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+# Advanced Attention
+
+## KV Caching
+
+The idea behind this is that during autoregression in autoregressive transformers,
+such as GPT, values for $K$ and $V$ are always recomputed, wasting computing power
+
+![autoregression without caching](./pngs/decoder-autoregression-no-cache.gif)
+> GIF taken from https://medium.com/@joaolages/kv-caching-explained-276520203249
+
+As we can notice, all tokens for previous steps gets recomputer, as well
+K and V values. So, a solution is to cache all keys and values until step $n$
+and compute only that value.
+
+![autoregression with caching](./pngs/decoder-autoregression-cache.gif)
+> GIF taken from https://medium.com/@joaolages/kv-caching-explained-276520203249
+
+Moreover, if we discard taking $QK^T$ for previous steps, we can just obtain
+the token we are interested in.
+
+To compute the size needed to have a $KV$ cache, let's go step by step:
+
+- For each layer, we need to store both $K$ and $V$ that are of the same
+  dimensions (in this context), the number of heads, so the *"number"*
+  of $K$ matrices, the head dimension and the number of tokens incoming,
+  the sequence lenght, and
+  we need to know the number of bytes for `d_type`, usually a `float16`, thus
+  2 Bytes:
+
+  $$
+  \text{Layer\_Space} = 2 \times \text{n\_heads} \times \text{d\_heads} \times
+  \text{seq\_len} \times \text{d\_type}
+  $$
+
+- Now, during training, we pass a minibatch, so we will have a tensor of
+  dimensions $N \times \text{seq\_length} \times \text{d\_model}$. When they
+  are processed by $W_K$ and $W_Q$, we will have to store times $N$ more
+  values:
+
+  $$
+  \text{Batch\_Layer\_Space} = \text{Layer\_Space} \times N
+  $$
+
+- If you have $L$ layers, during training, at the end you'll need space
+  equivalent to:
+
+  $$
+  \text{Model\_Space} = \text{Batch\_Layer\_Space} \times L
+  $$
+
+## Multi-Query and Grouped-Query Attention
+
+The idea here is that we don't need different keys and vectors, but only
+different queries. These approaches drastrically reduce memory consumption at
+a slight cost of accuracy.
+
+In **multi-query** approach, we have only one $K$ and $V$ with a number of
+queries equal to the number of heads. In the **grouped-query** approach, we have
+a hyperparameter $G$ that will determine how many $K$s and $V$s matrices are
+in the layer, while the number of queries remains equal to the number of attention
+heads. Then, queries will be grouped to some $K_g$ and $V_g$
+
+![grouped head attention](./pngs/grouped-head-attention.png)
+> Image take from [Multi-Query & Grouped-Query Attention](https://tinkerd.net/blog/machine-learning/multi-query-attention/#multi-query-attention-mqa)
+
+Now, the new layer size becomes:
+
+$$
+\text{Layer\_Space} = 2 \times G \times \text{d\_heads} \times
+\text{seq\_len} \times \text{d\_type}
+$$
+
+## Multi-Head Latent Attention
+
+The idea is to reduce memory consumption by `rank` factoring $Q$, $K$, and $V$
+computation. This means that each matrix will be factorized in 2 matrices so
+that:
+
+$$
+A = M_l \times M_r
+\\
+A \in \R^{in \times out}, M_l \in \R^{in \times rank},
+M_r \in \R^{rank \times out}
+$$
+
+The problem, though, is that this method introduces compression, and the lower
+$rank$ is, the more compression artifacts.
