Kernel Case Study: Flash Attention

mechanism is on the core of recent day transformers. However scaling the context window of those transformers was a significant problem, and it nonetheless is although we’re within the period of one million tokens + context window (Qwen 2.5 [1]). There are each appreciable compute and reminiscence certain complexities in these fashions after we scale the context window (A naive Attention Mechanism scales quadratically in each compute and reminiscence necessities). Revisiting Flash Consideration lets us perceive the complexities of optimizing the underlying operations on GPUs and extra importantly offers us a greater grip on pondering what’s subsequent.

Let’s rapidly revisit a naive consideration algorithm to see what’s occurring.

As you possibly can see if we aren’t being cautious then we are going to find yourself materializing a full NxM consideration matrix into the GPU HBM. That means the reminiscence requirement will go up quadratically to growing context size.

In the event you wanna study extra concerning the GPU reminiscence hierarchy and its variations, my previous post on Triton is an effective start line. This may even be helpful as we go alongside on this put up after we get to implementing the Flash Attention kernel in triton. The flash attention paper additionally has some actually good introduction to this.

Moreover, after we take a look at the steps concerned in executing this algorithm and its sample of accessing the gradual HBM, (which as defined later within the put up could possibly be a significant bottleneck as nicely) we discover just a few issues:

1. We’ve got Q, Ok and V within the HBM initially
2. We have to entry Q and Ok initially from the HBM to compute the dot product
3. We write the output scores again to the HBM
4. We entry it once more to execute the softmax, and optionally for Causal consideration, like within the case of LLMs, we should masks this output earlier than the softmax. The ensuing full consideration matrix is written once more into the HBM
5. We entry the HBM once more to execute the ultimate dot product, to get each the eye weights and the Worth matrix to jot down the output again to the gradual GPU reminiscence

I feel you get the purpose. We might neatly learn and write from the HBM to keep away from redundant operations, to make some potential beneficial properties. That is precisely the first motivation for the unique Flash Consideration algorithm.

Flash Consideration initially got here out in 2022 [2], after which a 12 months later got here out with some a lot wanted enhancements in 2023 as Flash Consideration v2 [3] and once more in 2024 with extra enhancements for Nvidia Hopper and Blackwell GPUs [4] as Flash Consideration v3 [5]. The unique consideration paper recognized that the eye operation continues to be restricted by reminiscence bandwidth moderately than compute. (Prior to now, there have been makes an attempt to scale back the computation complexity of Consideration from O(N**2) to O(NlogN) and decrease by means of approximate algorithms)

Flash consideration proposed a fused kernel which does all the above consideration operations in a single go, block-wise, to get the ultimate consideration output with out ever having to appreciate the total N**2 consideration matrix in reminiscence, making the algorithm considerably quicker. The time period `fused` merely means we mix a number of operations within the GPU SRAM earlier than invoking the a lot slower journey throughout the slower GPU reminiscence, making the algorithm performant. All of the whereas offering the precise consideration output with none approximations.

This lecture, from Stanford CS139, demonstrates brilliantly how we will consider the influence of a nicely thought out reminiscence entry sample can have on an algorithm. I extremely advocate you examine this one out in case you haven’t already.

Earlier than we begin diving into flash consideration to name it FA, lets?) in triton there’s something else that I wished to get out of the way in which.

Numerical Stability in exponents

Let’s take the instance of FP32 numbers. float32 (normal 32-bit float) makes use of 1 signal bit, 8 exponent bits, and 23 mantissa bits [6]. The biggest finite base for the exponent in float32 is 2¹²⁷≈1.7×10³⁸. Which suggests after we take a look at exponents, e⁸⁸ ≈ 1.65×10³⁸, something near 88 (though in actuality could be a lot decrease to maintain it protected) and we’re in bother as we might simply overflow. Right here’s a very interesting chat with OpenAI o1 shared by of us at AllenAI of their OpenInstruct repo. This though is speaking about stabilizing KL Divergence calculations within the setting of RLHF/RL, the concepts translate precisely to exponents as nicely. So to cope with the softmax scenario in consideration what we do is the next:

TRICK : Let’s additionally observe the next, in case you do that:

then you possibly can rescale/readjust values with out affecting the ultimate softmax worth. That is actually helpful when you might have an preliminary estimate for the utmost worth, however that may change after we encounter a brand new set of values. I do know I do know, stick with me and let me clarify.

