➩ Speech recognition.

➩ Handwritting recognition (online recognition: input is a sequence of pen positions, not an image).

➩ Text to speech.

➩ Machine translation.

➩ Image captioning (combines with convolutional networks).

➩ Time series prediction (for example, stock price prediction).

➩ Robot control (and other reinforcement learning tasks).

➩ \(a_{t+1} = g\left(a_{t}, x_{t+1}\right)\) and \(y_{t+1} = g\left(a_{t+1}\right)\)

➩ \(a_{t+2} = g\left(a_{t+1}, x_{t+2}\right)\) and \(y_{t+2} = g\left(a_{t+2}\right)\)

➩ \(a_{t+3} = g\left(a_{t+2}, x_{t+3}\right)\) and \(y_{t+3} = g\left(a_{t+3}\right)\)

➩ ...

➩ Each item can be a sequence with different number of elements \(t_{i}\), therefore, each item has different number of activation units \(a_{i,t}\), \(t = 1, 2, ..., t_{i}\).

➩ There can be either one output unit at the end of each item \(o = g\left(w^{\left(o\right)} \cdot a_{t_{i}} + b^{\left(o\right)}\right)\), or \(t_{i}\) output units one for each activation unit \(o_{t} = g\left(w^{\left(o\right)} \cdot a_{t} + b^{\left(o\right)}\right)\).

➩ Convolutional layers share weights over different regions of an image.

➩ Recurrent layers share weights over different times (positions in a sequence).

➩ In the case with one output unit at the end, \(\dfrac{\partial C_{i}}{\partial w^{\left(o\right)}} = \dfrac{\partial C_{i}}{\partial o_{i}} \dfrac{\partial o_{i}}{\partial w^{\left(o\right)}}\), and \(\dfrac{\partial C_{i}}{\partial w^{\left(a\right)}} = \dfrac{\partial C_{i}}{\partial o_{i}} \dfrac{\partial o_{i}}{\partial a_{t_{i}}} \dfrac{\partial a_{t_{i}}}{\partial w^{\left(a\right)}} + \dfrac{\partial C_{i}}{\partial o_{i}} \dfrac{\partial o_{i}}{\partial a_{t_{i}}} \dfrac{\partial a_{t_{i}}}{\partial a_{t_{i} - 1}} \dfrac{\partial a_{t_{i} - 1}}{\partial w^{\left(a\right)}} + ... + \dfrac{\partial C_{i}}{\partial o_{i}} \dfrac{\partial o_{i}}{\partial a_{t_{i}}} \dfrac{\partial a_{t_{i}}}{\partial a_{t_{i} - 1}} ... \dfrac{\partial a_{2}}{\partial a_{1}} \dfrac{\partial a_{1}}{\partial w^{\left(a\right)}}\), and \(\dfrac{\partial C_{i}}{\partial w^{\left(x\right)}} = \dfrac{\partial C_{i}}{\partial o_{i}} \dfrac{\partial o_{i}}{\partial a_{t_{i}}} \dfrac{\partial a_{t_{i}}}{\partial w^{\left(x\right)}} + \dfrac{\partial C_{i}}{\partial o_{i}} \dfrac{\partial o_{i}}{\partial a_{t_{i}}} \dfrac{\partial a_{t_{i}}}{\partial a_{t_{i} - 1}} \dfrac{\partial a_{t_{i} - 1}}{\partial w^{\left(x\right)}} + ... + \dfrac{\partial C_{i}}{\partial o_{i}} \dfrac{\partial o_{i}}{\partial a_{t_{i}}} \dfrac{\partial a_{t_{i}}}{\partial a_{t_{i} - 1}} ... \dfrac{\partial a_{2}}{\partial a_{1}} \dfrac{\partial a_{1}}{\partial w^{\left(x\right)}}\).

➩ The case with one output unit for each activation unit is similar.

➩ Technically adding small weights will not lead to vanishing gradient like multiplying small weights, so hidden units can be added together.

➩ The (long term) memory is updated by \(a^{\left(c\right)}_{t} = a^{\left(f\right)}_{t} \times a^{\left(c\right)}_{t-1} + a^{\left(i\right)}_{t} \times a^{\left(g\right)}_{t}\), where \(a^{\left(c\right)}\) is called the cell unit, \(a^{\left(f\right)}\) is the forget gate and controls how much memory to forget, \(a^{\left(i\right)}\) is the input gate and controls how much information to add to memory, \(a^{\left(g\right)}\) is the new values added to memory.

