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Transformer Inference Arithmetic | kipply’s weblog

Transformer Inference Arithmetic | kipply’s weblog

2023-05-13 03:17:21

This text presents detailed few-principles reasoning about massive language mannequin inference efficiency, with no experiments or troublesome math. The quantity of understanding that may be acquired this fashion is basically spectacular and sensible! A quite simple mannequin of latency for inference seems to be match for emprical outcomes. It is helped me make higher predictions and kind higher explanations about transformer inference.

This submit assumes some prior data about transformers, say at having understood most of The Illustrated Transformer however not having internalised all of it. Familiarity with this parameter counting submit which I developed together with this one might also be helpful.

Desk of Contents

  • kv cache explains the efficiency enchancment of caching self-attention vectors as part of inferencing, in addition to the doable tradeoffs and capability prices
  • capacity takes the storage price of kv cache and connects it to the storage price of mannequin weights and what capability means for efficiency.
  • model parallelism builds up an understanding particularly of tensor parallelism to obviously determine the price of communication
  • latency calculations pulls understanding from different ideas to create equations that function floorlines for inference velocity.
  • batch sizes discusses what impression batch measurement has on efficiency and what sizes could also be optimum.
  • flops counting steps by way of the transformer blocks and identifies which operations meaningfully contribute to flops velocity.
  • intermediate memory costs covers how the activations take further reminiscence and what that reminiscence bandwidth prices appears like from some actual benchmarks.
  • comparing against real benchmarks compares what we are able to calculate to what Nvidia FasterTransformer benchmarks report and identifies the discrepancies.

kv cache

For sampling, transformer inference consists of processing a offered immediate/context (which might occur in parallel), after which sampling further tokens one after the other (that is the place the autoregressiveness surfaces). Within the sampling, the transformer performs self-attention, which requires the kv values for every merchandise presently within the sequence (whether or not it was immediate/context or a generated token). These vectors are offered a matrix referred to as the kv cache, aka previous cache (the open supply GPT-2 implementation known as it previous). The previous cache can be formed like [batch, 2, num_heads, seq_len, features].

The aim of that is to keep away from recalculations of these vectors each time we pattern a token. With the computed (okay, v ) values, we are able to save fairly a little bit of computation at the price of some storage. Per token, the variety of bytes we retailer is

The primary issue of two is to account for the 2 vectors, (okay) and (v). We retailer that per every layer, and every of these values is a ( n_text{heads}instances d_text{head}) matrix. Then multiply by 2 once more for the variety of bytes (we’ll assume 16-bit codecs all through the submit).

The weights that we multiply by the token embeddings are (W_text{okay}, W_text{v} in mathbb{R}^{d_text{mannequin}instances d_text{mannequin}}) after which every token embedding is (t_text{e}in mathbb{R}^{1times d_text{mannequin}}). So then the flops to compute (okay) and (v) for all our layers is

We multiply (t_text{e}) by (W_text{okay}), which takes (2 cdot {d_text{mannequin}}^2) flops. We have now one other issue of two as we do this twice, as soon as every for (okay) and (v) after which repeat for (n_text{layers}).

What number of flops in a matmul?

The computation for a matrix-vector multiplication is (2mn) for (A in mathbb{R}^{mtimes n}, b in mathbb{R}^{n}). A matrix-matrix is (2mnp) for (A in mathbb{R}^{mtimes n}, B in mathbb{R}^{n instances p}). The (mn) issue makes a variety of sense, and the 2 comes from the truth that a matmuls are composed of multiply(1)-add(2) operations. Extra in these lecture notes.

This implies for a 52B parameter mannequin (taking Anthropic’s, the place (d_text{mannequin} = 8192) and (n_text{layers} = 64)). The flops are

Say we have now an A100 GPU, which does (312text{e}12) flops per second and (1.5text{e}12) bytes per second of reminiscence bandwidth. The next are numbers for simply the kv weights and computations.

