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gemma.cpp/backprop/backward-inl.h

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// Copyright 2024 Google LLC
// SPDX-License-Identifier: Apache-2.0
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
// https://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
// Implementation of the Vector-Jacobian Products (VJP) of the individual
// operations of the forward pass.
// Include guard for non-SIMD code.
#ifndef THIRD_PARTY_GEMMA_CPP_GEMMA_BACKWARD_INL_H_
#define THIRD_PARTY_GEMMA_CPP_GEMMA_BACKWARD_INL_H_
#include <stddef.h>
#include <cmath>
#include <vector>
#include "backprop/activations.h"
#include "backprop/prompt.h"
#include "gemma/common.h"
#include "hwy/base.h"
#include "hwy/contrib/thread_pool/thread_pool.h"
#endif // THIRD_PARTY_GEMMA_CPP_GEMMA_BACKWARD_INL_H_
// Include guard for (potentially) SIMD code.
#if defined(THIRD_PARTY_GEMMA_CPP_BACKWARD_TOGGLE) == defined(HWY_TARGET_TOGGLE)
#ifdef THIRD_PARTY_GEMMA_CPP_BACKWARD_TOGGLE
#undef THIRD_PARTY_GEMMA_CPP_BACKWARD_TOGGLE
#else
#define THIRD_PARTY_GEMMA_CPP_BACKWARD_TOGGLE
#endif
#include "gemma/ops.h"
#include "hwy/highway.h"
HWY_BEFORE_NAMESPACE();
namespace gcpp {
namespace HWY_NAMESPACE {
namespace hn = hwy::HWY_NAMESPACE;
template <size_t kCols, size_t kRows>
void MatMulVJP(const float* HWY_RESTRICT weights, // kRows * kCols,
const float* HWY_RESTRICT x, // num_tokens * kCols
const float* HWY_RESTRICT v, // num_tokens * kRows
size_t num_tokens,
float* HWY_RESTRICT grad_w, // kRows * kCols,
float* HWY_RESTRICT grad_x, // num_tokens * kCols
hwy::ThreadPool& pool) {
hwy::ZeroBytes(grad_x, num_tokens * kCols * sizeof(grad_x[0]));
for (size_t pos = 0; pos < num_tokens; ++pos) {
const size_t voffs = pos * kRows;
const size_t xoffs = pos * kCols;
for (size_t j = 0; j < kRows; ++j) {
MulByConstAndAdd(v[voffs + j], &x[xoffs], &grad_w[j * kCols], kCols);
MulByConstAndAdd(v[voffs + j], &weights[j * kCols], &grad_x[xoffs],
kCols);
}
}
}
template <size_t kHeads, size_t kCols, size_t kRows>
void MultiHeadMatMulVJP(
const float* HWY_RESTRICT weights, // kHeads * kRows * kCols
const float* HWY_RESTRICT x, // num_tokens * kHeads * kCols
const float* HWY_RESTRICT v, // num_tokens * kRows
size_t num_tokens,
float* HWY_RESTRICT grad_w, // kHeads * kRows * kCols
float* HWY_RESTRICT grad_x, // num_tokens * kHeads * kCols
hwy::ThreadPool& pool) {
hwy::ZeroBytes(grad_x, num_tokens * kHeads * kCols * sizeof(grad_x[0]));
for (size_t pos = 0; pos < num_tokens; ++pos) {
for (size_t j = 0; j < kRows; ++j) {
for (size_t h = 0; h < kHeads; ++h) {
MulByConstAndAdd(v[pos * kRows + j],
&x[pos * kHeads * kCols + h * kCols],
&grad_w[h * kRows * kCols + j * kCols], kCols);
MulByConstAndAdd(v[pos * kRows + j],
&weights[h * kRows * kCols + j * kCols],
&grad_x[pos * kHeads * kCols + h * kCols], kCols);
}
}
}
}
template <class D, HWY_IF_F32_D(D)>
