Layer-wise Training for Stacked Autoencoders

Stacked autoencoders, as we've introduced, aim to learn hierarchical feature representations by creating deep networks with multiple hidden layers. The idea is that each successive layer learns more complex and abstract features based on the outputs of the previous layer. However, training such deep neural networks directly from a random initialization can be challenging. Problems like vanishing or exploding gradients, or the optimization process getting stuck in poor local minima, were significant hurdles, especially in the earlier days of deep learning.

To address these challenges and to effectively initialize the weights of a deep stacked autoencoder, a technique known as greedy layer-wise pre-training was developed. This approach breaks down the complex problem of training a deep network into a sequence of simpler, more manageable steps, training one layer at a time.

The Greedy Layer-wise Pre-training Process

The core idea is to train each layer of the stacked autoencoder individually as a shallow autoencoder. Once a layer is trained, its learned weights are fixed, and its output (the encoded representation) is then used as the input to train the next layer. Let's walk through the steps:

1. Train the First Autoencoder (AE1):

   • Take your original input data, let's call it $X$.
   • Train a simple autoencoder with a single hidden layer (Encoder1, Decoder1) to reconstruct $X$. The goal is to minimize the reconstruction error, for example, $L(X, \text{Decoder1}(\text{Encoder1}(X)))$.
   • Once AE1 is trained, you save the weights of its encoder (Encoder1) and its decoder (Decoder1). The Encoder1 transforms $X$ into a lower-dimensional representation $H_1 = \text{Encoder1}(X)$.

2. Train the Second Autoencoder (AE2):

   • Take the output $H_1$ (the activations from the hidden layer of AE1) and use it as the input data for a new simple autoencoder (AE2, with Encoder2 and Decoder2).
   • Train AE2 to reconstruct $H_1$. That is, minimize $L(H_1, \text{Decoder2}(\text{Encoder2}(H_1)))$.
   • After AE2 is trained, save the weights of Encoder2 and Decoder2. Encoder2 transforms $H_1$ into a new, potentially even more compressed representation $H_2 = \text{Encoder2}(H_1)$.

3. Repeat for Additional Layers:

   • If you want more layers in your stacked autoencoder, you continue this process. For the $k$-th autoencoder (AE_k), its input will be $H_{k-1}$ (the activations from Encoder_{k-1}), and it will be trained to reconstruct $H_{k-1}$.

4. Assemble the Stacked Autoencoder:

   • After all individual layers have been pre-trained, you assemble the full stacked autoencoder.
   • The encoder part of the stacked autoencoder is formed by stacking all the pre-trained encoders in sequence: $\text{Encoder}_{\text{stack}} = \text{Encoder}_N \circ \dots \circ \text{Encoder}_2 \circ \text{Encoder}_1$.
   • The decoder part of the stacked autoencoder is formed by stacking all the pre-trained decoders in reverse order: $\text{Decoder}_{\text{stack}} = \text{Decoder}_1 \circ \text{Decoder}_2 \circ \dots \circ \text{Decoder}_N$.
   • So, the input $X$ goes through Encoder1, then Encoder2, and so on, to produce the final latent representation. This latent representation then goes through DecoderN, then Decoder(N-1), etc., back to Decoder1, to reconstruct the original input $X$.

5. Fine-tuning the Entire Stack:

   • The weights obtained from the greedy layer-wise pre-training provide a good initialization for the deep network.
   • The final step is to treat the entire stacked autoencoder as a single model and train it end-to-end. This means you feed the original input $X$ to the stacked encoder, pass the result through the stacked decoder to get $\hat{X}$, and then update all the weights in all layers (both encoders and decoders) simultaneously using backpropagation to minimize the global reconstruction error $L(X, \hat{X})$.
   • This fine-tuning step allows the weights across all layers to adjust slightly to work more cohesively for the overall reconstruction task, often leading to improved performance.

The term "greedy" is used because each layer is trained to be optimal for its local task (reconstructing its own input) without initially considering the overall, global objective of the entire deep stack.

Below is a diagram illustrating the greedy layer-wise pre-training process for a stacked autoencoder with two hidden layers, followed by assembly and fine-tuning.

The greedy layer-wise pre-training procedure. Each autoencoder layer is trained sequentially. Then, the encoders and decoders are assembled into a deep stack, which is subsequently fine-tuned end-to-end.

Why is Layer-wise Training Effective?

This strategy was particularly beneficial because:

• Good Initialization: It provides a much better starting point for the weights of a deep network compared to random initialization. Each layer is already primed to extract useful features from its input.
• Hierarchical Feature Learning: By training layer by layer, the network naturally learns a hierarchy of features. The first layer learns basic features from the raw data, the second layer learns more complex features from the first layer's features, and so on.
• Mitigating Optimization Issues: Training shallower networks one at a time is generally easier and less prone to issues like vanishing gradients that can plague the end-to-end training of very deep networks from scratch. Each layer's training guides the parameters to a sensible region of the search space.

Relevance in Modern Deep Learning

While the advent of better activation functions (like ReLU), advanced optimization algorithms (e.g., Adam, RMSprop), normalization techniques (like Batch Normalization), and new architectural designs (like residual connections) has made direct end-to-end training of deep networks more feasible, greedy layer-wise pre-training is not obsolete.

• It can still be a valuable technique when dealing with very deep autoencoders or when computational resources for extensive hyperparameter search for end-to-end training are limited.
• It provides an intuitive way to build and understand hierarchical feature extractors.
• In some scenarios, especially with limited data, pre-training can help regularize the model and lead to better generalization.

Understanding layer-wise pre-training gives you insight into how deep representations can be built incrementally and provides another tool in your toolkit for constructing and training effective stacked autoencoders for feature extraction.