Flops vs Reminiscence Boundedness

Flops vs reminiscence boundedness is one thing we take care of so much for transformer inference, however also in deep learning optimisation in general. To do the computations we do, we have to load weights which prices memory bandwidth. We assume (appropriately, this has been very nicely optimised) that we are able to begin the computations whereas we load the weights. Flop sure would then imply that there’s time when nothing is being handed by way of reminiscence, and reminiscence sure would imply that no floperations are occuring. Nvidia makes use of the time period math bandwidth which I discover actually cute. Technically, this delineation exist per kernel however may be abstracted to exist for teams of operations.

Not one of the mannequin structure issues anymore — we get a definite ratio right here of 208 given this {hardware} specification. Which means if we will compute kv for one token, it’s going to take the identical period of time to compute for as much as 208 tokens! Something under, we’re reminiscence bandwidth sure. Above, flops sure. If we used the remainder of our weights to do a full forwards cross (run the remainder of the transformer) on our context, it is also 208 (each the numerator and denominator get an element of 6 added). This will likely be reasoned totally in future sections. The intersection of the under diagram is at 208, although in actuality the reminiscence line does have a slight slope as a result of reminiscence price of intermediate calculations (mentioned within the final part).

For a 52B mannequin full forwards cross, that is (12cdot 2 cdot n_text{layers} cdot {d_text{mannequin}}^2 / 1.5text{e}12 approx 69) milliseconds for as much as 208 tokens (in observe, we might use 4 GPUs in parallel so it could really be ~17 milliseconds, extra in following sections). If we had 416 (double) tokens within the context, then it could take twice as lengthy, and 312 tokens would take 1.5 instances as lengthy.

Calculating for a kv cache token is strictly 1/sixth of the compute of passing the token by way of the mannequin. Normally, these forwards passes (what we expertise in getting logits, embeddings and coaching) are very low-cost due to the parallelism that’s doable versus sampling the place we’re pressured to learn by way of all of the weights for every token and do the autoregression.

This does not imply that 1/sixth of the time is saved! Let’s assume we’re flops sure. Then at every pattern step, we save (2 cdot 2 cdot n_text{tokens} cdot n_text{layers} cdot {d_text{mannequin}}^2 div 312text{e}12) flops whereas the decoding steps prices (2 cdot 12 cdot n_text{layers} cdot {d_text{mannequin}}^2 div 312text{e}12). Thus at every step we save 1/6 of the flops time multiplied by the variety of tokens in our sequence (massive!) — which will increase as we pattern tokens. It’s the case that and not using a kv cache, sampling can be quadratic in time complexity as we improve the variety of tokens.

This isn’t the entire story (given overheads and tradeoffs related to storing this cache). If we’re serving small batches we could also be reminiscence bandwidth sure moderately than flops, during which case we can’t even need to use the previous cache and can as an alternative fortunately do recomputations, spending the flops (we’ll already be paying the reminiscence price to do our sampling).


We have now a stable concept of the 2 issues we retailer in our GPUs — kv cache and weights. GPU capability does come into play for transformer inferencing efficiency and we have now all of the understanding we have to consider that now!

Nvidia A100 GPUs (that are typically talking, one of the best GPUs we are able to get for inference) have a normal of 40GB of capability. There are ones with 80GB and better reminiscence bandwidth (2e12 as an alternative of 1.5e12) however they don’t seem to be obtainable with any massive cloud providers but which implies they don’t seem to be actual to me!

Given the parameter rely, we are able to multiply by two to get bytes. So to calculate the dimensions of the weights for a 52B mannequin.

Oh no! This does not slot in one GPU! We might want no less than three GPUs simply to have all of the weights loaded in (will talk about how to do this sharding later). That leaves us (120-104 = 16GB) left for our kv cache. Is that sufficient? Again to our equation for kv cache reminiscence per token, once more with a 52B mannequin;

After which we might do (16/0.002 approx 8000) tokens can match into our kv cache with this GPU arrange, or that we may do as much as a batch measurement 4 the place every request has as much as 2048 tokens (and better sizes for much less tokens).