Setting the scene

Let’s take a small detour into matrix multiplication.

This exhibits a toy instance of a blocked matrix multiplication besides we have now blocks solely on the rows of A (inexperienced) and columns of B (Orange? Beige?). As you possibly can see above the output O1, O2, O3 and O4 are full (these positions want no extra calculations). We simply must fill within the remaining columns within the preliminary rows through the use of the remaining columns of B. Like under:

So we will fill these locations within the output with a block of columns from B and a block of rows from A at a time.

Connecting the dots

After I launched FA, I mentioned that we by no means should compute the total consideration matrix and retailer the entire thing. So right here’s what we do:

1. Compute a block of the eye matrix utilizing a block of rows from Q and a block of columns from Ok. When you get the partial consideration matrix compute just a few statistics and hold it within the reminiscence.

I’ve greyed O5 to O12 as a result of we don’t know these values but, as they should come from the following blocks. We then remodel Sb like under:

Now you might have setup for a partial softmax

However:

1. What if the true most is within the Oi’s which might be but to return?
2. The sum continues to be native, so we have to replace this each time we see new Pi’s. We all know the right way to hold monitor of a sum, however what about rebasing it to the true most?

Recall the trick above. All that we have now to do is to maintain a monitor of the utmost values we encounter for every row, and iteratively replace as you see new maximums from the remaining blocks of columns from Ok for a similar set of rows from Q.

We nonetheless don’t need to write our partial softmax matrix into HBM. We hold it for the subsequent step.

The ultimate dot product

The final step in our consideration computation is our dot product with V. To start out we’d have initialized a matrix filled with 0’s in our HBM as our output of form NxD. The place N is the variety of Queries as above. We use the identical block dimension for V as we had for Ok besides we will apply it row smart like under (The subscripts simply denote that that is solely a block and never the total matrix)

Discover how we’d like the eye scores from all of the blocks to get the ultimate product. But when we calculate the native rating and `accumulate` it like how we did to get the precise Ls we will kind the total output on the finish of processing all of the blocks of columns (Ok_b) for a given row block (Q_b).

Placing all of it collectively

Let’s put all these concepts collectively to kind the ultimate algorithm

To know the notation, __ij implies that it’s the native values for a given block of columns and rows and _i implies it’s for the worldwide output rows and Question blocks. The one half we haven’t defined to date is the ultimate replace to O_i. That’s the place we use all of the concepts from above to get the fitting scaling.

The entire code is offered as a gist here.

Let’s see what these initializations appear to be in torch:

def flash_attn_v1(Q, Ok, V, Br, Bc):
  """Flash Consideration V1"""
  B, N, D = Q.form
  M = Ok.form[1]
  Nr = int(np.ceil(N/Br))
  Nc = int(np.ceil(N/Bc))
  
  Q = Q.to('cuda')
  Ok = Ok.to('cuda')
  V = V.to('cuda')
  
  batch_stride = Q.stride(0)
  
  O = torch.zeros_like(Q).to('cuda')
  lis = torch.zeros((B, Nr, int(Br)), dtype=torch.float32).to('cuda')
  mis = torch.ones((B, Nr, int(Br)), dtype=torch.float32).to('cuda')*-torch.inf
  
  grid = (B, )
  flash_attn_v1_kernel[grid](
      Q, Ok, V,
      N, M, D,
      Br, Bc,
      Nr, Nc,
      batch_stride,
      Q.stride(1),
      Ok.stride(1),
      V.stride(1),
      lis, mis,
      O,
      O.stride(1),
  )
  return O

If you’re uncertain concerning the launch grid, checkout my introduction to Triton

Take a more in-depth take a look at how we initialized our Ls and Ms. We’re protecting one for every row block of Output/Question, every of dimension B_r. There are N_r such blocks in whole.

