scripts/nn/layers/lstm.dml

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/*
 * LSTM layer.
 */
source("nn/layers/sigmoid.dml") as sigmoid
source("nn/layers/tanh.dml") as tanh

forward = function(matrix[double] X, matrix[double] W, matrix[double] b, int T, int D,
                   boolean return_sequences, matrix[double] out0, matrix[double] c0)
    return (matrix[double] out, matrix[double] c,
            matrix[double] cache_out, matrix[double] cache_c, matrix[double] cache_ifog) {
  /*
   * Computes the forward pass for an LSTM layer with M neurons.
   * The input data has N sequences of T examples, each with D features.
   *
   * In an LSTM, an internal cell state is maintained, additive
   * interactions operate over the cell state at each timestep, and
   * some amount of this cell state is exposed as output at each
   * timestep.  Additionally, the output of the previous timestep is fed
   * back in as an additional input at the current timestep.
   *
   * Reference:
   *  - Long Short-Term Memory, Hochreiter, 1997
   *    - http://deeplearning.cs.cmu.edu/pdfs/Hochreiter97_lstm.pdf
   *
   * Inputs:
   *  - X: Inputs, of shape (N, T*D).
   *  - W: Weights, of shape (D+M, 4M).
   *  - b: Biases, of shape (1, 4M).
   *  - T: Length of example sequences (number of timesteps).
   *  - D: Dimensionality of the input features (number of features).
   *  - return_sequences: Whether to return `out` at all timesteps,
   *      or just for the final timestep.
   *  - out0: Outputs from previous timestep, of shape (N, M).
   *      Note: This is *optional* and could just be an empty matrix.
   *  - c0: Initial cell state, of shape (N, M).
   *      Note: This is *optional* and could just be an empty matrix.
   *
   * Outputs:
   *  - out: If `return_sequences` is True, outputs for all timesteps,
   *      of shape (N, T*M).  Else, outputs for the final timestep, of
   *      shape (N, M).
   *  - c: Cell state for final timestep, of shape (N, M).
   *  - cache_out: Cache of outputs, of shape (T, N*M).
   *      Note: This is used for performance during training.
   *  - cache_c: Cache of cell state, of shape (T, N*M).
   *      Note: This is used for performance during training.
   *  - cache_ifog: Cache of intermediate values, of shape (T, N*4M).
   *      Note: This is used for performance during training.
   */
  N = nrow(X)
  M = as.integer(ncol(W)/4)
  N1 = nrow(out0)
  if(N < N1) {
    # Allow for smaller out0 for last batch
    out0 = out0[1:N,]
    c0 = c0[1:N,]
  }
  out_prev = out0
  c_prev = c0
  c = c_prev
  out = matrix(0, rows=N, cols=ifelse(return_sequences,T*M, M))
  
  # caches to be used during the backward pass for performance
  cache_out = matrix(0, rows=T, cols=N*M)
  cache_c = matrix(0, rows=T, cols=N*M)
  cache_ifog = matrix(0, rows=T, cols=N*4*M)

  for (t in 1:T) {  # each timestep
    X_t = X[,(t-1)*D+1:t*D]  # shape (N, D)
    input = cbind(X_t, out_prev)  # shape (N, D+M)
    ifog = input %*% W + b  # input, forget, output, and g gates; shape (N, 4M)
    ifog[,1:3*M] = sigmoid::forward(ifog[,1:3*M])  # i,f,o gates squashed with sigmoid
    ifog[,3*M+1:4*M] = tanh::forward(ifog[,3*M+1:4*M])  # g gate squashed with tanh
    # c_t = f*prev_c + i*g
    c = ifog[,M+1:2*M]*c_prev + ifog[,1:M]*ifog[,3*M+1:4*M]  # shape (N, M)
    # out_t = o*tanh(c)
    out_t = ifog[,2*M+1:3*M] * tanh::forward(c)  # shape (N, M)