static HWY_INLINE hn::Vec<D> DGelu(D d, hn::Vec<D> v) {
const hn::Vec<D> kMul = hn::Set(d, 0.044715f);
const hn::Vec<D> kSqrt2OverPi = hn::Set(d, 0.797884560804236f);
const hn::Vec<D> kHalf = hn::Set(d, 0.5f);
const hn::Vec<D> kOne = hn::Set(d, 1.0f);
// kSqrtOverPi*3*kMul
const hn::Vec<D> kMulv2 = hn::Set(d, 0.1070322244f);
const hn::Vec<D> v2 = hn::Mul(v, v);
const hn::Vec<D> v3 = hn::Mul(v2, v);
const hn::Vec<D> arg = hn::Mul(kSqrt2OverPi, hn::MulAdd(kMul, v3, v));
const hn::Vec<D> tanh = hn::Tanh(d, arg);
const hn::Vec<D> cdf = hn::MulAdd(kHalf, tanh, kHalf);
const hn::Vec<D> dtanh = hn::Sub(kOne, hn::Mul(tanh, tanh));
const hn::Vec<D> darg = hn::MulAdd(kMulv2, v2, kSqrt2OverPi);
return hn::MulAdd(kHalf, hn::Mul(v, hn::Mul(dtanh, darg)), cdf);
}
static HWY_NOINLINE void SoftmaxVJP(const float* HWY_RESTRICT forward,
float* HWY_RESTRICT backward,
const size_t size) {
namespace hn = hwy::HWY_NAMESPACE;
using D = hn::ScalableTag<float>;
const D d;
const auto offset =
hn::Set(d, hn::Dot::Compute<0>(d, forward, backward, size));
hn::Transform1(
d, backward, size, forward,
[&offset](const auto d, const auto v, const auto y)
HWY_ATTR { return hn::Mul(y, hn::Sub(v, offset)); });
}
static HWY_NOINLINE void RMSNormVJP(
const float* HWY_RESTRICT weights, const float* HWY_RESTRICT x,
const float* HWY_RESTRICT v, size_t model_dim, size_t num_tokens,
float* HWY_RESTRICT grad_w, float* HWY_RESTRICT grad_x,
hwy::ThreadPool& pool) {
for (size_t pos = 0; pos < num_tokens; ++pos) {
const size_t offset = pos * model_dim;
constexpr float eps = 1e-6f;
float ss = SquaredL2(x + offset, model_dim);
ss = 1.0f / sqrtf(ss / StaticCast<float>(model_dim) + eps);
for (size_t i = 0; i < model_dim; ++i) {
grad_w[i] += v[offset + i] * x[offset + i] * ss;
}
const float ss3 = ss * ss * ss / StaticCast<float>(model_dim);
float tmp = 0.0f;
for (size_t i = 0; i < model_dim; ++i) {
tmp += (1.0f + weights[i]) * v[offset + i] * x[offset + i];
}
tmp *= ss3;
for (size_t i = 0; i < model_dim; ++i) {
grad_x[offset + i] = ss * (1.0f + weights[i]) * v[offset + i] -
tmp * x[offset + i];
}
}
}
static HWY_NOINLINE void InputEmbeddingVJP(
const float* weights, const std::vector<int>& prompt,
const float scaling, const float* HWY_RESTRICT v,
float* HWY_RESTRICT grad, size_t model_dim) {
HWY_ASSERT(!prompt.empty());
for (size_t pos = 0; pos < prompt.size() - 1; ++pos) {
int token = prompt[pos];
MulByConstAndAdd(scaling, v + pos * model_dim,
grad + token * model_dim, model_dim);
}
}
template <typename TConfig, template<typename> typename LayerT>
void LayerVJP(const LayerT<TConfig>& weights,
const ForwardLayer<float, TConfig>& forward,
const float* HWY_RESTRICT next_layer_grad,
size_t num_tokens,
LayerT<TConfig>& grad,
ForwardLayer<float, TConfig>& backward,
hwy::ThreadPool& pool) {
static constexpr size_t kModelDim = TConfig::kModelDim;
static constexpr size_t kQKVDim = TConfig::kQKVDim;
static constexpr size_t kHeads = TConfig::kHeads;