Within the instance above I used to be merely utilizing B_r = 2 and B_c = 2. However within the above code the initialization is predicated on the gadget capability. I’ve included the calculation for a T4 GPU. For every other GPU, we have to get the SRAM capability and modify these numbers accordingly. Now for the precise kernel implementation:

# Flash Consideration V1
import triton
import triton.language as tl
import torch
import numpy as np
import pdb

@triton.jit
def flash_attn_v1_kernel(
    Q, Ok, V,
    N: tl.constexpr, M: tl.constexpr, D: tl.constexpr,
    Br: tl.constexpr,
    Bc: tl.constexpr,
    Nr: tl.constexpr,
    Nc: tl.constexpr,
    batch_stride: tl.constexpr,
    q_rstride: tl.constexpr,
    k_rstride: tl.constexpr, 
    v_rstride: tl.constexpr,
    lis, mis,
    O,
    o_rstride: tl.constexpr):
    
    """Flash Consideration V1 kernel"""
    
    pid = tl.program_id(0)
    

    for j in vary(Nc):
        k_offset = ((tl.arange(0, Bc) + j*Bc) * k_rstride)[:, None] + (tl.arange(0, D))[None, :] + pid * M * D
        # Utilizing k_rstride and v_rstride as we're trying on the total row directly, for every ok v block 
        v_offset = ((tl.arange(0, Bc) + j*Bc) * v_rstride)[:, None] + (tl.arange(0, D))[None, :] + pid * M * D
        k_mask = k_offset < (pid + 1) * M*D
        v_mask = v_offset < (pid + 1) * M*D
        k_load = tl.load(Ok + k_offset, masks=k_mask, different=0)
        v_load = tl.load(V + v_offset, masks=v_mask, different=0)
        for i in vary(Nr):
            q_offset = ((tl.arange(0, Br) + i*Br) * q_rstride)[:, None] + (tl.arange(0, D))[None, :] + pid * N * D
            q_mask = q_offset < (pid + 1) * N*D
            q_load = tl.load(Q + q_offset, masks=q_mask, different=0)
            # Compute consideration
            s_ij = tl.dot(q_load, tl.trans(k_load))
            m_ij = tl.max(s_ij, axis=1, keep_dims=True)
            p_ij = tl.exp(s_ij - m_ij)
            l_ij = tl.sum(p_ij, axis=1, keep_dims=True)
            
            ml_offset = tl.arange(0, Br) + Br * i + pid * Nr * Br
            m = tl.load(mis + ml_offset)[:, None]
            l = tl.load(lis + ml_offset)[:, None]

            m_new = tl.the place(m < m_ij, m_ij, m)

            l_new = tl.exp(m - m_new) * l + tl.exp(m_ij - m_new) * l_ij

            o_ij = tl.dot(p_ij, v_load)

            output_offset = ((tl.arange(0, Br) + i*Br) * o_rstride)[:, None] + (tl.arange(0, D))[None, :] + pid * N * D
            output_mask = output_offset < (pid + 1) * N*D
            o_current = tl.load(O + output_offset, masks=output_mask)

            o_new = (1/l_new) * (l * tl.exp(m - m_new) * o_current + tl.exp(m_ij - m_new) * o_ij)

            tl.retailer(O + output_offset, o_new, masks=output_mask)
            tl.retailer(mis + ml_offset, tl.reshape(m_new, (Br,)))
            tl.retailer(lis + ml_offset, tl.reshape(l_new, (Br,)))

Let’s perceive whats taking place right here:

1. Create 1 kernel for every NxD matrix within the batch. In actuality we’d have another dimension to parallelize throughout, the pinnacle dimension. However for understanding the implementation I feel this might suffice.
2. In every kernel we do the next:
   1. For every block of columns in Ok and V we load up the related a part of the matrix (B_c x D) into the GPU SRAM (Present whole SRAM utilization = 2B_cD). This stays within the SRAM until we’re accomplished with all of the row blocks
   2. For every row block of Q, we load the block onto SRAM as nicely (Present whole SRAM Utilization = 2B_cD + BrD)
   3. On chip we compute the dot product (s_ij), compute the native row-maxes (m_ij), the exp (p_ij), and the expsum (l_ij)
   4. We load up the working stats for the i^th row block. Two vectors of dimension B_r x 1, which denotes the present international row-maxes (m_i) and the expsum (l_i). (Present SRAM utilization: 2B_cD + B_rD + 2B_r)
   5. We get the brand new estimates for the worldwide m_i and l_i.
   6. We load the a part of the output for this block of Q and replace it utilizing the brand new working stats and the exponent trick, we then write this again into the HBM. (Present SRAM utilization: 2B_cD + 2B_rD + 2B_r)
   7. We write the up to date working stats additionally into the HBM.
3. For a matrix of any dimension, aka any context size, at a time we are going to by no means materialize the total consideration matrix, solely part of it all the time.
4. We managed to fuse collectively all of the ops right into a single kernel, decreasing HBM entry significantly.