    # store
    if (return_sequences) {
      out[,(t-1)*M+1:t*M] = out_t
    }
    else {
      out = out_t
    }
    out_prev = out_t
    c_prev = c
    cache_out[t,] = matrix(out_t, rows=1, cols=N*M)  # reshape
    cache_c[t,] = matrix(c, rows=1, cols=N*M)  # reshape
    cache_ifog[t,] = matrix(ifog, rows=1, cols=N*4*M)  # reshape
  }
}

backward = function(matrix[double] dout, matrix[double] dc,
                    matrix[double] X, matrix[double] W, matrix[double] b, int T, int D,
                    boolean given_sequences, matrix[double] out0, matrix[double] c0,
                    matrix[double] cache_out, matrix[double] cache_c, matrix[double] cache_ifog)
    return (matrix[double] dX, matrix[double] dW, matrix[double] db,
            matrix[double] dout0, matrix[double] dc0) {
  /*
   * Computes the backward pass for an LSTM layer with M neurons.
   *
   * Inputs:
   *  - dout: Gradient wrt `out`.  If `given_sequences` is `True`,
   *      contains gradients on outputs for all timesteps, of
   *      shape (N, T*M). Else, contains the gradient on the output
   *      for the final timestep, of shape (N, M).
   *  - dc: Gradient wrt `c` (from later in time), of shape (N, M).
   *      This would come from later in time if the cell state was used
   *      downstream as the initial cell state for another LSTM layer.
   *      Typically, this would be used when a sequence was cut at
   *      timestep `T` and then continued in the next batch.  If `c`
   *      was not used downstream, then `dc` would be an empty matrix.
   *  - X: Inputs, of shape (N, T*D).
   *  - W: Weights, of shape (D+M, 4M).
   *  - b: Biases, of shape (1, 4M).
   *  - T: Length of example sequences (number of timesteps).
   *  - D: Dimensionality of the input features.
   *  - given_sequences: Whether `dout` is for all timesteps,
   *      or just for the final timestep.  This is based on whether
   *      `return_sequences` was true in the forward pass.
   *  - out0: Outputs from previous timestep, of shape (N, M).
   *      Note: This is *optional* and could just be an empty matrix.
   *  - c0: Initial cell state, of shape (N, M).
   *      Note: This is *optional* and could just be an empty matrix.
   *  - cache_out: Cache of outputs, of shape (T, N*M).
   *      Note: This is used for performance during training.
   *  - cache_c: Cache of cell state, of shape (T, N*M).
   *      Note: This is used for performance during training.
   *  - cache_ifog: Cache of intermediate values, of shape (T, N*4*M).
   *      Note: This is used for performance during training.
   *
   * Outputs:
   *  - dX: Gradient wrt `X`, of shape (N, T*D).
   *  - dW: Gradient wrt `W`, of shape (D+M, 4M).
   *  - db: Gradient wrt `b`, of shape (1, 4M).
   *  - dout0: Gradient wrt `out0`, of shape (N, M).
   *  - dc0: Gradient wrt `c0`, of shape (N, M).
   */
  N = nrow(X)
  M = as.integer(ncol(W)/4)
  N1 = nrow(out0)
  if(N != N1) {
    # Allow for smaller out0 for last batch 
    # out0 = out0[1:N,]
    # c0 = c0[1:N,]
    stop("Unsupported operation: The batch size of previous iteration " + N1 + " is different than the batch size of current iteration " + N)
  }
  dX = matrix(0, rows=N, cols=T*D)
  dW = matrix(0, rows=D+M, cols=4*M)
  db = matrix(0, rows=1, cols=4*M)
  dout0 = matrix(0, rows=N, cols=M)
  dc0 = matrix(0, rows=N, cols=M)
  dct = dc
  if (!given_sequences) {
    # only given dout for output at final timestep, so prepend empty douts for all other timesteps
    dout = cbind(matrix(0, rows=N, cols=(T-1)*M), dout)  # shape (N, T*M)
  }