➩ The (short term memory) state is updated by \(a^{\left(h\right)} = a^{\left(o\right)} \times g\left(a^{\left(c\right)}_{t}\right)\), where \(a^{\left(h\right)}\) is the usual recurrent unit called hidden state, \(a^{\left(o\right)}\) is the output gate and controls how much information from the memory to reflect in the next state.

➩ Each of the gates are computed based on the hidden state and the input features (or the previous layer hidden states if there are multiple LSTM layers): \(a^{\left(f\right)}_{t} = g\left(w^{\left(f\right)} \cdot x_{t} + w^{\left(F\right)} \cdot a^{\left(h\right)}_{t-1} + b^{\left(f\right)}\right)\), \(a^{\left(i\right)}_{t} = g\left(w^{\left(i\right)} \cdot x_{t} + w^{\left(I\right)} \cdot a^{\left(h\right)}_{t-1} + b^{\left(i\right)}\right)\), \(a^{\left(g\right)}_{t} = g\left(w^{\left(g\right)} \cdot x_{t} + w^{\left(G\right)} \cdot a^{\left(h\right)}_{t-1} + b^{\left(g\right)}\right)\), \(a^{\left(o\right)}_{t} = g\left(w^{\left(o\right)} \cdot x_{t} + w^{\left(O\right)} \cdot a^{\left(h\right)}_{t-1} + b^{\left(o\right)}\right)\).

➩ The memory is also updated through addition: \(a^{\left(h\right)}_{t} = \left(1 - a^{\left(z\right)}_{t}\right) \times a^{\left(h\right)}_{t-1} + a^{\left(z\right)}_{t} \times a^{\left(g\right)}_{t}\), where \(a^{\left(z\right)}\) is the update gate, and \(a^{\left(r\right)}\) is the reset gate.

➩ Each of the gates are computed in a similar way: \(a^{\left(z\right)}_{t} = g\left(w^{\left(z\right)} \cdot x_{t} + w^{\left(Z\right)} \cdot a^{\left(h\right)}_{t-1} + b^{\left(z\right)}\right)\), \(a^{\left(r\right)}_{t} = g\left(w^{\left(r\right)} \cdot x_{t} + w^{\left(R\right)} \cdot a^{\left(h\right)}_{t-1} + b^{\left(r\right)}\right)\), \(a^{\left(g\right)}_{t} = g\left(w^{\left(g\right)} \cdot x_{t} + w^{\left(G\right)} \cdot a^{\left(r\right)}_{t} \times a^{\left(h\right)}_{t-1} + b^{\left(g\right)}\right)\).

➩ \(a^{\left(h\right)}_{t} = g\left(w^{\left(x\right)} \cdot x_{t} + b^{\left(x\right)}\right)\) is not recurrent.

➩ There are value units \(a^{\left(v\right)}_{t} = w^{\left(v\right)} \cdot a^{\left(h\right)}_{t}\), key units \(a^{\left(k\right)}_{t} = w^{\left(k\right)} \cdot a^{\left(h\right)}_{t}\), and query units \(a^{\left(q\right)}_{t} = w^{\left(q\right)} \cdot a^{\left(h\right)}_{t}\), and attention context can be computed as \(g\left(\dfrac{a^{\left(q\right)}_{s} \cdot a^{\left(k\right)}_{t}}{\sqrt{m}}\right) \cdot a^{\left(v\right)}_{t}\) where \(g\) is the softmax activation.

➩ The attention matrix is usually masked so that a unit \(a_{t}\) cannot pay attention to another unit in the future \(a_{t+1}, a_{t+2}, a_{t+3}, ...\) by making the \(a^{\left(q\right)}_{s} \cdot a^{\left(k\right)}_{t} = -\infty\) when \(s \geq t\) so that \(e^{a^{\left(q\right)}_{s} \cdot a^{\left(k\right)}_{t}} = 0\) when \(s \geq t\).

➩ There can be multiple parallel attention units called multi-head attention.

➩ Fine tuning for pre-trained models freezes a subset of the weights (usually in earlier layers) and updates the other weights (usually the later layers) based on new datasets: Wikipedia.

➩ Prompt engineering does not alter the weights, and only provides more context (in the form of examples): Link, Wikipedia.

➩ Reinforcement learning from human feedback (RLHF) uses reinforcement learning techniques to update the model based on human feedback (in the form of rewards, and the model optimizes the reward instead of some form of costs): Wikipedia.