A **Transformer** is a model architecture that eschews recurrence and instead relies entirely on an [attention mechanism](https://paperswithcode.com/methods/category/attention-mechanisms-1) to draw global dependencies between input and output. Before Transformers, the dominant sequence transduction models were based on complex recurrent or convolutional neural networks that include an encoder and a decoder. The Transformer also employs an encoder and decoder, but removing recurrence in favor of [attention mechanisms](https://paperswithcode.com/methods/category/attention-mechanisms-1) allows for significantly more parallelization than methods like [RNNs](https://paperswithcode.com/methods/category/recurrent-neural-networks) and [CNNs](https://paperswithcode.com/methods/category/convolutional-neural-networks).

**Temporal Distribution Matching**, or **TDM**,  is a module used in the [AdaRNN](https://paperswithcode.com/method/adarnn) architecture to match the distributions of the discovered periods to build a time series prediction model $\mathcal{M}$ Given the learned time periods, the TDM module is designed to learn the common knowledge shared by different periods via matching their distributions. Thus, the learned model $\mathcal{M}$ is expected to generalize well on unseen test data compared with the methods which only rely on local or statistical information.

Within the context of AdaRNN, Temporal Distribution Matching aims to adaptively match the distributions between the [RNN](https://paperswithcode.com/methods/category/recurrent-neural-networks) cells of two periods while capturing the temporal dependencies. TDM introduces the importance vector $\mathbf{\alpha} \in \mathbb{R}^{\hat{V}}$ to learn the relative importance of $V$ hidden states inside the RNN, where all the hidden states are weighted with a normalized $\alpha$. Note that for each pair of periods, there is an $\mathbf{\alpha}$, and we omit the subscript if there is no confusion. In this way, we can dynamically reduce the distribution divergence of cross-periods.

Given a period-pair $\left(\mathcal{D}\_{i}, \mathcal{D}\_{j}\right)$, the loss of temporal distribution matching is formulated as:

$$
\mathcal{L}\_{t d m}\left(\mathcal{D}\_{i}, \mathcal{D}\_{j} ; \theta\right)=\sum_{t=1}^{V} \alpha\_{i, j}^{t} d\left(\mathbf{h}\_{i}^{t}, \mathbf{h}\_{j}^{t} ; \theta\right)
$$

where $\alpha\_{i, j}^{t}$ denotes the distribution importance between the periods $\mathcal{D}\_{i}$ and $\mathcal{D}\_{j}$ at state $t$.

All the hidden states of the RNN can be easily computed by following the standard RNN computation. Denote by $\delta(\cdot)$ the computation of a next hidden state based on a previous state. The state computation can be formulated as

$$
\mathbf{h}\_{i}^{t}=\delta\left(\mathbf{x}\_{i}^{t}, \mathbf{h}\_{i}^{t-1}\right)
$$

The final objective of temporal distribution matching (one RNN layer) is:

$$
\mathcal{L}(\theta, \mathbf{\alpha})=\mathcal{L}\_{\text {pred }}(\theta)+\lambda \frac{2}{K(K-1)} \sum\_{i, j}^{i \neq j} \mathcal{L}\_{t d m}\left(\mathcal{D}\_{i}, \mathcal{D}\_{j} ; \theta, \mathbf{\alpha}\right)
$$

where $\lambda$ is a trade-off hyper-parameter. Note that in the second term, we compute the average of the distribution distances of all pairwise periods. For computation, we take a mini-batch of $\mathcal{D}_{i}$ and $\mathcal{D}\_{j}$ to perform forward operation in RNN layers and concatenate all hidden features. Then, we can perform TDM using the above equation.

Temporal Distribution Matching

AdaRNN: Adaptive Learning and Forecasting of Time Series

Transformer

Attention Is All You Need

**InfoNCE**, where NCE stands for Noise-Contrastive Estimation, is a type of contrastive loss function used for [self-supervised learning](https://paperswithcode.com/methods/category/self-supervised-learning).

Given a set $X = ${$x\_{1}, \dots, x\_{N}$} of $N$ random samples containing one positive sample from $p\left(x\_{t+k}|c\_{t}\right)$ and $N − 1$ negative samples from the 'proposal' distribution $p\left(x\_{t+k}\right)$, we optimize:

$$ \mathcal{L}\_{N} = - \mathbb{E}\_{X}\left[\log\frac{f\_{k}\left(x\_{t+k}, c\_{t}\right)}{\sum\_{x\_{j}\in{X}}f\_{k}\left(x\_{j}, c\_{t}\right)}\right] $$

Optimizing this loss will result in $f\_{k}\left(x\_{t+k}, c\_{t}\right)$ estimating the density ratio, which is:

$$ f\_{k}\left(x\_{t+k}, c\_{t}\right) \propto \frac{p\left(x\_{t+k}|c\_{t}\right)}{p\left(x\_{t+k}\right)} $$

Source	Attention Is All You Need
Year	2000
Data Source	CC BY-SA - https://paperswithcode.com