This sucks as a result of we wish to have the ability to do greater batch sizes, however are capability restricted! Larger batch sizes are extra environment friendly when it comes to how a lot GPU time it takes to course of the identical request. Then again, at batch sizes this low we’re sure to be reminiscence sure, and may forego the kv cache and simply pay the flops price as an alternative.

For 4 GPUs, we might get (56/0.002 approx 23000). We positively need to go for the 4 GPUs since we’ll need to have the ability to do greater batch sizes, and it is foolish to to divide powers of two over three GPUs. But it surely’s not simply batch measurement! If we have now excessive quantity, then we might have a number of cases of our fashions. We roughly need every occasion to have the ability to do as massive as a batch measurement as doable, as we pay the price of storing the weights anyway.

There’s some further house utilized by intermediate calculation steps, however they’re negligible.

mannequin parallelism

I am not going to construct up full understanding of mannequin parallelism and all of the implementation particulars, as a result of many have done so. However we are going to construct out the elements of the understanding which can be helpful to determine to make efficiency choices and calculate communication prices!

The result of mannequin parallelism, is that the price of passing all of the weights by way of by way of reminiscence and the flops are all divided over the diploma (variety of accelerators we use).

We’ll assume tensor parallel (mannequin parallel) the place we are going to break up down the center of the mannequin. Every accelerator will execute as a lot as it may with its shards of the weights and can talk each time synchronisation is required. A extra naive manner is pipeline parallel, the place every GPU will maintain onto a fraction of the layers. This does efficiently even out the load loading price, however has the apparent foolish that every one however one GPU will likely be idling! In coaching you would pipeline by way of it (as the primary batch strikes onto the subsequent GPU, begin on a brand new batch on the primary GPU) however it would not work out for a single pattern request (although you would nonetheless do it for a number of requests). Pipeline additionally would not exhaust the reminiscence bandwidth, which is definitely okay in the event you’re flops sure anyway. The one place the place pipeline parallel does higher is communications. A pipeline parallel mannequin would talk (d_text{mannequin}) per accelerator, whereas a mannequin parallel does (Ncdot d_text{mannequin}) per layer the place (N) is the variety of accelerators.

Right here we introduce the final fixed for our A100 GPUs which is a communication bandwith of 300GB/s. The doc marks it as 600GB/s as a result of Nvidia is including up 300GB/s into every chip and 300GB/s out concurrently moderately than utilizing a bidirectional quantity (which will likely be extra intuitive for our calculations).

On this diagram, we begin by following the yellow brick street the place we insert our token embeddings into the underside of the mannequin. The purple containers define how our weights can be break up throughout the accelerators, and we work with a particularly tiny mannequin so we are able to draw every thing to scale. A normal concept is that if we have now two matrices (X) and (Y) we are able to shard each of them and multiply the shards. This does not really full the matmul of (Xcdot Y), and a simple technique to inform (apart from our capacity to multiply matrices) is that if we concatenated the results of multiplying the shards, we get too massive of a matrix. As an alternative, we’d need to talk, compute a shard sum, talk that sum again out after which concatenate for the output of (X cdot Y).

For consideration the parallelism is intuitive from the truth that we have now a number of heads. We undergo a lot of the consideration layer with out communication as a result of our consideration heads are concatenated to multiply by (W_o). After we multiply by (v), we multiply the outcome by our shard of (W_o) to get a shard of (o_s in mathbb{R}^{d_text{mannequin}instances n_text{heads}/N}). Then every accelerator will talk its personal shard to all of the others, and all of the others will talk their shards again. That is ((N-1)d_text{mannequin}/N) of comms price. Every accelerator will do an excellent share of the addition to get the output projection, then do the identical communication they did final time and the person hosts will do the concatenation (roughly instantaneous).

The MLP layer is by nature very related! Similar to we have now (W_o) to undertaking our multi-headed consideration outcomes again right down to a vector of size (d_text{mannequin}), we have now (W_1in mathbb{R}^{4times d_text{mannequin}}) and (W_2in mathbb{R}^{d_text{mannequin}instances 4}) to make a dimension 4 instances bigger after which undertaking it again down. The identical two communications are accomplished on the finish of the MLP.

