Incorrect. Self-attention is a highly parallel algorithm that makes it a great candidate for being a memory-bound workload once you have enough compute.
Both datacenter grade CPUs and GPUs have enough compute to carry out the self-attention computation but it is only the latter that has enough hi-bandwidth memory to make the algorithm really perform. If this hadn't been the case, the theory behind flash-attention wouldn't materialize, and it does, and reason being that (main) memory is slow.
Transformers are deep feedforward networks that happen to also have attention. Causal LMs are super memory bound during inference due to kv caching as all of those linear layers need to be loaded onto the core to transform only a single token per step.
This obviously depends on the hardware and the shape of the LLM model itself but, generally speaking, it's quite the opposite. The idea of batching is to grow the compute bandwidth per single request, bigger batch sizes with much more compute will put more stress to the underlying (cache, RAM) memory subsystem, no?
For N self-attention layers, there will be N compute (tensor) units doing the computation in parallel. To retire the computation, each compute unit will need to LOAD/STORE from and to the chip memory. At batch size B, this only becomes a bigger scale, e.g. B * (N, LOAD/STORE).
If you have a batch of size 1, for every token you need to load the entire model from memory into cache as you go through it. If it is 32 you can produce 32 tokens while doing the same amount of loading from VRAM.
That's not how it works because if what you're saying had been true then the self-attention memory complexity would be O(1), e.g. regardless of the batch size. This obviously isn't the case since each batch computation necessitates it's own load/store memory bandwidth. I suggest reading one of the transformers papers to really understand how it works.
