Maths of AI: Machine learning and Deep learning as search optimization problems

In my teaching, I find it easier to explain machine learning and deep learning in terms of maths. Machine learning is a stochastic problem i.e. the solution is affected by randomness. A deterministic solution does not exist.

In this sense, its easier to describe machine learning as a search problem. We can think of Learning and optimization as two aspects of finding an optimal solution from the (vast)? search space.??Machine learning as a search optimization problem can be understood by breaking it down into its core components: the "search" and the "optimization."

The Search:

In machine learning, the search refers to the quest for the best possible model or algorithm that can make the most accurate predictions or decisions based on input data. The machine sifts through a vast space of possible models, parameters, or solutions to find the one that works best.?

The Optimization:

Optimization is about finding the most optimal parameters or settings for a given model. This is typically done through a process known as "training" where the model makes predictions on a set of data and then adjusts its parameters to improve its accuracy based on the errors it made. The objective is to minimize these errors, often represented as a loss function, which quantifies how far off the model's predictions are from the actual values. The process of optimization involves algorithms like gradient descent, which iteratively adjusts the parameters to find the minimum of this loss function.

When you combine the search for the right model with the optimization of its parameters, you get the full machine learning problem. It's a two-level problem:

Outer Level (Model Selection): Searching through different models or types of algorithms to find the best one.

Inner Level (Parameter Optimization): Optimizing the parameters of the chosen model to make the best possible predictions.

This process isn't easy. It involves challenges like overfitting (where the model performs well on the training data but poorly on unseen data), underfitting (where the model is too simple to capture the complexity of the data), and the curse of dimensionality (where the search space becomes exponentially large as the number of features increases).

Over time, various techniques have been developed to make this search and optimization process more efficient and effective. These include regularizations (to prevent overfitting), cross-validation (for better model evaluation), and various optimization algorithms beyond gradient descent (like stochastic gradient descent, Adam, etc.).

Deep learning, a subset of machine learning, can also be framed as a search optimization problem, but with its unique characteristics and complexities.?

In deep learning, the "search" involves finding the best possible neural network architecture and set of parameters (weights and biases) that enable the network to accurately represent and predict complex patterns and relationships in data. The search space in deep learning is typically much larger than in traditional machine learning because of the depth and complexity of the networks involved. Each layer of neurons adds a new dimension to the search, making the space exponentially larger and the search more challenging.

Deep learning models are composed of many layers of neurons, each transforming the input in a non-linear way. The "depth" of these models is what gives them their power but also what makes the optimization problem so complex. The architecture of the network itself (how many layers, how many neurons in each layer, what types of layers, etc.) is part of the search. Finding the right architecture is often done through experimentation, heuristic techniques, or more recently, through the use of large language models.

Optimizing a deep learning model means adjusting its millions (or even billions) of parameters so that the model performs well. This is typically done through backpropagation and gradient descent or variations thereof. The loss function, which measures the difference between the model's predictions and the actual data, guides this optimization. However, due to the high dimensionality and complexity of the models, this landscape is riddled with local minima, plateaus, and other challenging terrain that makes optimization a tough journey.

Various techniques and methodologies are employed to navigate the optimization landscape of deep learning effectively:

Regularization techniques like dropout, L2 regularization, and early stopping are used to prevent overfitting and help the model generalize better to unseen data.

Advanced optimizers like Adam, RMSprop, and SGD with momentum are designed to navigate the complex optimization landscape more effectively than standard gradient descent.

Learning rate schedules and adaptive learning rates are used to adjust how the model learns over time, helping it to settle into global minima.

Deep learning's effectiveness is also heavily dependent on large amounts of data. More data helps in more accurately defining the optimization landscape and guiding the model toward the global minimum. It also helps the model generalize better to new, unseen data.

Viewing deep learning as a search optimization problem helps in understanding the nature of learning in these complex networks. It highlights the need for efficient search strategies and robust optimization techniques. It also underscores the challenges like avoiding overfitting, navigating high-dimensional spaces, and selecting the right model architecture.

Thus, deep learning as a search optimization problem is about finding the best neural network architecture and parameters that allow the network to learn and make predictions from complex data. It involves navigating a vast, complex optimization landscape with the help of advanced techniques and algorithms to find the model that can best capture and represent the underlying patterns in the data.

I find that many people who are new to machine learning and deep learning find it easier to think of it as a search and optimization problem.?

Happy new year!

Image source:

https://pixabay.com/photos/grassland-nature-rural-expanse-7625729/?