Solving Regularized Exp, Cosh and Sinh Regression Problems

State-of-the-art language models like Transformer [69], BERT [21], GPT-3 [6], PaLM [18], and OPT [80] exhibit greater proficiency in natural language processing when compared to smaller models or traditional techniques. These models have the capacity to understand and generate intricate language, proving beneficial in various applications such as language translation, sentiment analysis, and question answering. LLMs can be customized for multiple purposes without necessitating their reconstruction from scratch. An instance of this is ChatGPT, an OpenAI-developed chat software that employs GPT-3’s full potential. The latest iteration, GPT-4 [53], has the potential to surpass GPT-3 in its impressive capabilities, including text generation, question answering, and language translation. This development could lead to significant implications in the field of NLP, with new applications potentially emerging in areas such as virtual assistants, chatbots, and automatic content creation. However, even though deep learning has a swift incline in popularity, we hold the belief that there exist discrepancies in our comprehension of the concept of attention and the reasoning behind its effectiveness.

The primary technical foundation behind LLMs is the attention matrix [69, 55, 21, 6, 3, 77]. An attention matrix is a matrix that features rows and columns aligning with individual words or ”tokens” and their relationships within a given text. Its purpose is to measure the critical nature of each token in a sequence in relation to the intended output. The attention matrix is learned during training. These parameters are optimized to maximize the model’s accuracy in predicting the desired output. Through the attention mechanism, each input token is evaluated based on its importance or relevance to the desired output. This is achieved by weighing the token score, which is based on a similarity function comparing the current output state and input states.

More formally, the attention matrix can be expressed by considering two matrices, $Q$ and $K$, containing query and key tokens, respectively. Both $Q$ and $K$ hold values in the $n\times d$ dimensional space. The attention matrix can be denoted by the square matrix $A$ which is of size $n\times n$. This matrix establishes a relationship between the input tokens in the sequence where every entry represents the attention weight or score between a particular input token (query token $Q$) and an output token (key token $K$). It is essential to mention that diagonal entries of this matrix display self-attention scores, signifying the importance of each token with respect to itself. A majority of the methods used for effective computation of attention matrices are divided into two primary categories based on their approach. One approach involves leveraging sparsity, as seen in Reformer [45], while the other involves utilizing the low-rank attributes of the attention matrices, as observed in Linformer [73] and Performer [16].

During training, our primary goal is to tackle the issue of multiple attention regression by utilizing the exponential function and its corresponding equation: $\min_{X}\|D^{-1}\exp(AX)-B\|_{2}$, where $D^{-1}$ denotes the normalization factor. However, upon further investigation, it has been brought to our attention that the single regression scenario has not been thoroughly studied. As a result, this study is centered on the situation of single regression.

In our setting, the presence of $D$ is unnecessary due to the fact that $Ax$ is a column vector (but not a matrix). Thus, we have opted to address the optimization problem with regards to $\min_{x}\|\exp(Ax)-b\|_{2}$. It is important to note that our approach is not exclusive to the exponential function, and can also be extended to other hyperbolic functions such as $\cosh$ and $\sinh$.¹¹1$\cosh(x):=\frac{1}{2}(e^{x}+e^{-x})$ and $\sinh(x):=\frac{1}{2}(e^{x}-e^{-x})$

In this section, we provide preliminaries to be used in our paper. In Section 2.1 we introduce notations we use. In Section 2.2 we provide some facts about approximate computations and exact computations. In Section 2.3, we provide some trivial facts regarding gradient and hessian. In Section 2.4, we computed the gradient and hessian of a regularization term $L_{\operatorname{reg}}$.

In this section, we provide detailed analysis of $L_{\exp}$. In Section 3.1 we define the loss function $L_{\exp}$ based on $\exp(x)$. In Section 3.2 we compute the gradient of $L_{\exp}$ by detail. In Section 3.3 we compute the hessian of $L_{\exp}$ by detail. In Section 3.4, we summarize the result of Section 3.2 and Section 3.3 and aquire the gradient $\nabla L_{\exp}$ and hessian $\nabla^{2}L_{\exp}$ for $L_{\exp}$. In Section 3.5 we define $L_{\exp,\operatorname{reg}}$ by adding the regularization term $L_{\operatorname{reg}}$ in Section 2.4 to $L_{\exp}$ and compute the gradient $\nabla L_{\exp,\operatorname{reg}}$ and hessian $\nabla^{2}L_{\exp,\operatorname{reg}}$ of $L_{\exp,\operatorname{reg}}$. In Section 3.6 we proved that $\nabla^{2}L_{\exp,\operatorname{reg}}\succ 0$ and thus showed that $L_{\exp,\operatorname{reg}}$ is convex. In Section 3.7 we provide the upper bound for $\|\nabla^{2}L_{\exp,\operatorname{reg}}(x)-\nabla^{2}L_{\exp,\operatorname{reg}}(y)\|$ and thus proved $\nabla^{2}L_{\exp,\operatorname{reg}}$ is lipschitz.