---
slug: thoughts/optimization
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tags:
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title: "ml optimization"
date: 2024-10-31
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---
## softmax

$$
\text{softmax(y)}_i = \frac{e^{y_i}}{\sum_{i} e^{y_i}}
$$

where $y \in \mathbb{R}^k$

## `exp()`

[Abdelkhalik et al. (2022)](#bib-abdelkhalik2022demystifyingnvidiaamperearchitecture), [RDNA3 instruction sets of V\_LDEXP\_F32](https://www.amd.com/content/dam/amd/en/documents/radeon-tech-docs/instruction-set-architectures/rdna3-shader-instruction-set-architecture-feb-2023_0.pdf)

Usually a lot better comparing to `2**t` simply for [numerical stability](https://aarnphm.xyz/thoughts/optimization/../../thoughts/university/twenty-three-twenty-four/compsci-4x03/Equations) reasons

For ARM the design specially [instructions](https://developer.arm.com/documentation/ddi0602/2024-09/SVE-Instructions/FEXPA--Floating-point-exponential-accelerator-) set for it!

```cpp title="pseudocode-exp-fexpa.cpp"
// Pseudocode representing the computation flow:
float32x4_t exp_sve2(float32x4_t x) {
    // Step 1: Range reduction
    // N = round(x * log2(e))
    // r = x - N * ln(2)    [reduced argument]
 
    // Step 2: FEXPA instruction provides 2^N approximation
    // In hardware: FEXPA Z0.S, Z1.S
    float32x4_t exp_approx; // Result of FEXPA
 
    // Step 3: Polynomial evaluation for exp(r)
    // Typically uses Horner's method with reduced precision
    // coefficients since we're starting with a good approximation
    float32x4_t exp_r = evaluate_polynomial(r);
 
    // Step 4: Combine results
    return exp_approx * exp_r;
}
```

Advantages of FEXPA:

- single instruction latency for initial approximation
- vectorized ops for batch processing

On GPU we can utilise bit-shift `1<<x` or CUDA’s exp2

Optimization in `llama.cpp`: [ggerganov/llama.cpp#7154](https://github.com/ggerganov/llama.cpp/pull/7154)

## RoPE

([Su et al., 2023, p. pg.V](#bib-su2023roformerenhancedtransformerrotary))

## sigmoid

$$
\text{sigmoid}(x) = \frac{1}{1+e^{-x}}
$$

## ReLU

$$
\text{FFN}(x, W_{1}, W_{2}, b_{1}, b_{2}) = max(0, xW_{1}+b_{1})W_{2} + b_{2}
$$

A version in T5 without bias:

$$
\text{FFN}_\text{ReLU}(x,W_{1},W_{2}) = max(xW_{1},0)W_{2}
$$

## Swish

[Ramachandran et al. (2017)](#bib-ramachandran2017searchingactivationfunctions) introduces an alternative to ReLU that works better on deeper models across different tasks.

$$
f(x) = x \cdotp \text{sigmoid}(\beta x) \\ \because \beta : \text{ constant parameter}
$$

## Gated Linear Units and Variants

> component-wise product of two linear transformations of the inputs, one of which is sigmoid-activated.

[Shazeer (2020)](#bib-shazeer2020gluvariantsimprovetransformer) introduces a few GELU activations to yield improvements in [Transformers](https://aarnphm.xyz/thoughts/optimization/../../thoughts/Transformers) architecture.

$$
\begin{aligned} \text{GLU}(x,W,V,b,c) &= \sigma(xW+b) \otimes (xV+c) \\ \text{Bilinear}(x,W,V,b,c) &= (xW+b) \otimes (xV+c) \end{aligned}
$$

GLU in other variants:

$$
\begin{aligned} \text{ReGLU}(x,W,V,b,c) &= \max(0, xW+b) \otimes (xV+c) \\ \text{GEGLU}(x,W,V,b,c) &= \text{GELU}(xW+b) \otimes (xV+c) \\ \text{SwiGLU}(x,W,V,b,c) &= \text{Swish}_\beta(xW+b) \otimes (xV+c) \end{aligned}
$$

