Importance of Parameter Initialization

Optimization algorithms like SGD, Momentum, and Adam intelligently update model weights based on gradients calculated during backpropagation. However, these algorithms require a starting point. The initial values assigned to the weights and biases of a neural network before training begins play a surprisingly significant role in the training process. This is the concept of parameter initialization.

Think of the loss landscape of a neural network as a complex terrain with hills, valleys, and plateaus. The goal of optimization is to navigate this terrain to find a low point (a good solution). Where you start your descent, the initial weight configuration, heavily influences whether you find a good valley quickly, get stuck on a plateau, or even fall off a metaphorical cliff (encountering numerical instability).

Why Does Initialization Matter?

Poor initialization can lead to several training difficulties:

Symmetry Breaking: If all weights connected to a layer are initialized to the same value (e.g., zero), all neurons in that layer will produce the same output during the forward pass. Consequently, during backpropagation, they will all receive the identical gradient signal and update in the exact same way. This symmetry prevents the neurons from learning different features, effectively reducing the capacity of the layer. Initializing weights with small random values is the first step to break this symmetry.
Vanishing Gradients: If weights are initialized too small, the activations and gradients can progressively shrink as they propagate through the network (especially backward during gradient calculation). In deep networks, this can cause gradients in the earlier layers to become extremely close to zero. When gradients vanish, the weights in those layers stop updating, effectively halting learning for a significant portion of the network. This is particularly problematic with activation functions like sigmoid or tanh, which saturate (output values flatten out near 0 or 1, or -1 and 1, respectively) and have small gradients in their saturated regions.
Exploding Gradients: Conversely, if weights are initialized too large, the activations and gradients can grow exponentially as they propagate through the network. This leads to massive gradient values, causing unstable updates that overshoot optimal points or even result in numerical overflow (represented as NaN - Not a Number). Exploding gradients make training diverge completely.

Consider the effect on activations. We ideally want the outputs of neurons (activations) across different layers to have roughly similar variances. If the variance keeps increasing layer after layer, activations might become very large, leading to saturation or exploding gradients. If the variance keeps decreasing, activations might shrink towards zero, contributing to vanishing gradients.

Activation distributions in a layer based on initialization. Poor initialization can lead to activations clustering near zero (vanishing) or saturating near the bounds of activation functions like tanh (e.g., -1 or 1). Good initialization aims for a healthier spread.

Proper initialization aims to set the initial weights such that the activations and gradients stay within a reasonable range throughout the network, promoting faster convergence and avoiding these issues. It helps ensure that the signal flows effectively both forward (for predictions) and backward (for learning).

In the following sections, we will examine specific initialization strategies like Xavier/Glorot and He initialization, which are designed based on mathematical principles to maintain appropriate activation and gradient variances, significantly improving the trainability of deep networks.

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References

Understanding the difficulty of training deep feedforward neural networks, Xavier Glorot, Yoshua Bengio, 2010 Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, Vol. 9 (PMLR) DOI: 10.5598/AISTATS.2010.53 - Introduces Xavier initialization, a method for setting initial weights to help maintain activation variance and mitigate vanishing/exploding gradients in deep networks.
Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification, Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun, 2015 Proceedings of the IEEE International Conference on Computer Vision (ICCV) DOI: 10.1109/ICCV.2015.122 - Proposes He initialization, a strategy specifically designed for neural networks using ReLU activations to address issues like vanishing gradients by preserving signal strength.
Deep Learning, Ian Goodfellow, Yoshua Bengio, and Aaron Courville, 2016 (MIT Press) - Provides a comprehensive theoretical and practical discussion of parameter initialization techniques, including the motivations and challenges (see Chapter 8.4).
Neural Networks Part 2: Setting up the Data and the Preprocessing, Loss, Optimization, Andrej Karpathy, Justin Johnson, and Fei-Fei Li, 2023 (Stanford University) - Offers a practical overview of parameter initialization, explaining common problems and effective strategies for deep neural networks.