From Equilibrium Propagation (EqProp) to Functional Gradient Descent (FGD)

- Equilibrium Propagation (EqProp)

- Functional Gradient Descent (FGD)

- Neural Implicit Surfaces (NIS)

- Forward Propagation

- Backward Propagation (Backpropagation)

- Gradient Descent

Equilibrium Propagation (EqProp)

Equilibrium Propagation (EqProp) is an alternative to the backpropagation method commonly used to train neural networks. Inspired by physics, EqProp allows neural networks to learn by iteratively settling into “low-energy” states, similar to how physical systems naturally settle into states of equilibrium. This approach is seen as a more efficient and biologically realistic way to train AI models, particularly for future hardware that mimics the brain.

Equilibrium Propagation treats a neural network like a dynamic system that seeks balance or equilibrium. When an input is fed into the network, it begins to settle into an initial stable state. Once it reaches this equilibrium, a slight change or "perturbation" is introduced to push the network towards the desired outcome, creating a new equilibrium. The model learns by updating its weights based on the difference between the original and the new equilibrium states.

Unlike backpropagation, which requires calculating gradients and propagating errors layer by layer, EqProp operates locally. Each neuron or unit in the network adjusts itself based on local signals, which makes the model inherently more efficient, particularly on neuromorphic hardware—hardware designed to function similarly to the human brain. This efficiency makes EqProp suitable for hardware with limited resources, such as mobile devices, or in scenarios where energy efficiency is a priority, like in autonomous robots or wearables. EqProp’s biologically-inspired approach also brings us closer to understanding how learning might occur in the brain, where local interactions between neurons adjust their connections based on input stimuli.

In energy-efficient AI applications, like battery-operated devices or edge computing, EqProp could allow AI systems to learn while consuming less power. For instance, a wearable health device might use EqProp to continuously learn from sensor data about the wearer’s health, adapting to new patterns without draining the battery quickly.

Functional Gradient Descent (FGD)

Functional Gradient Descent (FGD) is an optimization technique that operates in “function space” rather than the traditional “parameter space.” Unlike regular gradient descent, where the model tries to adjust fixed parameters to fit the data, FGD focuses on adjusting entire functions, making it particularly useful for complex systems where outputs change continuously, like those in physics or economics.

In traditional gradient descent, the algorithm adjusts the weights of a model layer by layer, changing these fixed numbers to reduce error and fit the training data. In Functional Gradient Descent, however, we treat the solution as a function that changes and evolves based on the data, which allows for more flexibility. Think of it like painting on a canvas: instead of adjusting only the brush's position (parameters), we are adjusting the brushstrokes as continuous movements on the entire canvas (function space).

FGD does not see each step as a fixed adjustment but as part of a larger evolving solution. This approach is especially useful when dealing with complex systems, such as those governed by differential equations (e.g., predicting weather patterns, market fluctuations), where the relationship between variables is constantly changing. FGD enables models to adapt in real time to new data and conditions without retraining from scratch, making it suitable for dynamic environments where conditions are constantly shifting, like robotics or autonomous systems.

In financial modeling, FGD can be used to adaptively adjust pricing models for complex financial instruments, such as derivatives, where market conditions are volatile and unpredictable. FGD can optimize the model by learning to adjust the pricing function continuously in response to market changes, leading to more accurate valuations and better risk management.

Neural Implicit Surfaces (NIS)

Neural Implicit Surfaces (NIS) are a technique for representing complex 3D shapes and surfaces using neural networks. Instead of using traditional 3D models that store data as meshes or grids (collections of points and lines), NIS represents a 3D shape as a continuous function inside the network, making it highly efficient and capable of handling intricate shapes with less storage.

In Neural Implicit Surfaces, the neural network learns a continuous function that determines whether any given point in space is inside, on, or outside the surface of an object. To build a 3D object, NIS doesn’t store all of the object's points (like pixels in an image); instead, it calculates them as needed. Given a coordinate in 3D space, the neural network outputs a scalar (single number) that tells us the distance or location relative to the object’s surface. If the output value meets a specific condition (for instance, zero), then the point is on the surface of the object.