In the end we do (4 cdot (N – 1)d_text{mannequin}/N) bytes of communication. The kv cache is break up throughout heads by GPU.

latency calculations

We have mentioned the capability pretty totally, mapped out comms within the mannequin parallelism part and mentioned normal compute steps. Now we’ll construct it into equations that estimate latency!

Our latency calculations are principally concerning the flops vs reminiscence boundedness. If we have now a small variety of multiplies to do per parameter, then perhaps we’ll be throttled by reminiscence bandwidth. Flops are elevated by each batch measurement and variety of parameters, whereas reminiscence is simply elevated by variety of parameters.

For comms, it isn’t about boundedness, however moderately about including a latency time period and a throughput time period (the 300GB/s). One thing tough concerning the latency facet of this determine is that it isn’t reported, so one of the best I can do is guess “roughly small”, which is roughly 8 microseconds per message despatched as discovered on this Citadel paper however it’s for V100 NVLink.

Due to the compute components, to calculate the latency of a single token decoding step we might have two formulation – one for reminiscence bandwidth sure (small batch) and one other for flops sure (massive batch). For giant batch, we’ll drop the latency issue for communications.

Equations for a small batch (say 1, so we are able to drop the batch issue) can be; (the place (N) is the variety of accelerators and (P) is the variety of parameters and (b) is “byte” as a unit)

There’s (2 cdot P) as a result of we have to cross all of the parameters by way of the reminiscence, and every parameter is 2 bytes. (A_text{bm}) is the accelerator reminiscence bandwidth, and this price is break up throughout accelerators. For comms, we have now ( 4 cdot n_text{layers} ) communications per layer, and the latency per every request. Comms will often come out to be comparatively small so for the compute sure case we can’t want to concentrate to it anyway. There’s additionally a throughput price in comms which additionally rounds away.

There’s one other sometimes-significant issue right here which is the learn time for the kv cache, which I am going to omit of the equation now because it is dependent upon variety of context tokens, which might even differ inside a batch and whole variety of tokens we need to pattern. This might be calculated as reminiscence bandwidth time. One other lacking reminiscence bandwidth time is the learn of the unembeddings to calculate logits at every sampling step, which is ( in mathbb{R}^{d_text{mannequin}instances n_text{vocab}}).

As beforehand talked about, the reminiscence doesn’t really keep fixed, moderately some further reminiscence is used per batch for intermediate activations. The explanation we do not issue this in is just because it is exhausting to rely because it varies so much by the software program stack, compiler optimisations, and so forth.

For giant batches (say 512), the place (B) is the batch measurement;

The place (A_f) is the flops of the accelerator and (A_c) is the comms bandwidth. We do (2cdot P) flops of operations, which may be intuited by the truth that we matmul by way of all of the parameters, and as talked about earlier, a matrix-vector multiplication is (2mn) given (A in mathbb{R}^{mtimes n}, b in mathbb{R}^{n}).

For comms, we see the 4 (I am going to spherical that (N-1) issue to (N)) communications every of a (d_{mannequin}) measurement vector per layer as defined within the mannequin parallelism part. We swapped out the latency calculation for a throughput one. Then it is all divided by the comms bandwidth.

Let’s play with a bigger mannequin, a Gopher sized 260B mannequin on 16 GPUs. For a small batch, it is 22 ms per token generated. The throughput price for the comms which we are able to calculate with the equation for giant batch is roughly 35 microseconds, assuring us that it was protected to drop.

For a big batch of 512, for a complete of 53 ms per token generated (per batch, so within the 62ms 512 tokens are generated). The latency price on comms right here would’ve additionally been 3ms (latency will not be multiplied by batch because the message may be ready collectively) which is considerably important to drop however it’s high quality if we assuming parallel comms and compute.

The upper worth between the comms and compute is taken as we’re assuming that it’s parallel. Thus, we’d need to keep away from having comms being higher than compute (that is the mechanism that forestalls us from approaching latency zero as we insert extra chips, ultimately the comms will begin taking increasingly more time). It isn’t assured that every one methods will do that in parallel, and definitely not completely in parallel.

