Adam Epsilon & Convergence In Gaussian Splatting

Introduction

Hey guys! Let's dive into a fascinating discussion about optimizer settings, specifically the default epsilon value in the Adam optimizer, and how it might be affecting the convergence behavior in certain scenarios, particularly within the context of Gaussian Splatting. This is super important because, as developers and researchers, we always want our models to train efficiently and effectively. If a small setting like the epsilon is causing hiccups, we need to know about it! So, let's break down the issue, understand why it matters, and explore potential solutions. We'll be looking at a real-world example from the Gaussian Splatting repository and discussing how seemingly minor tweaks can have significant impacts on training dynamics.

The core of the discussion revolves around the default Adam epsilon value of 1e-8. While this value is commonly used and often works well, there are situations where it can be too small, especially when dealing with gradients that have specific characteristics. In the realm of Gaussian Splatting, opacity gradients are one such example. When the epsilon value is significantly larger than the second momentum term (v) in the Adam optimizer, it can skew the update calculations and lead to slower convergence. Imagine you're trying to fine-tune a musical instrument, and a tiny adjustment makes a big difference in the sound quality. Similarly, in optimization, a seemingly small parameter like epsilon can dramatically affect how quickly and accurately our models learn. We'll explore this dynamic in detail and see why opacity recovery, in particular, can be sluggish under these conditions. This is not just about theoretical possibilities; it's about real-world observations and practical implications for model training. We'll also touch on how these observations might influence other related parameters and default choices, such as the delete_opacity_threshold. Adjusting one setting often means re-evaluating others, creating a ripple effect that needs careful consideration. The goal here is not just to identify a problem but to understand the interconnectedness of optimization parameters and how they collectively shape the training landscape. So, buckle up, and let's get into the nitty-gritty of optimizer settings and their impact on convergence!

Understanding the Adam Optimizer and Epsilon

To really grasp the issue, let's quickly recap how the Adam optimizer works and what epsilon does. Adam (Adaptive Moment Estimation) is a popular optimization algorithm known for its efficiency and adaptability. It's like having a smart, self-adjusting tool in your toolbox that can navigate complex landscapes to find the lowest point – in our case, the minimum of the loss function. Adam combines the concepts of momentum and RMSprop to adapt the learning rates for each parameter individually. This means that parameters that have been updated more consistently receive smaller learning rate adjustments, while those with infrequent updates get larger adjustments. This dynamic adaptation is crucial for handling the diverse gradients encountered in deep learning models. The algorithm maintains two moving averages: the first momentum (m), which is an exponentially decaying average of past gradients, and the second momentum (v), which is an exponentially decaying average of the squared gradients. These moving averages help Adam estimate both the direction and scale of the gradient, allowing it to navigate efficiently through the optimization landscape. It’s like having a compass and a map that not only points you in the right direction but also tells you how steep the terrain is.

Now, where does epsilon come into play? Epsilon (ε) is a small constant added to the denominator of the update equation to prevent division by zero. Think of it as a safety net that keeps the optimizer from crashing when it encounters very small or zero values in the second momentum. Without epsilon, the update step could become extremely large or undefined, leading to instability and divergence. In essence, epsilon ensures numerical stability. However, even though its primary role is to prevent division by zero, the magnitude of epsilon can also affect the optimization process. A very small epsilon, like the default 1e-8, can sometimes dominate the update when the second momentum (v) is also small. This is where things get interesting. When epsilon is significantly larger than v, it can dampen the adaptive learning rate, causing the optimizer to take smaller steps than it should. It’s like trying to run with a slight drag on your feet – you can still move, but you're not as efficient as you could be. This can be particularly problematic in situations where we need to make significant adjustments, such as when opacities are being reset and need to