n$ ; 2, 3, 4, and 36 hidden units for $a^n b^n c^n$ ; and 3, 4, 5, and 36 hidden units for $a^n b^n c^n d^n$ .  Following the traditional approach adopted by BIBREF7 , BIBREF12 , BIBREF9 and many other studies, we train our neural network as follows. At each time step, we present one input character to our model and then ask it to predict the set of next possible characters, based on the current character and the prior hidden states. Given a vocabulary $\mathcal {V}^{(i)}$ of size $d$ , we use a one-hot representation to encode the input values; therefore, all the input vectors are $d$ -dimensional binary vectors. The output values are $(d+1)$ -dimensional though, since they may further contain the termination symbol $\dashv $ , in addition to the symbols in $\mathcal {V}^{(i)}$ . The output values are not always one-hot encoded, because there can be multiple possibilities for the next character in the sequence, therefore we instead use a $k$ -hot representation to encode the output values. Our objective is to minimize the mean-squared error (MSE) of the sequence predictions. 
 Question: What training settings did they try?
answer:
Training and testing are done in alternating steps: In each epoch, for training, we first present to an LSTM network 1000 samples in a given language, which are generated according to a certain discrete probability distribution supported on a closed finite interval. We then freeze all the weights in our model, exhaustively enumerate all the sequences in the language by their lengths, and determine the first $k$ shortest sequences whose outputs the model produces inaccurately.  experimented with 1, 2, 3, and 36 hidden units for $a^n b^n$ ; 2, 3, 4, and 36 hidden units for $a^n b^n c^n$ ; and 3, 4, 5, and 36 hidden units for $a^n b^n c^n d^n$ .  Following the traditional approach adopted by BIBREF7 , BIBREF12 , BIBREF9 and many other studies, we train our neural network as follows. At each time step, we present one input character to our model and then ask it to predict the set of next possible characters, based on the current character and the prior hidden states. Given a vocabulary $\mathcal {V}^{(i)}$ of size $d$ , we use a one-hot representation to encode the input values; therefore, all the input vectors are $d$ -dimensional binary vectors. The output values are $(d+1)$ -dimensional though, since they may further contain the termination symbol $\dashv $ , in addition to the symbols in $\mathcal {V}^{(i)}$ . The output values are not always one-hot encoded, because there can be multiple possibilities for the next character in the sequence, therefore we instead use a $k$ -hot representation to encode the output values. Our objective is to minimize the mean-squared error (MSE) of the sequence predictions.


question:
To investigate whether our BERT-based model can transfer knowledge beyond language, we consider image features as simple visual tokens that can be presented to the model analogously to textual token embeddings. In order to make the $o_j$ vectors (of dimension $2048+4=2052$) comparable to BERT embeddings (of dimension 768), we use a simple linear cross-modal projection layer $W$ of dimensions $2052\hspace{-1.00006pt}\times \hspace{-1.00006pt}768$. The $N$ object regions detected in an image, are thus represented as $X_{img} = (W.o_1,...,W.o_N)$. Once mapped into the BERT embedding space with $W$, the image is seen by the rest of the model as a sequence of units with no explicit indication if it is a text or an image embedding. 
 Question: How are multimodal representations combined?
answer:

The image feature vectors are mapped into BERT embedding dimensions and treated like a text sequence afterwards.