These numbers are positively a lot decrease than what we are able to get with actual sand, because it assumes optimum {hardware} utilization, would not think about softmaxes, assumes zero comms latency and ignores many different smaller components. Nonetheless, all of the reasoning behind this math is helpful for enthusiastic about the place to go optimise efficiency what deltas incoming optimisations will trigger.

batch sizes

Batch measurement is a vital issue of our efficiency, particularly in the direction of understanding efficiency for particular usages.

Within the earlier part, we have now two calculations for when one thing reminiscence bandwidth sure versus flops sure. To determine which is at play we are able to examine these numbers;

We’re coping with the identical ratio we discovered within the kv cache part. The min batch measurement for reminiscence bandwidth sure is (A_text{bw}/A_f = 208). This can be a helpful ratio! If we have now the load to do it, we desire flops sure because it’s extra compute environment friendly. Although it is also the case that if we’re flops sure, making the batch measurement bigger does not imply something is getting quicker.

To calculate when the capability goes from principally kv cache to principally weights is trivial, and in addition is not a binary in the identical manner (nothing particular occurs when your kv cache begins taking over extra reminiscence than your weights). Nothing particular actually occurs with comms both. In some unspecified time in the future in growing the batch measurement, the throughput begins dwarfing the latency so we dropped that issue. As noticed beforehand, the latency turns into insignificant a lot later (our 512 batch on 52B communication price was nonetheless 11% latency).

One thing oversimplified about comms is that it occurs at 4 completely different steps, which implies we do not simply need our compute time to be longer than our comms time, we wish it to be the case at every step (if we are able to parallelise the compute and comms). For that, we have now a weirder ratio: flops per byte of comms. This is a pleasant chart of our computations, which will even be helpful within the part under.

(q, okay, v) (o) (w_1) (w_2)
flops (3B({d_text{mannequin}}^2)) (B({d_text{mannequin}}^2)) (4B({d_text{mannequin}}^2)) (4B({d_text{mannequin}}^2))
bytes of comms (B(d_text{mannequin})) (B(d_text{mannequin})) (B(d_text{mannequin})) (B(d_text{mannequin}))
flops/byte (3(d_text{mannequin})) (d_text{mannequin}) (4(d_text{mannequin})) (4(d_text{mannequin}))

(312text{e}12 div 300text{e}9 = 1040), which is our flops per byte of comms for our A100s. We would like the values within the final row to be bigger than our {hardware} flops per byte in order that we keep flops sure (assuming we’re not reminiscence sure right here). For any mannequin with an embedding dimension over 1024 (per chip), we’re protected! For 512, it is a little bit awkward.

A low-load API could end in smaller batch sizes, resulting in cheap choices like dropping the kv cache. If an API had the load for giant batches it could in all probability need to serve the bottom batch measurement that will get flop sure even when there’s capability left in order that it may optimise for per-request-latency. In say mass-inferencing jobs like AlphaCode we’d need to insert as many chips as we are able to after which do the most important batch we are able to do with that capability. I say “could” so much right here however I really suppose these are absolute and all three sorts of circumstances.

flops counting


We do (2cdot P) flops of operations, which may be intuited by the truth that we matmul by way of all of the parameters.

That is right reasoning, however we are able to break it down by strolling by way of all of the transformer steps and verify that we get (2P).

The next calculations are per token, per layer. I describe (W_q, W_k, W_v in mathbb{R}^{d_text{mannequin}instances d_text{mannequin}}) the place it is extra correct to say we have now (W_q^i, W_k^i, W_v^i in mathbb{R}^{d_text{mannequin}instances d_text{head}}), the place (i) goes as much as (n_text{heads}). However for the sake of calculating latency, I simplify (W_q, W_k, W_v) to incorporate all of the heads.

