Intro

What’s a functor? You might use it daily without realizing it. For example, calling map on a collection means you’re using a functor.

This post explains functors from two perspectives: category theory and programming. By this end, you will have a deeper understanding of the concept.

Functor in category theory

A Recap of Self-attention Mechanism

In self-attention, the query ($\mathbf q_m$), key ($\mathbf k_n$), and value ($\mathbf v_n$) vectors are computed as follows:

$$ \begin{equation} \begin{aligned} \mathbf q_m&=f_q(\mathbf x_m,m)\\ \mathbf k_n&=f_k(\mathbf x_n,n)\\ \mathbf v_n&=f_v(\mathbf x_n,n) \end{aligned} \end{equation} $$

Here, the $\mathbf x_i$ is the $i$-th token embedding, while $n$ and $m$ denote different positions.

The attention score between position $m$ and $n$ is computed as:

$$ \alpha_{m,n}=\frac{exp(\frac{\mathbf q_m^T\mathbf k_n}{\sqrt d})}{\sum_{j=1}^Nexp(\frac{\mathbf q_m^T\mathbf k_j}{\sqrt d})} $$

Intro

It’s been 8 years since the famous transformer architecture was first proposed. You might have noticed that some modifications to the original design - for instance, most large language models (LLMs) now use RMSNorm¹ instead of LayerNorm. Today I will briefly introduce RMSNorm, but first, let’s recap LayerNorm.

LayerNorm Recap

$$ \mathbf y=\frac{\mathbf x-E[\mathbf x]}{\sqrt{Var(\mathbf x)+\epsilon}}*\gamma+\beta $$

The equation above shows how LayerNorm works. If we ignore the scaling factors ($\gamma, \beta$), LayerNorm’s behavior becomes intuitive: it transforms each input $\mathbf x$ into a feature vector with zero mean and unit standard deviation .

Intro

During my daily coding practice, I encountered an interesting problem - 1682. Flight Routes Check. Solving this problem requires finding all strongly connected components (SCCs) in a directed graph. After some research, I discovered Kosaraju’s algorithm, which solves this problem in linear time. That is, the time complexity is

$$ O(V+E) $$

Where $V$ refers to the nodes and $E$ refers to the edges in the graph.

By interesting, I mean that Kosaraju’s algorithm is easy to implement yet a bit tricky to understand fully. In my opinion, knowing why it works matters more than just memorizing how to code it. That’s why I’m sharing this short post - to break down the key insights.

Motivations

In PyTorch, multiple APIs exist for matrix multiplication operations. However, these functions often lead to memorization challenges. Additionally, many of these APIs require explicit dimension manipulation (e.g., permuting, reshaping)

Does there exist a magic API that can cover all the use cases? A potential unified solution is the torch.einsum API.

What is torch.einsum ?

The syntax of torch.einsum is

Intro

As a machine learning engineer, I work with vast amounts of data, performing tasks such as data cleaning and information extraction. These tasks are typically data-parallel and do not involve race conditions. Such workloads are usually referred to as CPU-intensive tasks. Typically, these tasks can be formulated as map(fn, data).

Python is limited by its Global Interpreter Lock (GIL). As a result, using multithreading could not improve the performance. Instead, multiprocessing should be used. Usually, you would not manually manage all the launched processes but would instead use a process pool. Today, I’d like to discuss two different APIs related to multiprocessing: multiprocessing.Pool and concurrent.futures.ProcessPoolExecutor, and provide a simple comparison.

The key to call graph construction

For an OOP programming language, the key challenge in call graph construction is handling the virtual call, as it may involve multiple target methods, as shown in the following table¹.

The method call and method signature

Let’s take Java as an example; a method call may have this form.

Motivations

There is a type of problem where you are given an array $arr$ and $Q$ queries. Each query is represented as $query(l, r)$, which asks for the sum of the elements in the subarray $[l, r]$, i.e., $arr[l] + arr[l + 1] + … + arr[r]$.

If we handle each query using a brute-force approach, the time complexity will be $O(N)$. Thus, solving $Q$ queries would require $O(NQ)$. Is there a more efficient approach?

Intro

What’s Multi-Head Attention (MHA)

In last post I have explained how the self-attention mechanism works. Today let’s take a step further and explore multi-head attention (MHA), which is the full version of self-attention as described in the original paper¹. Since I have covered most of the foundation concepts in last post, this post will be short. :)

Previously, we mentioned that the self-attention mechanism has three import matrices.

$$ \mathbf Q,\mathbf K,\mathbf V\in\mathcal{R}^{n\times d} $$