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zW4GC7g|@)d@C7esSVI$_p8g^rIVSnG{X>Im|8ZQG z>EBj1{Kf1LBUv0(jkE?yV|=-XsgWG8;Z!v3KlCMUcg(wVBspE##f^v5w$fm~m-Gkp zT5Ll^|G&LPT5}XDH$+a(J(gd*_No4p#ia~{(`XGR#C4)}JQN04+6>qXUaX7tYufxA ztk>5X93(qaD4QPrzpnkGl|M{qcRiVmH1}StJV#5VHSlm1z{{L#nc2J$y~whA6vF18HBfw5dFarQ`)jv#TrInA z@Splo@b^Lfn~nY8uZViN%4+{JkROcP(-n-k4V!QOo|r%DG=~ie4QD}8_H9|%&Oo}i zj_pYsxGp5_6U_fQiGQ&lHm^pML(I?Lsbx;cQ36s6{1ssN3rJll&3r9?nESf*m4kkh zce#EuK62jp8pQCX{=wzL+(7v}t5R9vLCYRc&>`WBWAcsQi=tjIn?t~-t7C{N*K&;b E4* Date: Mon, 24 Jun 2024 00:00:51 +0800 Subject: [PATCH 02/17] Update 03Extend.md --- 04Inference/05Optimize/03Extend.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/04Inference/05Optimize/03Extend.md b/04Inference/05Optimize/03Extend.md index 51158842..ba4d5793 100644 --- a/04Inference/05Optimize/03Extend.md +++ b/04Inference/05Optimize/03Extend.md @@ -84,6 +84,8 @@ Recomputation(重算)则是一种以计算能力为代价来节省存储空 主要的算法实现: +![flashAttention 算法方式](images/03Extend08.png) + 步骤一:计算分子块的大小 首先,我们需要获取 GPU 硬件 SRAM 的大小,我们假设为 M。为了让 Q、K、V 在计算中可以存放在 SRAM 中,我们需要设定分块的大小尺寸。 From fedf753e18912bb943faf21b5fe3f4bfd397ac8a Mon Sep 17 00:00:00 2001 From: engin-work <542673108@qq.com> Date: Mon, 24 Jun 2024 19:31:30 +0800 Subject: [PATCH 03/17] Rename 03Extend04.png to 03Extend040.png --- .../images/{03Extend04.png => 03Extend040.png} | Bin 1 file changed, 0 insertions(+), 0 deletions(-) rename 04Inference/05Optimize/images/{03Extend04.png => 03Extend040.png} (100%) diff --git a/04Inference/05Optimize/images/03Extend04.png b/04Inference/05Optimize/images/03Extend040.png similarity index 100% rename from 04Inference/05Optimize/images/03Extend04.png rename to 04Inference/05Optimize/images/03Extend040.png From 745c82c4b542ce323703290e7980539eabec8de4 Mon Sep 17 00:00:00 2001 From: engin-work <542673108@qq.com> Date: Mon, 24 Jun 2024 19:32:10 +0800 Subject: [PATCH 04/17] Rename 03Extend08.png to 03Extend04.png --- .../images/{03Extend08.png => 03Extend04.png} | Bin 1 file changed, 0 insertions(+), 0 deletions(-) rename 04Inference/05Optimize/images/{03Extend08.png => 03Extend04.png} (100%) diff --git a/04Inference/05Optimize/images/03Extend08.png b/04Inference/05Optimize/images/03Extend04.png similarity index 100% rename from 04Inference/05Optimize/images/03Extend08.png rename to 04Inference/05Optimize/images/03Extend04.png From f26c36de79dfd769ea325b193f3b734a6503086d Mon Sep 17 00:00:00 2001 From: engin-work <542673108@qq.com> Date: Mon, 24 Jun 2024 19:33:22 +0800 Subject: [PATCH 05/17] Rename 03Extend07.png to 03Extend070.png --- .../images/{03Extend07.png => 03Extend070.png} | Bin 1 file changed, 0 insertions(+), 0 deletions(-) rename 04Inference/05Optimize/images/{03Extend07.png => 03Extend070.png} (100%) diff --git a/04Inference/05Optimize/images/03Extend07.png