static constexpr size_t kSeqLen = TConfig::kSeqLen;
static constexpr size_t kFFHiddenDim = TConfig::kFFHiddenDim;
static const float kQueryScale =
static_cast<float>(1.0 / sqrt(static_cast<double>(kQKVDim)));
HWY_ASSERT(num_tokens <= kSeqLen);
MatMulVJP<kFFHiddenDim, kModelDim>(
weights.linear_w.data(), forward.ffw_hidden_gated.data(), next_layer_grad,
num_tokens, grad.linear_w.data(), backward.ffw_hidden_gated.data(),
pool);
for (size_t pos = 0; pos < num_tokens; ++pos) {
const size_t hidden_offset = pos * kFFHiddenDim * 2;
const float* HWY_RESTRICT f_out = forward.ffw_hidden.data() + hidden_offset;
const float* HWY_RESTRICT f_out_mul = f_out + kFFHiddenDim;
const float* HWY_RESTRICT b_out_gated =
backward.ffw_hidden_gated.data() + pos * kFFHiddenDim;
float* HWY_RESTRICT b_out = backward.ffw_hidden.data() + hidden_offset;
float* HWY_RESTRICT b_out_mul = b_out + kFFHiddenDim;
namespace hn = hwy::HWY_NAMESPACE;
using DF = hn::ScalableTag<float>;
using VF = hn::Vec<DF>;
DF df;
for (size_t i = 0; i < kFFHiddenDim; i += Lanes(df)) {
const auto y = Load(df, f_out + i);
const auto x = Load(df, f_out_mul + i);
const auto v = Load(df, b_out_gated + i);
hn::Store(hn::Mul(v, Gelu(df, y)), df, b_out_mul + i);
hn::Store(hn::Mul(v, hn::Mul(x, DGelu(df, y))), df, b_out + i);
}
}
MatMulVJP<kModelDim, kFFHiddenDim * 2>(
weights.gating_einsum_w.data(),
forward.bf_pre_ffw_rms_out.data(), backward.ffw_hidden.data(),
num_tokens, grad.gating_einsum_w.data(),
backward.bf_pre_ffw_rms_out.data(), pool);
RMSNormVJP(weights.pre_ffw_norm_scale.data(),
forward.attention_out.data(),
backward.bf_pre_ffw_rms_out.data(),
kModelDim, num_tokens,
grad.pre_ffw_norm_scale.data(),
backward.attention_out.data(), pool);
for (size_t pos = 0; pos < num_tokens; ++pos) {
AddFrom(next_layer_grad + pos * kModelDim,
backward.attention_out.data() + pos * kModelDim, kModelDim);
}
hwy::ZeroBytes(backward.qkv.data(),
num_tokens * (kHeads + 2) * kQKVDim * sizeof(backward.qkv[0]));
MultiHeadMatMulVJP<kHeads, kQKVDim, kModelDim>(
weights.attn_vec_einsum_w.data(), forward.att_out.data(),
backward.attention_out.data(), num_tokens,
grad.attn_vec_einsum_w.data(), backward.att_out.data(), pool);
for (size_t head = 0; head < kHeads; ++head) {
for (size_t pos = 0; pos < num_tokens; ++pos) {
const size_t aoffset = head * kSeqLen + pos * kHeads * kSeqLen;
const float* HWY_RESTRICT f_head_att = forward.att.data() + aoffset;
const float* HWY_RESTRICT b_att_out =
backward.att_out.data() + (pos * kHeads + head) * kQKVDim;
float* HWY_RESTRICT b_head_att = backward.att.data() + aoffset;
for (size_t pos2 = 0; pos2 <= pos; ++pos2) {
const size_t v2offs = (pos2 * (kHeads + 2) + kHeads + 1) * kQKVDim;
const float* HWY_RESTRICT f_v2 = forward.qkv.data() + v2offs;
float* HWY_RESTRICT b_v2 = backward.qkv.data() + v2offs;
b_head_att[pos2] = Dot(b_att_out, f_v2, kQKVDim);
MulByConstAndAdd(f_head_att[pos2], b_att_out, b_v2, kQKVDim);
