Stochastic Gradient Descent and the Global Optimum in Non-Convex Functions

In the context of optimization, particularly in machine learning and deep learning, Stochastic Gradient Descent (SGD) is a widely used algorithm to find the global minimum of complex non-convex functions. The question arises: if every critical point of a non-convex function is a global minimum, does SGD always find the global optimum? This article delves into the factors that influence the success of SGD in such scenarios.

Nature of the Function and Critical Points

For the function in question, if every critical point is a global minimum, this means the function has no local minima that are different from global minima. A critical point is a point where the first derivative is zero. This property ensures that any critical point discovered by SGD is indeed a global optimum. However, this property alone does not guarantee convergence to the global minimum without further considerations.

Convergence of SGD

SGD works by iteratively updating the parameters based on the gradient of the loss function in a stochastic manner. The key to convergence lies in the appropriate selection of hyperparameters, such as the learning rate. If the learning rate is too high, the updates will overshoot the minimum, and if it is too low, the algorithm may converge too slowly. Balancing this is crucial for efficient convergence.

Exploration of the Parameter Space

In non-convex functions, the challenge lies in how thoroughly SGD explores the parameter space. A critical issue is the possible presence of multiple local minima and saddle points. If the learning rate is too high, SGD may overshoot and fail to find a minimum. Conversely, if the rate is too low, the algorithm may get stuck in flat regions of the objective function, also known as saddle points, which can hamper convergence to a global minimum.

Initialization

The starting point (initialization) of SGD can significantly influence its trajectory. Good initialization strategies can help SGD converge more efficiently to a global minimum. However, even with a poor initialization, if the algorithm is given enough time, it may still reach the global minimum. The key is the combination of appropriate learning rate, initialization, and the overall structure of the objective function.

Stochastic Nature of SGD

The stochastic nature of SGD introduces randomness in the updates, which can help it escape from flat regions and explore the parameter space more broadly. This randomness can be beneficial in finding a global minimum. However, it is not foolproof; an excessively high step size can lead to erratic behavior and prevent convergence.

Trivial Counterexample

While the above conditions seem favorable, it is important to note that setting the step size to an excessively high value (e.g., infinity) would result in the algorithm jumping around randomly and not necessarily converging. This highlights the need for appropriate parameter tuning and the importance of considering practical constraints.

Conclusion

In summary, if every critical point of a non-convex function is a global minimum, SGD should be able to find the global optimum provided that the method is implemented correctly with appropriate hyperparameter tuning. The success of SGD depends on several factors, including the learning rate, initialization, and the structure of the objective function. Careful selection of these parameters, along with thorough exploration of the parameter space, can help ensure efficient and robust convergence to the global minimum.