b/04Inference/05Optimize/images/03Extend070.png similarity index 100% rename from 04Inference/05Optimize/images/03Extend07.png rename to 04Inference/05Optimize/images/03Extend070.png From fc52461aabce371c4196783c0a3add0dd392822d Mon Sep 17 00:00:00 2001 From: engin-work <542673108@qq.com> Date: Mon, 24 Jun 2024 19:33:40 +0800 Subject: [PATCH 06/17] Rename 03Extend06.png to 03Extend060.png --- .../images/{03Extend06.png => 03Extend060.png} | Bin 1 file changed, 0 insertions(+), 0 deletions(-) rename 04Inference/05Optimize/images/{03Extend06.png => 03Extend060.png} (100%) diff --git a/04Inference/05Optimize/images/03Extend06.png b/04Inference/05Optimize/images/03Extend060.png similarity index 100% rename from 04Inference/05Optimize/images/03Extend06.png rename to 04Inference/05Optimize/images/03Extend060.png From b4a92f28c6f4625f2eb3f0688fc8f01e6065f141 Mon Sep 17 00:00:00 2001 From: engin-work <542673108@qq.com> Date: Mon, 24 Jun 2024 19:34:02 +0800 Subject: [PATCH 07/17] Rename 03Extend05.png to 03Extend050.png --- .../images/{03Extend05.png => 03Extend050.png} | Bin 1 file changed, 0 insertions(+), 0 deletions(-) rename 04Inference/05Optimize/images/{03Extend05.png => 03Extend050.png} (100%) diff --git 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2001 From: engin-work <542673108@qq.com> Date: Mon, 24 Jun 2024 20:09:31 +0800 Subject: [PATCH 14/17] Rename 03Extend060.png to 03Extend15.png --- .../images/{03Extend060.png => 03Extend15.png} | Bin 1 file changed, 0 insertions(+), 0 deletions(-) rename 04Inference/05Optimize/images/{03Extend060.png => 03Extend15.png} (100%) diff --git a/04Inference/05Optimize/images/03Extend060.png b/04Inference/05Optimize/images/03Extend15.png similarity index 100% rename from 04Inference/05Optimize/images/03Extend060.png rename to 04Inference/05Optimize/images/03Extend15.png From acab0600f71ed3a9cada5faa6e393dacde242edd Mon Sep 17 00:00:00 2001 From: engin-work <542673108@qq.com> Date: Mon, 24 Jun 2024 20:09:41 +0800 Subject: [PATCH 15/17] Rename 03Extend070.png to 03Extend16.png --- .../images/{03Extend070.png => 03Extend16.png} | Bin 1 file changed, 0 insertions(+), 0 deletions(-) rename 04Inference/05Optimize/images/{03Extend070.png => 03Extend16.png} (100%) diff --git a/04Inference/05Optimize/images/03Extend070.png b/04Inference/05Optimize/images/03Extend16.png similarity index 100% rename from 04Inference/05Optimize/images/03Extend070.png rename to 04Inference/05Optimize/images/03Extend16.png From 98e030be791d568c95ac0f63d5b6fd7937951ebc Mon Sep 17 00:00:00 2001 From: engin-work <542673108@qq.com> Date: Mon, 24 Jun 2024 20:09:56 +0800 Subject: [PATCH 16/17] Update 03Extend.md --- 04Inference/05Optimize/03Extend.md | 100 ++++++++++++++++++++++++----- 1 file changed, 83 insertions(+), 