Xavier Initialization for RNN Stability | Generated by AI
Why Xavier Initialization in RNNs?
RNNs (Recurrent Neural Networks) rely on Xavier (also called Glorot) initialization for their weight matrices—like Wxh (input-to-hidden), Whh (hidden-to-hidden, the recurrent weights), and Why (hidden-to-output)—to maintain stable training dynamics. The specific form in your code uses a Gaussian (normal) distribution scaled by sqrt(1 / fan_in), where fan_in is the number of input neurons to that layer. This keeps the variance of the weights around 1 / fan_in, ensuring incoming signals don’t get amplified or squashed too aggressively.
Here’s why this is crucial for RNNs, and why a simple uniform random draw from [0, 1] would cause problems:
1. Preserving Signal Variance Across Layers and Time Steps
- In feedforward networks, Xavier helps keep the variance of activations roughly constant as signals propagate forward (and gradients backward). Without it, deep layers could see activations explode (become huge) or vanish (drop to near-zero), making training impossible.
- RNNs are like “deep” networks unrolled over time: The recurrent weight
Whhmultiplies the hidden state at each time step, creating a chain of multiplications (e.g., for sequence length T, it’s like T layers deep). If weights inWhhhave variance >1, gradients explode exponentially backward (bad for long sequences). If <1, they vanish. - Xavier’s scaling (e.g.,
* sqrt(1. / hidden_size)forWhh) ensures the expected variance of the hidden state stays ~1, preventing this. For [0,1] uniform init:- Mean ~0.5 (biased positive, causing drifts).
- Variance ~1/12 ≈ 0.083—too small for large
hidden_size(e.g., 512), leading to vanishing signals quickly.
2. Adapting to Layer Dimensions
- Xavier accounts for fan_in (inputs to the layer) and sometimes fan_out (outputs). For
Wxh, scaling bysqrt(1 / input_size)normalizes based on vocabulary size or embedding dims. ForWhh, it’s based onhidden_sizeto match the recurrent loop. - [0,1] ignores dimensions: In a large hidden layer (e.g., 1000 units), summing many [0,1] inputs without scaling would overwhelm the next layer. In small layers, it’d underflow.
3. Gradient Flow in Backpropagation Through Time (BPTT)
- RNN training uses BPTT, where gradients flow backward over T steps. Poor init leads to the “vanishing gradient problem” (common in vanilla RNNs, why LSTMs/GRUs exist). Xavier mitigates this by starting with balanced magnitudes.
- Empirical studies (e.g., Glorot & Bengio’s original paper) show Xavier reduces epochs needed for convergence by 2-10x in deep/recurrent nets.
Alternatives and When [0,1] Might “Work”
- [0,1] could “work” for tiny toy models (e.g., single-layer, short sequences) but fails at scale—think exploding losses or NaNs.
- Modern tweaks: He init (for ReLUs, scales by
sqrt(2 / fan_in)) or orthogonal init (for RNNs, preserves norms exactly). But Xavier is a safe default for sigmoid/tanh activations common in RNNs.
In short, Xavier is about stability first: It sets weights small-but-not-too-small, dimension-aware, to let gradients flow freely. Random [0,1] is too crude—like flooring a car without balancing the tires.
For deeper dives:
Understanding the Difficulty of Training Deep Feedforward Neural Networks (Glorot & Bengio, 2010)
Recurrent Neural Network Regularization (Zaremba et al., on RNN init issues)