+
+What we are going to compress are the weight matrices for $Q$, $K$ and $V$:
+
+$$
+\begin{aligned}
+    Q = X \times W_Q^L \times W_Q^R
+\end{aligned} \\
+X \in \R^{N\times S \times d_{\text{model}}},
+W_Q \in \R^{d_{\text{model}} \times (n_\text{head} \cdot d_\text{head})}
+\\
+W_Q^L\in \R^{d_{\text{model}} \times rank},
+W_Q^R\in \R^{rank \times (n_\text{head} \cdot d_\text{head})}
+\\
+\text{}
+\\
+W_Q \simeq W_Q^L \times W_Q^R
+$$
+
+For simplicity, we didn't write equations for $K$ and $V$ that are basically
+the same. However, now we may think that we have just increased the number of
+operations from 1 matmul to 2 per each matrix. But the real power lies
+when we take a look at the actual computation:
+
+$$
+\begin{aligned}
+H_i &= \text{softmax}\left(
+    \frac{
+        XW_Q^LW_{Q,i}^R \times (XW_{KV}^LW_{K,i}^R)^T
+    }{
+        \sqrt{\text{d\_model}}
+    }
+\right) \times X W_{KV}^L W_{V,i}^R \\
+&= \text{softmax}\left(
+    \frac{
+        XW_Q^LW_{Q,i}^R \times W_{K,i}^{R^T} W_{KV}^{L^T} X^T
+    }{
+        \sqrt{\text{d\_model}}
+    }
+\right) \times X W_{KV}^L W_{V,i}^R \\
+&= \text{softmax}\left(
+    \frac{
+        C_{Q} \times W_{Q,i}^R  W_{K,i}^{R^T} \times C_{KV}^T
+    }{
+        \sqrt{\text{d\_model}}
+    }
+\right) \times C_{KV} W_{V,i}^R \\
+\end{aligned}
+$$
+
+As it can be seen, $C_Q$ and $C_{KV}$ do not depend on the head, so they can be
+computed once and shared across all heads, then we have
+$W_{Q,i}^R \times W_{K,i}^{R^T}$ that while it depends on the head number, it
+can be computer ahead of time as it does not depend on the input.
+
+So, for each attention head we need just 3 matmuls plus another 2 happening
+at runtime. Moreover, if we want to, we can still apply caching over $C_{KV}$
+
+## Decoupled RoPE for Multi-Latent Head Attention
+
+Since `RoPE` is a positional embedding used **during** attention, this causes
+problems if we use a standard Multi Latent Head Attention. In fact, it
+shoudl be used on both, separately though, matrices $Q$ and $K$.
+
+Since they come from $C_Q \times W_{Q, i}^R$ and
+$W_{KV, i}^{R^T} \times C_{KV}^T$, this means that we can't cache
+$W_{Q,i}^R \times W_{K,i}^{R^T}$ anymore.
+
+A solution is to cache head matrices, but not their product, and compute
+ new pieces that will be used in `RoPE` and then concatenated
+to the actual query and keys:
+
+$$
+\begin{aligned}
+Q_{R,i} &= RoPE(C_Q \times W_{QR,i}^R) \\
+K_{R,i} &= RoPE(X \times W_{KR,i}^L)
+\end{aligned}
+$$
+
+These matrices will then be concatenated with the reconstruciton of $Q$ and $K$.
+
+## References
+
+- [KV Caching Explained: Optimizing Transformer Inference Efficiency](https://huggingface.co/blog/not-lain/kv-caching)
+- [Transformers KV Caching Explained](https://medium.com/@joaolages/kv-caching-explained-276520203249)
+- [How to calculate size of KV cache](https://www.rohan-paul.com/p/how-to-calculate-size-of-kv-cache)
+- [Multi-Query & Grouped-Query Attention](https://tinkerd.net/blog/machine-learning/multi-query-attention/#multi-query-attention-mqa)
+- [Understanding Multi-Head Latent Attention](https://planetbanatt.net/articles/mla.html)
+- [https://machinelearningmastery.com/a-gentle-introduction-to-multi-head-latent-attention-mla/](https://machinelearningmastery.com/a-gentle-introduction-to-multi-head-latent-attention-mla/)
+- [DeepSeek-V3 Explained 1: Multi-head Latent Attention](https://medium.com/data-science/deepseek-v3-explained-1-multi-head-latent-attention-ed6bee2a67c4)
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