FFN for transformers layers would become:

$$
\begin{aligned} \text{FFN}_\text{GLU}(x,W,V,W_{2}) &= (\sigma (xW) \otimes xV)W_{2} \\ \text{FFN}_\text{Bilinear}(x,W,V,W_{2}) &= (xW \otimes xV)W_{2} \\ \text{FFN}_\text{ReGLU}(x,W,V,W_{2}) &= (\max(0, xW) \otimes xV)W_{2} \\ \text{FFN}_\text{GEGLU}(x,W,V,W_{2}) &= (\text{GELU}(xW) \otimes xV)W_{2} \\ \text{FFN}_\text{SwiGLU}(x,W,V,W_{2}) &= (\text{Swish}_\beta(xW) \otimes xV)W_{2} \end{aligned}
$$

_note_: reduce number of hidden units $d_\text{ff}$ (second dimension of $W$ and $V$ and the first dimension of $W_{2}$) by a factor of $\frac{2}{3}$ when comparing these layers

## JumpReLU

[Erichson et al. (2019)](#bib-erichson2019jumpreluretrofitdefensestrategy) proposes JumpRELU to address robustness through adversarial examples.

[Rajamanoharan et al. (2024)](#bib-rajamanoharan2024jumpingaheadimprovingreconstruction) then apply this to improves construction fielity as [Gated SAE](https://aarnphm.xyz/thoughts/optimization/../../thoughts/sparse-autoencoder#gated-sae)

$$
J(z) \coloneqq z H(z - \kappa) = \begin{cases} 0 & \text{if } z \leq \kappa \\ z & \text{if } z > \kappa \end{cases}
$$

[](https://aarnphm.xyz/thoughts/optimization/../../thoughts/images/JumpReLU.mp4)

## momentum

See also [SGD](https://aarnphm.xyz/thoughts/optimization/../../thoughts/university/twenty-four-twenty-five/sfwr-4ml3/Stochastic-gradient-descent), [Cornell’s CS6787](https://www.cs.cornell.edu/courses/cs6787/2017fa/Lecture3.pdf), [gradient descent](https://aarnphm.xyz/thoughts/optimization/../../thoughts/gradient-descent)

In the case of quadratic function: $f(x) = \frac{1}{2} x^2$, then $x_{t+1} = x_t - \alpha x_t = (1-\alpha)x_t$

Think of convergence rate

$$
\mid x_{t+1} - 0 \mid = \mid 1 - \alpha \mid \mid x_t - 0 \mid
$$

![](https://aarnphm.xyz/thoughts/optimization/../../thoughts/images/convergence-vs-step-side-momentum.webp)

If we set different curvature ($f(x) = 2x^2$) thus $x_{t+1} = x_t - 4 \alpha x_t = (1-4 \alpha)x_t$

> [!tip] step size
>
> step size depends on curvature for one-dimensional quadratics
>
> more curvature means smaller ideal step size

_how would this play for general quadratics?_

for PSD symmetric $A$

$$
f(x) = \frac{1}{2} x^T Ax
$$

with gradient descent has update step

$$
x_{t+1} = x_t - \alpha A x_t = (I - \alpha A)x_t
$$

convergence rate would be

$$
\begin{aligned} \max_{x} \frac{\|(I - \alpha A)x\|}{\|x\|} &= \max_{x} \frac{1}{\|x\|} \left\| \left( I - \alpha \sum_{i=1}^{n} \lambda_i u_i u_i^T \right) x \right\| \\[8pt] &= \max_{x} \frac{\|\sum_{i=1}^{n} (1- \alpha \lambda_i) u_i u_i^T x\|}{\|\sum_{i=1}^{n} u_i u_i^T x\|} \\ &= max_i \mid 1- \alpha \lambda_i \mid \\ &=max(1-\alpha \lambda_{\text{min}}, \alpha \lambda_{\text{max}} - 1) \end{aligned}
$$

> [!math] optimal convergence rate
>
> optimal value occurs when
>
> $$
> 1 - \alpha \lambda_{\text{min}} = \alpha \lambda_{\text{max}} - 1 \Rightarrow \alpha = \frac{2}{\lambda_{\text{max}} + \lambda_{\text{min}}}
> $$
>
> with rate
>
> $$
> \frac{\lambda_{\text{max}} - \lambda_{\text{min}}}{\lambda_{\text{max}} + \lambda_{\text{min}}}
> $$