  • Computing qkv
    • Multiply (t_e in mathbb{R}^{1times d_text{mannequin}}) by (W_q, W_k, W_v in mathbb{R}^{d_text{mannequin}instances d_text{mannequin}})
    • Flop rely: ({2 cdot 3 cdot d_text{mannequin}}^2)
  • Calculate z
    • That is (textual content{softmax}((qcdot okay)divsqrt{d_text{head}}) cdot v = z)
    • No matrices are multiplied, the variety of flops is a few issue of (d_text{mannequin}).
  • Multiply by the output projection matrix
    • Multiply (W_o in mathbb{R}^{d_text{mannequin}instances d_text{mannequin}}), by (z in mathbb{R}^{d_text{mannequin}times1})
    • Flop rely: (2 cdot {d_text{mannequin}}^2)
  • Feed-forward
    • We have now our MLP weights (W_1 in mathbb{R}^{4times d_text{mannequin}}, W_2 in mathbb{R}^{d_text{mannequin}instances 4} ) for 2 linear transformations (there is a ReLU within the center, which small).
    • Flop rely: (2cdot 8 cdot {d_text{mannequin}}^2 )
  • Another issues
    • There are sometimes layernorm that occur after every consideration, the place the weights there are a vector of size (d_text{mannequin}).
    • There’s one other linear layer after which a softmax that sits on high, which is our output (token) embedding or unembedding or de-embedding or embedding(^{-1}).
    • The unique transformer has a cosine absolute positional encoding scheme, which is an addition operation on the token embedding.

Including up all of the flops!

Subbing in our 8192 mannequin, we must always get about 100B flops;

103079215104 over two is about 51.5B. We’re a lil beneath (we get 51.5B as an alternative of 52B) however that is as a result of token (un)embeddings are almost a billion parameters. It could be cheap to do the latency calculations with (2cdot 12cdot n_text{layers} cdot {d_text{mannequin}}^2) as an alternative of (2cdot P), however it’s lower than a 2% distinction.

What concerning the the calculation of (z) and all the opposite steps I did not rely? These are all vector-vector (and even vector-scalar) operations, so they’re constructed round an element of (d_text{mannequin}) moderately than ({d_text{mannequin}}^2). Even when we had 100 of those operations per layer, it could come out to 100 million flops, which is 0.1% of the variety of flops we counted.

intermediate reminiscence prices

Data Movement Is All You Need (which is usually about optimising low stage information motion for transformers, and is not a very related learn) has a pleasant manner of classifying operations. We have now tensor contractions, that are the massive matmuls we have principally cared about (together with the linear layers). Then there are statistical normalisations, the softmax and layernorm. Lastly, which this submit has utterly ignored until now are element-wise operators, that are issues like biases, dropouts and activations.

So how will we calculate the latency of these matmuls, layernorms, and so forth? The reported flops on our {hardware} is specificially for the multiply-add operations so it could not be proper to rely it in there even when we may rely the flops. Shock! It is solely to price reminiscence to do the softmax learn/writes as that is what the bandwidth to flops ratio favours. That is the latency issue that has been alluded to!

I will break character on the first-principles facet of this and talk about Desk A.1 from the Data Movement Is All You Need paper. Right here we see that the latency for softmax is definitely barely greater than the calculations for qkv (that are a 1/3 of the time). This can be a little regarding!

For a similar purpose the softmax is reminiscence sure, so is the multiplication of qk, ReLU and dropout are additionally fairly costly.

GPU Kernel Fusion

See Also

GPUs execute operations in models of “kernels”. Kernel fusion signifies that one thing that was often 2 kernels can develop into one, and the first use is to reuse hundreds into reminiscence and scale back redundant hundreds and shops. For instance, a multiply-add is one kernel. If it had been two, then one would load+add+retailer and the second would load+multiply+retailer. We may save a variety of journeys by doing load+add+multiply+retailer.

We are able to additionally inform the softmax right here will not be completely fused by counting the variety of read-writes we must always want. In principle it may simply be one learn and one write (the usual is uh, four so I am bullying a bit). For qk, it could be two reads and one write (the 2 reads can in all probability be saved). The three to at least one ratio then, signifies that the softmax is doing extra reminiscence passes than is perfect. I say this, as a result of this expresses how a lot this counting is software program dependents and wishes experiments to estimate, since in principle the associated fee may very well be zero.