Closing SRAM utilization stands though at 4BD + 2B, the place B was initially calculated as M/4d the place M is the SRAM capability. Undecided if am lacking one thing right here. Please remark if you understand why that is the case!

Block Sparse Consideration and V2 and V3

I’ll hold this quick as these variations hold the core thought however discovered higher and higher methods to do the identical.

For Block Sparse Consideration,

1. Think about we had masks for every block like within the case of causal consideration. If for a given block we have now the masks all set to zero then we will merely skip your entire block with out computing something actually. Saving FLOPs. That is the place the foremost beneficial properties had been seen. To place this into perspective, within the case of BERT pre-training the algorithm will get a 15% enhance over the perfect performing coaching setup on the time, whereas for GPT-2 we get a 3x over huggingface coaching implementation and ~ 2x over a Megatron setup.

2. You’ll be able to actually get the identical efficiency in GPT2 in a fraction of the time, actually shaving off days from the coaching run, which is superior!

In V2:

1. Discover how presently we will solely do parallelization on the batch and head dimension. However in case you merely simply flip the order to have a look at all of the column blocks for a given row block then we get the next benefits:
   1. Every row block turns into embarrassingly parallel. How you understand that is by trying on the illustrations above. You want all of the column blocks for a given row block to totally kind the eye output. In the event you had been to run all of the column blocks in parallel, you’ll find yourself with a race situation that can attempt to replace the identical rows of the output on the similar time. However not in case you do it the opposite manner round. Though there are atomic add operators in triton which might assist, they could doubtlessly set us again.
   2. We are able to keep away from hitting the HBM to get the worldwide Ms and Ls. We are able to initialize one on the chip for every kernel.
   3. Additionally we shouldn’t have to scale all of the output replace phrases with the brand new estimate of L. We are able to simply compute stuff with out dividing by L and on the finish of all of the column blocks, merely divide the output with the newest estimate of L, saving some FLOPS once more!
2. A lot of the development additionally comes within the type of the backward kernel. I’m omitting all of the backward kernels from this. However they’re a enjoyable train to try to implement, though they’re considerably extra complicated.

Listed here are some benchmarks:

The precise implementations of those kernels must take note of numerous nuances that we encounter in the actual world. I’ve tried to maintain it easy. However do check them out here.

Extra not too long ago in V3:

1. Newer GPUs, particularly the Hopper and Blackwell GPUs, have low precision modes (FP8 in Hopper and GP4 in Blackwell), which might double and quadruple the throughput for a similar energy and chip space and extra specialised GEMM (Common Matrix Multiply) kernels, which the earlier model of the algorithm fails to capitalize on. It is because there are various operations that are non-GEMM, like softmax, which reduces the utilization of those specialised GPU kernels.
2. The FA v1 and v2 are primarily synchronous. Recall within the v2 description I discussed that we’re restricted when column blocks attempt to write to the identical output pointers, or when we have now to go step-by-step utilizing the output from the earlier steps. Effectively these fashionable GPUs could make use particular directions to interrupt this synchrony.

3. Additionally they tailored the algorithm to focus on these specialised low precision Tensor cores on these new units, considerably growing the FLOPs.

Some extra benchmarks:

Conclusion

There may be a lot to admire of their work right here. The ground for this technical ability stage typically appeared excessive owing to the low stage particulars. However hopefully instruments like Triton might change the sport and get extra folks into this! The longer term is shiny.

References

[1] Qwen 2.5-7B-Instruct-1M Huggingface Model Page

[2] Tri Dao, Daniel Y. Fu, Stefano Ermon, Atri Rudra, and Christopher Re, FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness

[3] Tri Dao, FlashAttention-2: Faster Attention with Better Parallelism and Work Partitioning

[4] NVIDIA Hopper Architecture Page

[5] Jay Shah, Ganesh Bikshandi, Ying Zhang, Vijay Thakkar, Pradeep Ramani, Tri Dao, FlashAttention-3: Fast and Accurate Attention with Asynchrony and Low-precision

[6] Single-precision floating-point format, Wikipedia