This approach means the network doesn’t need to store vast amounts of data for every part of the shape. Instead, it simply needs the parameters for the function it has learned, allowing for smaller storage requirements and smoother, more detailed models. Neural Implicit Surfaces can be particularly useful when building models that need to be flexible and adaptable, like those used in video games, virtual reality (VR), or medical imaging.

In medical imaging, NIS can be used to create 3D models of organs from only a few scans, such as CT or MRI slices. By learning the shape of an organ, NIS can "fill in the gaps" and create a detailed model without needing an exhaustive 3D scan, saving time and computational power. This can be used by doctors to analyze an organ’s shape or size in 3D, helping in diagnosis or surgery planning.

Forward Propagation

Forward Propagation is the process of passing input data through a neural network to produce an output. It’s the “forward” pass in the learning process, where data moves from the input layer, through hidden layers, to the output layer, generating predictions.

In a neural network, each layer consists of nodes (or “neurons”), and each node is connected to nodes in the next layer by weights. When data enters the network at the input layer, it goes through a series of transformations as it moves through each layer. Each node in the network takes the input data, applies a weight to it, sums up the values, and then passes the result through an activation function (like ReLU or Sigmoid). This process allows the network to capture complex patterns in the data.

Forward Propagation is responsible for producing the network's predictions based on the current state of the weights. If it’s a classification task, for example, the output might be probabilities for each class. The accuracy of the prediction depends on the weights in the network, which are adjusted later through Backward Propagation and Gradient Descent to minimize errors.

In image recognition, if a neural network is given an image of a cat as input, Forward Propagation would process the image through layers to output a prediction—such as “cat” with a certain confidence level. This initial prediction might not be accurate if the model is untrained, but it forms the starting point for learning.

Backward Propagation (Backpropagation)

Backward Propagation, commonly called Backpropagation, is a process used to update the weights of a neural network based on the error in the predictions. It’s the “backward” pass in the learning process, where the network adjusts itself to minimize the difference between the predicted output and the actual output.

After Forward Propagation generates an output, the model compares it to the actual target (the correct answer). The difference between the two, called the loss or error, is calculated using a loss function (such as Mean Squared Error or Cross-Entropy). Backpropagation then takes this error and moves backward through the network, layer by layer, calculating how much each weight contributed to the error.

The goal of Backpropagation is to understand the contribution of each weight to the overall error, so that adjustments can be made. It does this using partial derivatives (from calculus) to measure the effect of changing each weight on the error. This step-by-step gradient calculation is called the chain rule, where Backpropagation computes the gradient (direction and rate of change) for each weight in the network.

Once these gradients are known, the network can update its weights accordingly to reduce the error in the next round. This weight update process is handled by Gradient Descent, which optimizes the network’s predictions over time.

In spam email detection, after Forward Propagation predicts whether an email is spam or not, Backpropagation evaluates the prediction error. If the model incorrectly labels a spam email as non-spam, Backpropagation calculates the gradients for each word or feature in the email that contributed to the mistake, and adjusts the weights so that the model can better detect spam in future emails.

Gradient Descent

Gradient Descent is an optimization algorithm used in training machine learning models. It is the process of adjusting model parameters (like weights in a neural network) to minimize error by following the “slope” of the error function. Think of it as a method to “descend” or reduce the error in the model’s predictions.

The goal of Gradient Descent is to find the optimal set of weights for the neural network by minimizing the loss function (the difference between predicted and actual output). During each training iteration, the model calculates the gradient of the loss function with respect to each weight in the network. A gradient is a vector that tells us the direction in which to move each weight to reduce the error the most efficiently.

In simple terms, Gradient Descent takes a "step" in the direction that reduces the error. The size of this step is controlled by the learning rate, a hyperparameter that determines how quickly the model updates its weights. If the learning rate is too high, the model might “overshoot” the optimal point; if it’s too low, training may take too long to converge.

There are several variations of Gradient Descent, including Stochastic Gradient Descent (SGD), where updates are made after each individual example, and Mini-Batch Gradient Descent, where updates are made after a small batch of examples. These variations help models generalize better and improve efficiency in large datasets.

In linear regression, Gradient Descent is used to find the best-fit line for a set of data points. The model starts with random values for the slope and intercept, then uses Gradient Descent to adjust these values step by step, minimizing the distance between the predicted and actual points, until it arrives at the line that best represents the data.