softmax.h

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// Copyright © 2023-2024 Apple Inc.
 
template <typename T>
inline T softmax_exp(T x) {
  // Softmax doesn't need high precision exponential cause x is gonna be in
  // (-oo, 0] anyway and subsequently it will be divided by sum(exp(x_i)).
  return fast::exp(x);
}
 
template <typename T, typename AccT = T, int N_READS = SOFTMAX_N_READS>
[[kernel]] void softmax_single_row(
    const device T* in,
    device T* out,
    constant int& axis_size,
    uint gid [[threadgroup_position_in_grid]],
    uint _lid [[thread_position_in_threadgroup]],
    uint simd_lane_id [[thread_index_in_simdgroup]],
    uint simd_group_id [[simdgroup_index_in_threadgroup]]) {
  int lid = _lid;
 
  constexpr int SIMD_SIZE = 32;
 
  threadgroup AccT local_max[SIMD_SIZE];
  threadgroup AccT local_normalizer[SIMD_SIZE];
 
  AccT ld[N_READS];
 
  in += gid * axis_size + lid * N_READS;
  if (lid * N_READS + N_READS <= axis_size) {
    for (int i = 0; i < N_READS; i++) {
      ld[i] = AccT(in[i]);
    }
  } else {
    for (int i = 0; i < N_READS; i++) {
      ld[i] = ((lid * N_READS + i) < axis_size) ? AccT(in[i])
                                                : Limits<AccT>::finite_min;
    }
  }
  if (simd_group_id == 0) {
    local_max[simd_lane_id] = Limits<AccT>::finite_min;
    local_normalizer[simd_lane_id] = 0;
  }
  threadgroup_barrier(mem_flags::mem_threadgroup);
 
  // Get the max
  AccT maxval = Limits<AccT>::finite_min;
  for (int i = 0; i < N_READS; i++) {
    maxval = (maxval < ld[i]) ? ld[i] : maxval;
  }
  maxval = simd_max(maxval);
  if (simd_lane_id == 0) {
    local_max[simd_group_id] = maxval;
  }
  threadgroup_barrier(mem_flags::mem_threadgroup);
  if (simd_group_id == 0) {
    maxval = simd_max(local_max[simd_lane_id]);
    if (simd_lane_id == 0) {
      local_max[0] = maxval;
    }
  }
  threadgroup_barrier(mem_flags::mem_threadgroup);
  maxval = local_max[0];
 
  // Compute exp(x_i - maxval) and store the partial sums in local_normalizer
  AccT normalizer = 0;
  for (int i = 0; i < N_READS; i++) {
    AccT exp_x = softmax_exp(ld[i] - maxval);
    ld[i] = exp_x;
    normalizer += exp_x;
  }
  normalizer = simd_sum(normalizer);
  if (simd_lane_id == 0) {
    local_normalizer[simd_group_id] = normalizer;
  }
  threadgroup_barrier(mem_flags::mem_threadgroup);
  if (simd_group_id == 0) {
    normalizer = simd_sum(local_normalizer[simd_lane_id]);
    if (simd_lane_id == 0) {
      local_normalizer[0] = normalizer;
    }
  }
  threadgroup_barrier(mem_flags::mem_threadgroup);
  normalizer = 1 / local_normalizer[0];
 
  // Normalize and write to the output
  out += gid * axis_size + lid * N_READS;
  if (lid * N_READS + N_READS <= axis_size) {
    for (int i = 0; i < N_READS; i++) {
      out[i] = T(ld[i] * normalizer);
    }
  } else {
    for (int i = 0; i < N_READS; i++) {
      if ((lid * N_READS + i) < axis_size) {
        out[i] = T(ld[i] * normalizer);
      }
    }
  }
}
 
template <typename T, typename AccT = T, int N_READS = SOFTMAX_N_READS>
[[kernel]] void softmax_looped(
    const device T* in,
    device T* out,
    constant int& axis_size,
    uint gid [[threadgroup_position_in_grid]],
    uint lid [[thread_position_in_threadgroup]],
    uint lsize [[threads_per_threadgroup]],
    uint simd_lane_id [[thread_index_in_simdgroup]],
    uint simd_group_id [[simdgroup_index_in_threadgroup]]) {
  in += gid * axis_size;
 
  constexpr int SIMD_SIZE = 32;
 
  threadgroup AccT local_max[SIMD_SIZE];
  threadgroup AccT local_normalizer[SIMD_SIZE];
 
  // Get the max and the normalizer in one go
  AccT prevmax;
  AccT maxval = Limits<AccT>::finite_min;
  AccT normalizer = 0;
  for (int r = 0; r < static_cast<int>(ceildiv(axis_size, N_READS * lsize));
       r++) {
    int offset = r * lsize * N_READS + lid * N_READS;
    AccT vals[N_READS];
    if (offset + N_READS <= axis_size) {
      for (int i = 0; i < N_READS; i++) {
        vals[i] = AccT(in[offset + i]);
      }
    } else {
      for (int i = 0; i < N_READS; i++) {
        vals[i] = (offset + i < axis_size) ? AccT(in[offset + i])
                                           : Limits<AccT>::finite_min;
      }
    }
    prevmax = maxval;
    for (int i = 0; i < N_READS; i++) {
      maxval = (maxval < vals[i]) ? vals[i] : maxval;
    }
    normalizer *= softmax_exp(prevmax - maxval);
    for (int i = 0; i < N_READS; i++) {
      normalizer += softmax_exp(vals[i] - maxval);
    }
  }
  // Now we got partial normalizer of N_READS * ceildiv(axis_size, N_READS *
  // lsize) parts. We need to combine them.
  //    1. We start by finding the max across simd groups
  //    2. We then change the partial normalizers to account for a possible
  //       change in max
  //    3. We sum all normalizers
  prevmax = maxval;
  maxval = simd_max(maxval);
  normalizer *= softmax_exp(prevmax - maxval);
  normalizer = simd_sum(normalizer);
 
  // Now the normalizer and max value is correct for each simdgroup. We write
  // them shared memory and combine them.
  prevmax = maxval;
  if (simd_lane_id == 0) {
    local_max[simd_group_id] = maxval;
  }
  threadgroup_barrier(mem_flags::mem_threadgroup);
  maxval = simd_max(local_max[simd_lane_id]);
  normalizer *= softmax_exp(prevmax - maxval);
  if (simd_lane_id == 0) {
    local_normalizer[simd_group_id] = normalizer;
  }
  threadgroup_barrier(mem_flags::mem_threadgroup);
  normalizer = simd_sum(local_normalizer[simd_lane_id]);
  normalizer = 1 / normalizer;
 
  // Finally given the normalizer and max value we can directly write the
  // softmax output
  out += gid * axis_size;
  for (int r = 0; r < static_cast<int>(ceildiv(axis_size, N_READS * lsize));
       r++) {
    int offset = r * lsize * N_READS + lid * N_READS;
    if (offset + N_READS <= axis_size) {
      for (int i = 0; i < N_READS; i++) {
        out[offset + i] = T(softmax_exp(in[offset + i] - maxval) * normalizer);
      }
    } else {
      for (int i = 0; i < N_READS; i++) {
        if (offset + i < axis_size) {
          out[offset + i] =
              T(softmax_exp(in[offset + i] - maxval) * normalizer);
        }
      }
    }
  }
}