17 deletions(-) diff --git a/04Inference/05Optimize/03Extend.md b/04Inference/05Optimize/03Extend.md index ba4d5793..5d3b76a7 100644 --- a/04Inference/05Optimize/03Extend.md +++ b/04Inference/05Optimize/03Extend.md @@ -84,43 +84,79 @@ Recomputation(重算)则是一种以计算能力为代价来节省存储空 主要的算法实现: -![flashAttention 算法方式](images/03Extend08.png) +![flashAttention 算法方式](images/03Extend04.png) -步骤一:计算分子块的大小 +### 步骤一:计算分子块的大小 首先,我们需要获取 GPU 硬件 SRAM 的大小,我们假设为 M。为了让 Q、K、V 在计算中可以存放在 SRAM 中,我们需要设定分块的大小尺寸。 其次,在 SRAM 上需要存在的数据包括,Q 子块,K 子块,V 子块,其次还应包括计算过程中的中间输出 O,O 的大小应该与 Q、K、V 子块大小一致。 -所以,在这里我们计算出子块的列大小 Bc =[M/4d], d 为矩阵维度。 +![flashAttention 算法方式](images/03Extend05.png) -步骤二:初始化输出矩阵 O +所以,在这里我们计算出子块的列大小 Bc =[M/4d], d 为矩阵维度。当然,需要注意的是,上面的设置子块的大小并非唯一的,只有保证子块大小不超过SRAM的大小即可。 -SRAM 上的输出 O 矩阵赋值为全 0,它将作为一个累加器保存 softmax 的累积分母。 +### 步骤二:初始化输出矩阵 O -步骤三:切分子块 +![flashAttention 算法方式](images/03Extend06.png) + +SRAM 上的输出 O 矩阵赋值为全 0,它将作为一个累加器保存 softmax 的累积分母,其中 l 也类似。m 用于记录每一行行最大分数,其初始化为-inf。 + +### 步骤三:切分子块 + +![flashAttention 算法方式](images/03Extend07.png) 将 Q 划分成 Tr 个 Bolck,K、V 划分成 Tc 个 Block,初始化 attention output O,并划分成 Tr 个 Block。 -步骤四:外循环加载 K、V 内循环加载 Q 子块 +### 步骤四:外循环加载 K、V 内循环加载 Q 子块 -![flashattention cycle](images/03Extend04.png) +![flashattention cycle](images/03Extend08.png) -如图所示: +上图完美解释了这个循环过程, 1. 外循环:对于每一个 Block Key 和 Value,从 HBM 加载进 SRAM 2. 内循环:对于每个 Block Query,从 HBM 加载进 SRAM 3. 在 SRAM 上完成 Block S 的计算 -步骤五:实现分块 SoftMax 算法 +![flashattention cycle](images/03Extend09.png) + +这里要注意的是,Oi, li, mi其中存储的可能是上一个循环计算的中间结果。 + +### 步骤五:实现分块 SoftMax 算法 下面我们看看原版的证明公式: -假如有切片向量 x = [x^(1), x^(2)],切片后 softmax 的计算方式: +#### 标准 softmax 计算方式 + +![flashattention cycle](images/03Extend10.png) + +在实际硬件中,因为浮点数表示的范围是有限的,对于 float32 和 bfloat16 来说,当 z ≥ 89 时,exp(z) 就会变成inf,发生数据上溢的问题。 + +为了确保数值计算的稳定性,避免溢出问题,通常采用一种称为“safe softmax”的计算策略。在此方法中,通过减去最大值来缩放输入数据,以保证数值的相对稳定性。 + +所以说,现有所有的深度学习框架中都采用了“safe softmax”这种计算方式,其计算公式如下: + +![flashattention cycle](images/03Extend11.png) + +计算举例:a = [0.1, 0.2, 0.3, 0.4]; m(a) = 0.4 + +则可以得到:f(a) = [e^(0.1-0.4), e^(0.2-0.4), e^(0.3-0.4), e^(0.4-0.4)] + +然后计算得到:l(a) = Σf(a), softmax(a) = f(a) / l(a) + +从上面可以看出,首先在分子上“safe softmax”需要获取当前区间的最大值来缩放输入数据,而在分母上需要累加所有的分子 f(a)。 + +由于 flashAttention 已经采取了分块计算的策略,也就意味着在计算 softmax 时,并不能拿到所有数据列的最大值和全部 f(a) 的和。 + +#### flashAttention 改进方式 + +虽然softmax与K的列是耦合的,但如果分开计算每个子块的softmax再将最后的结果进行收集转换是否可以等价呢?下面我们看看原版的证明公式: -![softmax 计算方式 1](images/03Extend05.png) +1. 假如有切片向量x = [x^(1), x^(2)],切片后softmax 的计算方式: -update m(x),根据更新后的 m(x),根据上一步计算结果重新计算 f(x), l(x)。假设存在 x^(3), 那么便可以将 x^(1)和 x^(2)合并成一个序列,重复步骤 1 即可。 +![softmax 计算方式 1](images/03Extend12.png) + +2. update m(x),根据更新后的 m(x),根据上一步计算结果重新计算 f(x), l(x)。假设存在 x^(3), 那么便可以将 x^(1)和 x^(2)合并成一个序列,重复步骤 1 即可。 