LSTM Gate Computation
Compute the LSTM gates for one timestep.
Given previous hidden state ht−1, previous cell state ct−1, and input xt, compute all four gates by concatenating [ht−1,xt] into a combined vector, then:
- Forget gate: f=σ(Wf⋅[h,x]+bf)
- Input gate: i=σ(Wi⋅[h,x]+bi)
- Candidate: c~=tanh(Wc⋅[h,x]+bc)
- Cell state: ct=f⊙ct−1+i⊙c~
- Output gate: o=σ(Wo⋅[h,x]+bo)
- Hidden state: ht=o⊙tanh(ct)
Return (h_t, c_t) both as lists rounded to 4 decimal places.
Example:
h_prev = [0] c_prev = [0] x = [1] W_f = [[0.5, 0.5]] b_f = [0] W_i = [[0.5, 0.5]] b_i = [0] W_c = [[0.5, 0.5]] b_c = [0] W_o = [[0.5, 0.5]] b_o = [0]
([0.1743], [0.2876])
- First, we concatenate ht−1 and xt into a combined vector [h,x]=[0,1].
- Then, we compute the forget gate: f=σ(Wf⋅[h,x]+bf)=σ([0.5,0.5]⋅[0,1]+0)=σ(0.5), the input gate: i=σ(Wi⋅[h,x]+bi)=σ(0.5), and the candidate: c~=tanh(Wc⋅[h,x]+bc)=tanh(0.5).
- Next, we calculate the cell state: ct=f⊙ct−1+i⊙c~, and the output gate: o=σ(Wo⋅[h,x]+bo)=σ(0.5).
- The final output is calculated as ht=o⊙tanh(ct), resulting in ht=[0.1743] and ct=[0.2876].
Constraints:
- h_prev, c_prev: 1D lists (hidden_size)
- x: 1D list (input_size)
- W_f, W_i, W_c, W_o: 2D lists (hidden_size x (hidden_size + input_size))
- b_f, b_i, b_c, b_o: 1D lists (hidden_size)
- Return tuple (h_t, c_t) rounded to 4 decimal places
Background Knowledge
Long Short-Term Memory (LSTM) networks are a type of Recurrent Neural Network (RNN) designed to handle the vanishing gradient problem that occurs when training traditional RNNs. LSTMs are capable of learning long-term dependencies in data, making them particularly useful for sequence modeling tasks such as language translation, text generation, and speech recognition. The core component of an LSTM is the memory cell, which allows the network to store and retrieve information over long periods of time.
The LSTM gates are the key to controlling the flow of information into and out of the memory cell. There are four gates in total: the forget gate, input gate, candidate gate, and output gate. Each gate is responsible for a specific function: the forget gate determines what information to discard from the previous cell state, the input gate determines what new information to add to the cell state, the candidate gate generates a new candidate value for the cell state, and the output gate determines what information to output based on the current cell state and hidden state.
The mathematical operations involved in computing the LSTM gates are based on linear algebra and activation functions. The sigmoid function, σ(x)=1+e−x1, is used to introduce non-linearity into the model, while the hyperbolic tangent function, tanh(x)=ex+e−xex−e−x, is used to generate the candidate value for the cell state. The element-wise multiplication operator, ⊙, is used to compute the Hadamard product of two vectors.
Algorithm/Approach
The general approach to solving this problem involves the following steps:
- Compute the combined vector by concatenating the previous hidden state and input
- Compute the four LSTM gates using the combined vector and the corresponding weights and biases
- Compute the new cell state using the forget gate, input gate, and candidate gate
- Compute the new hidden state using the output gate and the new cell state
This approach requires a good understanding of linear algebra, activation functions, and the mathematical operations involved in computing the LSTM gates.
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