  t = T
  for (iter in 1:T) {  # each timestep in reverse order
    X_t = X[,(t-1)*D+1:t*D]  # shape (N, D)
    dout_t = dout[,(t-1)*M+1:t*M]  # shape (N, M)
    out_t = matrix(cache_out[t,], rows=N, cols=M)  # shape (N, M)
    ct = matrix(cache_c[t,], rows=N, cols=M)  # shape (N, M)
    if (t == 1) {
      out_prev = out0  # shape (N, M)
      c_prev = c0  # shape (N, M)
    }
    else {
      out_prev = matrix(cache_out[t-1,], rows=N, cols=M)  # shape (N, M)
      c_prev = matrix(cache_c[t-1,], rows=N, cols=M)  # shape (N, M)
    }
    input = cbind(X_t, out_prev)  # shape (N, D+M)
    ifog = matrix(cache_ifog[t,], rows=N, cols=4*M)
    i = ifog[,1:M]  # input gate, shape (N, M)
    f = ifog[,M+1:2*M]  # forget gate, shape (N, M)
    o = ifog[,2*M+1:3*M]  # output gate, shape (N, M)
    g = ifog[,3*M+1:4*M]  # g gate, shape (N, M)

    dct = dct + o*tanh::backward(dout_t, ct)  # shape (N, M)
    do = tanh::forward(ct) * dout_t  # output gate, shape (N, M)
    df = c_prev * dct  # forget gate, shape (N, M)
    dc_prev = f * dct  # shape (N, M)
    di = g * dct  # input gate, shape (N, M)
    dg = i * dct  # g gate, shape (N, M)

    di_raw = i * (1-i) * di
    df_raw = f * (1-f) * df
    do_raw = o * (1-o) * do
    dg_raw = (1-g^2) * dg
    difog_raw = cbind(di_raw, df_raw, do_raw, dg_raw)  # shape (N, 4M)

    dW = dW + t(input) %*% difog_raw  # shape (D+M, 4M)
    db = db + colSums(difog_raw)  # shape (1, 4M)
    dinput = difog_raw %*% t(W)  # shape (N, D+M)
    dX[,(t-1)*D+1:t*D] = dinput[,1:D]
    dout_prev = dinput[,D+1:D+M]  # shape (N, M)
    if (t == 1) {
      dout0 = dout_prev  # shape (N, M)
      dc0 = dc_prev  # shape (N, M)
    }
    else {
      dout[,(t-2)*M+1:(t-1)*M] = dout[,(t-2)*M+1:(t-1)*M] + dout_prev  # shape (N, M)
      dct = dc_prev  # shape (N, M)
    }
    t = t - 1
  }
}

init = function(int N, int D, int M)
    return (matrix[double] W, matrix[double] b, matrix[double] out0, matrix[double] c0) {
  /*
   * Initialize the parameters of this layer.
   *
   * Note: This is just a convenience function, and parameters
   * may be initialized manually if needed.
   *
   * We use the Glorot uniform heuristic which limits the magnification
   * of inputs/gradients during forward/backward passes by scaling
   * uniform weights by a factor of sqrt(6/(fan_in + fan_out)).
   *  - http://jmlr.org/proceedings/papers/v9/glorot10a/glorot10a.pdf
   *
   * Inputs:
   *  - N: Number of examples in batch.
   *  - D: Dimensionality of the input features (number of features).
   *  - M: Number of neurons in this layer.
   *
   * Outputs:
   *  - W: Weights, of shape (D+M, 4M).
   *  - b: Biases, of shape (1, 4M).
   *  - out0: Empty previous timestep output matrix, of shape (N, M).
   *  - c0: Empty initial cell state matrix, of shape (N, M).
   */
  fan_in = D+M
  fan_out = 4*M
  scale = sqrt(6/(fan_in+fan_out))
  W = rand(rows=D+M, cols=4*M, min=-scale, max=scale, pdf="uniform")
  b = matrix(0, rows=1, cols=4*M)
  out0 = matrix(0, rows=N, cols=M)
  c0 = matrix(0, rows=N, cols=M)
}