计算举例:a = [0.1, 0.2, 0.3, 0.4] = [a1, a2] @@ -130,17 +166,47 @@ update m(x),根据更新后的 m(x),根据上一步计算结果重新计算 最终得到 softmax(a) = f(a) / l(a) -通过上述的转换可知,softmax 与分块 softmax 是在数学上是等价的关系。不过由于真实计算中次数变多,精度上也可能存在一定丢失。 +需要注意的是,可以利用GPU多线程同时并行计算多个block的softmax。 + +可见通过上述的转换可知,softmax 与分块 softmax是在数学上是等价的关系。不过由于真实计算中次数变多,精度上也可能存在一定丢失。 + +在介绍完 flashAttention 中 softmax 的改进后,我们继续围绕论文中的代码进行分析: + +![softmax 计算方式 1](images/03Extend13.png) + +首先,根据上一步计算的子块 Sij,来计算当前块的行最大值 mij,当前块 Pij (即 softmax 的分子),lij 为 Pij 的累积值。 + +其次,计算子块与子块间的最大值 m^new 和多个子块的 Pij 的累积值 l^new。 + +最后,根据 softmax 公式计算最终的 softmax,经结果写到 SRAM 的 Oi 中,并写出到 HBM,同时将最后的最后的l^new 赋值给 li 写出到 HBM,m^new 赋值到 mi 写出到 HBM,开始下一轮循环。 + +到此前向计算就算完成,我们可以通过下图来总结下flashAttention的前向计算过程,这里就不做过多解释了。 + +![softmax 计算方式 1](images/03Extend14.png) 步骤六:反向计算 +从上面的前向过程中,我们知道前向过程中只将 Oi, li, mi 写出到了 HBM,而S和P的保存则主要在反向的重算中实现。 + 1. 前向过程会保留 Q,K,V,O, l, m 在 HBM 中,dO 由反向计算获取后,按照前向相同的分块模式重新分块。 2. 初始化 dQ,dK,dV 为全 0 矩阵,并按照对等 Q,K,V 的分割方式分割 dQ,dK,dV。 3. 分别从 HBM 中 Load K V block on SRAM,再 Load Q block on SRAM。根据前向过程重新计算对应 block 的 S 和 P;按分块矩阵的方式分别计算对应梯度,完成参数更新。 -代码实现:https://github.com/openai/triton/blob/main/python/tutorials/06-fused-attention.py#L17 +最终可以看到,在将三个kernel进行合并后,flashAttention v1实现了中间计算完全基于SRAM的目的。 + +### flashAttention性能分析 + +FlashAttention 节省的访存次数计算如下: + +首先,K,V (Nxd)的每个block都需要Load 进SRAM,因此该过程的HBM访问次数为 O(Nxd)。 + +其次,Q 也需要分 block Load 进 SRAM,该过程一共持续外循环 Tc 次,因此该过程的 HBM 访问次数为 O(TcNd) + +最后,而 Tc = N/(Bc) = 4Nd/M(向上取整)。因此 flash attention 的 HBM 访问次数为 O(N^2d^2/M) + +下面来看 flash Attention 代码的具体实现:https://github.com/openai/triton/blob/main/python/tutorials/06-fused-attention.py#L17 ```python def _fwd_kernel( @@ -234,7 +300,7 @@ Flash Attention 的优点在于充分考虑了在计算任务中 IO 的重要性 具体示例如下: -![数据节点转换](images/03Extend06.png) +![数据节点转换](images/03Extend15.png) 内存优化是一种计算机系统优化技术,主要目的是提高系统的运行性能,通过更有效地使用和管理内存资源来达到这个目的。 @@ -242,7 +308,7 @@ Inplace operation:是一种内存优化手段,它在当前的内存块上直 Memory sharing:是另一种内存优化策略。它在内存使用上进行优化,当两个数据的内存大小相同,且有一个数据参与计算后不再需要时,我们可以让后一个数据直接覆盖前一个数据的内存。这样做的好处是可以减少内存的开销,节省内存空间,提高内存的使用效率。 -![内存优化](images/03Extend07.png) +![内存优化](images/03Extend16.png) ## 小结与思考 From 9ea14f3997b05e28d8d4a70b7140cb8e80f8af52 Mon Sep 17 00:00:00 2001 From: engin-work <542673108@qq.com> Date: Tue, 25 Jun 2024 19:51:16 +0800 Subject: [PATCH 17/17] Update 03Extend.md --- 04Inference/05Optimize/03Extend.md | 1 + 1 file changed, 1 insertion(+) diff --git a/04Inference/05Optimize/03Extend.md b/04Inference/05Optimize/03Extend.md index 5d3b76a7..90cafcc8 100644 --- a/04Inference/05Optimize/03Extend.md +++ b/04Inference/05Optimize/03Extend.md @@ -93,6 +93,7 @@ Recomputation(重算)则是一种以计算能力为代价来节省存储空 其次,在 SRAM 上需要存在的数据包括,Q 子块,K 子块,V 子块,其次还应包括计算过程中的中间输出 O,O 的大小应该与 Q、K、V 子块大小一致。 ![flashAttention 算法方式](images/03Extend05.png) +1. Set block sizes B_c = [M / 4d], B_r = min([M / 4d],d) 所以,在这里我们计算出子块的列大小 Bc =[M/4d], d 为矩阵维度。当然,需要注意的是,上面的设置子块的大小并非唯一的,只有保证子块大小不超过SRAM的大小即可。