It is also price noting that the share of time these operations take will get smaller rapidly as mannequin measurement will increase because the reminiscence will improve on the order of (d_text{mannequin}) whereas the flops improve on the order of ({d_text{mannequin}}^2) — per layer. The paper is a 336M param mannequin, (d_text{mannequin} = 1024, n_text{layers} = 24).

I added up the latency of all of the values within the “Ours” column that had been reminiscence sure, together with the element-wise operations. The result’s that these intermediate steps take 43% of the time. In a mannequin of measurement 52B (the place (d_text{mannequin}) is 8 instances bigger, we see these operations develop into much less important.

The period of those reminiscence sure intermediate operations will take 8 instances longer because the operations are vectors of size (d_text{mannequin}). Nonetheless, the variety of flops will improve by 64 instances, which implies the flop time will increase by 64 instances.

So utilizing the optimisations in that paper, a 52B mannequin inference latency can be about 5% of those intermediate calculations we did not issue into latency.

evaluating towards actual benchmarks

I work at a language modelling firm that has its personal infrastructure and present benchmarks however uh, IP is tough. There’s a sadly small variety of public benchmarks obtainable for mannequin parallel inferencing? It looks as if the one public engines for this are Nvidia FasterTransformer and Microsoft Deepspeed with different benchmarks in all probability scattered in papers I do not know exist. Anywho, we are able to confirm our calculations towards some actual benchmarks!

As a result of I solely need to use 2 GPUs, I’ve run a 13B parameter mannequin with FasterTransformer, which does a bunch of excellent kernel fusing and gives us with tensor parallelism. 13B is 40 layers, 40 heads, every of dim 128 for a dim measurement of 5120. I’ve screenshots of the profiles in here and there are a bunch of fascinating issues in there which may make one other submit.

We’ll begin with a 512 context size, batch measurement 1 and 10 tokens outputted. For a small batch for one token on 2 GPUs we count on 8.4ms, and about 1ms of comms. For 1 GPU, that will be 16.8ms and 0 comms. (2x40x12x5120^2/1.5e12)

Excuse my mangled important figures, I in all probability ought to’ve stored the mem bandwidth to 1.555 as an alternative of 1.5.

Our empirical outcome for 1 GPU is 22.0ms, which means our guess was 76% there. We are able to really safely account for all of this, the place we all know some proportion will go to intermediate activations, and that we do not really get 100% of our theoretical reminiscence bandwidth. For these dimensions, a profile tells us we stand up to about 90% of our full reminiscence bandwidth (the place I examine the anticipated price of a matmul to the period of a single matmul kernel and rounded up because the bandwidth utilization varies fairly a bit relying on the tensors being loaded). Counting that in, we count on to take 18.5ms. Including up the price of intermediate activations (which we are able to do from a profile) we get one other 2.2ms, getting us to twenty.7 ms! To account for the final 1.4 ms there are another sub-millisecond operations like token embeddings, doing top-(okay|p), much less internet bandwidth than 90% (I could not be bothered to truly common every thing I took the best bw utilization I may discover) and even kernal launch instances.

Our emprical outcome for two GPUs is 13.5. We’re farther off this time, for under 62% of the way in which there. We’d verify the profile once more to see the reminiscence bandwidth (which we count on to be barely worse, as smaller tensors have a tendency to have the ability to get much less of the bandwidth). This time, it would not fairly get to 90, extra like 87, getting us to 9.5ms. The intermediate activations take the same period of time (2ms), getting us 11.7ms. With the remaining 1.5 ms then, we’re looking for comms! That is simply coated by our calculated 1ms of comms not being parallelised. From the profile, our comms take 40-50microseconds per layer, for a complete of 1.7ish ms of comms time, which accounts for every thing fairly nicely!