}
}
}
for (size_t head = 0; head < kHeads; ++head) {
for (size_t pos = 0; pos < num_tokens; ++pos) {
const size_t aoffset = head * kSeqLen + pos * kHeads * kSeqLen;
const float* HWY_RESTRICT f_head_att = forward.att.data() + aoffset;
float* HWY_RESTRICT b_head_att = backward.att.data() + aoffset;
SoftmaxVJP(f_head_att, b_head_att, pos + 1);
}
}
for (size_t head = 0; head < kHeads; ++head) {
for (size_t pos = 0; pos < num_tokens; ++pos) {
const size_t qoffs = (pos * (kHeads + 2) + head) * kQKVDim;
const size_t aoffs = head * kSeqLen + pos * kHeads * kSeqLen;
const float* HWY_RESTRICT f_q = forward.qkv.data() + qoffs;
const float* HWY_RESTRICT b_head_att = backward.att.data() + aoffs;
float* HWY_RESTRICT b_q = backward.qkv.data() + qoffs;
for (size_t pos2 = 0; pos2 <= pos; ++pos2) {
const size_t k2offs = (pos2 * (kHeads + 2) + kHeads) * kQKVDim;
const float* HWY_RESTRICT f_k2 = forward.qkv.data() + k2offs;
float* HWY_RESTRICT b_k2 = backward.qkv.data() + k2offs;
MulByConstAndAdd(b_head_att[pos2], f_k2, b_q, kQKVDim);
MulByConstAndAdd(b_head_att[pos2], f_q, b_k2, kQKVDim);
}
}
}
for (int pos = 0; pos < static_cast<int>(num_tokens); ++pos) {
float* HWY_RESTRICT b_kv =
backward.qkv.data() + (pos * (kHeads + 2) + kHeads) * kQKVDim;
Rope(b_kv, kQKVDim, -pos);
}
for (size_t head = 0; head < kHeads; ++head) {
for (size_t pos = 0; pos < num_tokens; ++pos) {
float* HWY_RESTRICT b_q =
backward.qkv.data() + (pos * (kHeads + 2) + head) * kQKVDim;
MulByConst(kQueryScale, b_q, kQKVDim);
Rope(b_q, kQKVDim, -pos);
}
}
MatMulVJP<kModelDim, (kHeads + 2) * kQKVDim>(
weights.qkv_einsum_w.data(), forward.pre_att_rms_out.data(),
backward.qkv.data(), num_tokens,
grad.qkv_einsum_w.data(), backward.pre_att_rms_out.data(), pool);
RMSNormVJP(weights.pre_attention_norm_scale.data(),
forward.input.data(),
backward.pre_att_rms_out.data(),
kModelDim, num_tokens,
grad.pre_attention_norm_scale.data(),
backward.input.data(), pool);
for (size_t pos = 0; pos < num_tokens; ++pos) {
AddFrom(backward.attention_out.data() + pos * kModelDim,
backward.input.data() + pos * kModelDim, kModelDim);
}
}
static HWY_NOINLINE void SoftcapVJP(const float cap,
const float* HWY_RESTRICT forward,
float* HWY_RESTRICT backward,
const size_t size) {
namespace hn = hwy::HWY_NAMESPACE;
using D = hn::ScalableTag<float>;
const D d;
const auto one = hn::Set(d, 1.0f);
const auto vcap = hn::Set(d, cap);
const auto vinv_cap = hn::Div(hn::Set(d, 1.0f), vcap);
hn::Transform1(d, backward, size, forward,
[&](const auto d, const auto v, const auto y) HWY_ATTR {
const auto scaled = hn::Mul(vinv_cap, y); // = tanh
return hn::Mul(v, hn::Sub(one, hn::Mul(scaled, scaled)));
});
}
static HWY_NOINLINE void CrossEntropyLossGrad(
const float* HWY_RESTRICT x, float* HWY_RESTRICT grad,
const Prompt& prompt, size_t vocab_size) {
HWY_ASSERT(!prompt.tokens.empty());
const float scaling = -1.0 / std::log(2.0);
size_t num_tokens = prompt.tokens.size() - 1;