We denote $\kappa = \frac{\lambda_{\text{max}}}{\lambda_{\text{min}}}$ as **condition number** of matrix A

> [!abstract] poorly conditioned
>
> Problems with larger condition numbers converge slower.
>
> Intuitively these are problems that are _highly curved in some directions, but flat others_

### Polyak

abbreviation: “heavy ball method”

idea: add an extra momentum term to gradient descent

$$
x_{t+1} = x_t - \alpha \nabla f(x_t) + \beta (x_t - x_{t-1})
$$

tl/dr: if current gradient step is in same direction as previous step, then move a little further in the same direction

> [!math]- momentum for 1D quadratics
>
> $$
> f(x) = \frac{\lambda}{2} x^{2}
> $$
>
> momentum GD gives
>
> $$
> \begin{aligned} x_{t+1} &= x_t - \alpha \lambda x_t + \beta (x_t - x_{t-1}) \\ &= (1+\beta - \alpha \lambda) x_t - \beta x_{t-1} \end{aligned}
> $$
>
> characterizing momentum:
>
> - start with $x_{t+1} = (1+\beta -\alpha \lambda) x_t - \beta x_{t-1}$
> - trick: let $x_t = \beta^{t/2}z_t$
>
> $$
> z_{t+1} = \frac{1 + \beta - \alpha \lambda}{\sqrt{\beta}} z_t - z_{t-1}
> $$
>
> let $u = \frac{1+\beta -\alpha \lambda}{2 \sqrt{\beta}}$, then
>
> $$
> z_{t+1} = 2 u z_t - z_{t-1}
> $$
>
> _degree-$\textbf{t}$ polynomial in $\textbf{u}$_

### Nesterov

<!--transclude of thoughts/Nesterov-momentum start-->

[Nesterov-momentum](https://aarnphm.xyz/thoughts/optimization/../../thoughts/Nesterov-momentum)

<!--transclude of thoughts/Nesterov-momentum end-->

## Bibliographie

- Abdelkhalik, H., Arafa, Y., Santhi, N., & Badawy, A.-H. (2022). _Demystifying the Nvidia Ampere Architecture through Microbenchmarking and Instruction-level Analysis_. arXiv preprint arXiv:2208.11174 [\[arXiv\]![arXiv](https://aarnphm.xyz/thoughts/optimization/../../static/favicons/arxiv.avif)](https://arxiv.org/abs/2208.11174)
- Erichson, N. B., Yao, Z., & Mahoney, M. W. (2019). _JumpReLU: A Retrofit Defense Strategy for Adversarial Attacks_. arXiv preprint arXiv:1904.03750 [\[arXiv\]![arXiv](https://aarnphm.xyz/thoughts/optimization/../../static/favicons/arxiv.avif)](https://arxiv.org/abs/1904.03750)
- Rajamanoharan, S., Lieberum, T., Sonnerat, N., Conmy, A., Varma, V., Kramár, J., & Nanda, N. (2024). _Jumping Ahead: Improving Reconstruction Fidelity with JumpReLU Sparse Autoencoders_. arXiv preprint arXiv:2407.14435 [\[arXiv\]![arXiv](https://aarnphm.xyz/thoughts/optimization/../../static/favicons/arxiv.avif)](https://arxiv.org/abs/2407.14435)
- Ramachandran, P., Zoph, B., & Le, Q. V. (2017). _Searching for Activation Functions_. arXiv preprint arXiv:1710.05941 [\[arXiv\]![arXiv](https://aarnphm.xyz/thoughts/optimization/../../static/favicons/arxiv.avif)](https://arxiv.org/abs/1710.05941)
- Shazeer, N. (2020). _GLU Variants Improve Transformer_. arXiv preprint arXiv:2002.05202 [\[arXiv\]![arXiv](https://aarnphm.xyz/thoughts/optimization/../../static/favicons/arxiv.avif)](https://arxiv.org/abs/2002.05202)
- Su, J., Lu, Y., Pan, S., Murtadha, A., Wen, B., & Liu, Y. (2023). _RoFormer: Enhanced Transformer with Rotary Position Embedding_. arXiv preprint arXiv:2104.09864 [\[arXiv\]![arXiv](https://aarnphm.xyz/thoughts/optimization/../../static/favicons/arxiv.avif)](https://arxiv.org/abs/2104.09864)