I believe for each of these operations, the counting of intermediate activations was a bit greater than it needs to be, as a result of the profile gave constantly slightly-higher latencies than the uncooked benchmarking run. The output of the benchmark run was 180.86 ms (context time: 45.45 ms) and 283.60 ms (context time: 63.17 ms).

However what concerning the forwards cross? I count on the forwards cross to take num_tokens/flops_to_bw_ratio instances so long as a decoding step. It’s because we have now to ship all the tokens to all of the GPUs, and every GPU will do their heads of consideration on it and retailer kv. Let’s use the up to date reminiscence bandwidth, 312e12/(1.5e12x0.9)=231. Trying on the 1 GPU setup, the place 22 is our anticipated decoding step, we see the 22*(512/231) = 48 which isn’t fairly the claimed 63. For two GPUs we get 13.5*(512/231) = 30ms, even worse!

For the one gpu, among the lacking time ought to simply be kv storing. Trying on the profiles, that is 18 microseconds per layer, 0.7ms. There are some Memsets for 0.2ms. We count on the flop time (that is flops sure!) for one in every of our MLP multiplies to be 512x4x5120^2×2/312e12 = 344 microseconds. In observe, that is 476 on the lowest which implies we get 72% of the flops we count on? For the projection within the consideration we count on we 512×5120^2×2/312e12 = 86 microseconds. In profiles we discover this to be 159 on the lowest, which is 54%. Yikes! I panicked for a bit, however uh that is apparently simply the flops we count on? See Determine 14 in this paper the place a 512x4000x4000 finally ends up getting lower than 150TFLOPs/s.

workout routines

  1. Given batch measurement, context size and next_n, how can we calculate the financial savings of utilizing kv cache?

  2. What overheads does the kv cache add in reminiscence time?

  3. Can we be reminiscence sure on our forwards cross however flops sure at every sampling step?

  4. What tradeoffs and calculations ought to we contemplate for utilizing extra GPUs than is critical for capability? Say for instance, a 52B mannequin on 8 or 16 GPUs as an alternative of 4.

  5. We got here up with formulation to calculate time to foretell one token. How would we calculate the time to do a whole pattern, from doing the forwards cross on the context to predicting all of the tokens requested?

  6. Within the capacity part, I say the reminiscence of intermediate calculations are negligble. How small are they precisely?

  7. Within the batch sizes part, we went a bit off subject and talked concerning the flops per byte of communication. What are the tradeoffs if we had an embedding dimension measurement of 512?

  8. We assume GPUs connected to the identical host right here, however may talk GPUs between hosts like we do in coaching. AWS has 400gb/s. What about it!

  9. In model parallelism, we may in observe talk all of the shards after which have every accelerator do all of the addition, as an alternative of only a share of their addition. What are the latency implications there?

  10. Attempt calculating the big batch velocity for a 52B on 4xGPUs at batch measurement 256. The compute needs to be about 21ms and comms needs to be about 4ms.

  11. Contemplate the operation of taking the vector out of the final layer and multiplying it by the unembedding matrix, storing the logits after which doing top-k or top-p sampling (which requires a form). How lengthy ought to this take for a 52B mannequin, and what can we parallelise right here?

  12. How can we shard the token embeddings? Would shard the enter token embeddings in another way from the unembeddings? Layernorms? What further communication does this incur?


Wish to lengthen credit score and because of individuals who make a constructive impression on this submit in various capacities. James Bradbury, Eric Zhang, Taylor Rogalski, Horace He, Julian Schrittwieser, Reiner Pope, Jim Wu, Mohammad Bavarian, Tudor Brindus and Adrien Morisot with James main by a protracted shot.


Please cite as:

Chen, Carol. "Transformer Inference Arithmetic",, 2022.

hey kipply you need to higher perceive our massive mannequin inferencing latency

sure that is a terrific concept i am going to look into it!

cool i might like to see the profile

if i sit in a darkish room on my own lengthy sufficient i feel i can clarify all of the milliseconds


The architectures and latencies expressed on this submit are these of publicly identified or theoretical fashions and benchmarks and don’t essentially replicate the architectures or latencies of my employer’s fashions.

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