hwy::ZeroBytes(grad, num_tokens * vocab_size * sizeof(grad[0]));
for (size_t pos = 0; pos < num_tokens; ++pos) {
if (pos + 1 < prompt.context_size) {
continue;
}
const int next_token = prompt.tokens[pos + 1];
grad[pos * vocab_size + next_token] =
scaling / x[pos * vocab_size + next_token];
}
}
template <typename TConfig, template<typename...> typename WeightsT,
template<typename> typename LayerT>
void CrossEntropyLossBackwardPass(const Prompt& prompt,
const WeightsT<TConfig>& weights,
const ForwardPass<float, TConfig>& forward,
WeightsT<TConfig>& grad,
ForwardPass<float, TConfig>& backward,
hwy::ThreadPool& pool) {
static constexpr size_t kVocabSize = TConfig::kVocabSize;
static constexpr size_t kModelDim = TConfig::kModelDim;
static constexpr size_t kLayers = TConfig::kLayers;
const float kEmbScaling = EmbeddingScaling<TConfig>();
static_assert(!TConfig::kAbsolutePE);
static_assert(TConfig::kPostNorm == PostNormType::None);
static_assert(TConfig::kKVHeads == 1);
HWY_DASSERT(prompt.context_size > 0);
HWY_DASSERT(prompt.context_size < prompt.tokens.size());
const size_t num_tokens = prompt.tokens.size() - 1;
CrossEntropyLossGrad(forward.probs.data(), backward.logits.data(), prompt,
kVocabSize);
for (size_t pos = 0; pos < num_tokens; ++pos) {
SoftmaxVJP(forward.probs.data() + pos * kVocabSize,
backward.logits.data() + pos * kVocabSize,
kVocabSize);
}
if constexpr (TConfig::kFinalCap > 0.0f) {
for (size_t pos = 0; pos < num_tokens; ++pos) {
SoftcapVJP(TConfig::kFinalCap, forward.logits.data() + pos * kVocabSize,
backward.logits.data() + pos * kVocabSize, kVocabSize);
}
}
MatMulVJP<kModelDim, kVocabSize>(
weights.embedder_input_embedding.data(), forward.final_norm_output.data(),
backward.logits.data(), num_tokens,
grad.embedder_input_embedding.data(), backward.final_norm_output.data(),
pool);
RMSNormVJP(weights.final_norm_scale.data(),
forward.final_layer_output.data(),
backward.final_norm_output.data(),
kModelDim, num_tokens,
grad.final_norm_scale.data(),
backward.final_layer_output.data(), pool);
for (int layer = static_cast<int>(kLayers) - 1; layer >= 0; --layer) {
auto type = TConfig::kLayerConfig[layer];
// TODO(szabadka) Implement Griffin layer vjp.
HWY_ASSERT(type == LayerAttentionType::kGemma);
float* next_layer_grad = layer + 1 < kLayers
? backward.layers[layer + 1].input.data()
: backward.final_layer_output.data();
LayerVJP<TConfig, LayerT>(
*weights.GetLayer(layer), forward.layers[layer], next_layer_grad,
num_tokens, *grad.GetLayer(layer), backward.layers[layer], pool);
}
InputEmbeddingVJP(weights.embedder_input_embedding.data(), prompt.tokens,
kEmbScaling, backward.layers[0].input.data(),
grad.embedder_input_embedding.data(), kModelDim);
}
// NOLINTNEXTLINE(google-readability-namespace-comments)
} // namespace HWY_NAMESPACE
} // namespace gcpp
HWY_AFTER_NAMESPACE();
#endif // NOLINT
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