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Minimize Computation Time by Using Convex Optimization in Machine Learning Algorithms

Key Takeaways

  • Today, convex optimization is used in image processing, communication systems, navigation, control, robotics, machine learning, and structural analysis.

  • If you are looking to train a system using machine learning algorithms, convex optimization will save a lot of computation time.

  • An optimization problem can be considered convex if it satisfies the following conditions (provided the objective is to minimize the function):

    • Objection function is a convex function 

    • Inequality constraints are convex sets 

    • Equality constraints are affine. 

Machine learning algorithm

Optimization is an important part of the machine learning algorithm

There are several optimization techniques such as continuous optimization, constrained optimization, discrete optimization, global optimization, linear programming, and convex optimization. For those who use scientific computation and optimization techniques in their work, the concept of convex optimization seems brilliant. Today, convex optimization is used in image processing, communication systems, navigation, control, robotics, machine learning, and structural analysis. Many types of engineers have greatly benefited from the use of convex optimization. 

Convex Optimization

In engineering, we optimize problems either in a general way or through case studies. Having a broad understanding of the problem helps engineers find a solution for a large set of similar problems. However, in certain optimization problems, we may be uncertain about the data. The knowledge available about the data set can be simply that it belongs to a certain set of values and all the constraints hold good for the same set. For optimization problems, gaining a proper understanding of the objective and constraints, and then finding an optimal solution satisfying all of them can be challenging.

If the objective function, equality, and inequality constraints are convex functions, then your optimization problem is convex. Linear functions are convex functions, therefore, linear programming problems are also convex problems. Even though convex optimization shares the properties of linear programming problems, the former is far more general than the latter. The convex optimization solution can be obtained at a faster rate, even when the number of variables and constraints is greater.

Before applying convex optimization to your problem, you need to check whether the problem itself is convex. It is difficult to recognize convexity, but understanding the convex function definition will be helpful.

Convex Functions

Convex function

Convex optimization can be applied to convex functions

A function, f(x), is convex when a chord drawn from any point, (x, f(x)) to (y, f(y)), lies on or above the graph of function f.  Mathematically, it can be represented by the following equation:

x,y ∈ f ⇒  𝛼 x + (1-𝛼) y ∈ f  for all 0 ≤𝛼 ≤ 1

The exponential functions eax and the power function xa are some examples of convex functions. The linear function is both convex and concave, and can be considered as a special case of a convex function. The least squares and quadratic function is another set of convex functions. 

An optimization problem can be considered convex if it satisfies the following conditions (provided the objective is to minimize the function):

  1. Objection function is a convex function. 

  2. Inequality constraints are convex sets. 

  3. Equality constraints are affine. 

In convex optimization, the feasible region and the intersection of the constraints always form a convex region. If the optimization problem satisfies all these properties, then there will be at most one optimal solution. The surprising fact is that many complex optimization problems can be reliably and efficiently solved using convex optimization.

Maximizing an Objective Function

We have discussed minimizing an objective function, now we will discuss maximizing the objective function where the function is concave—if all the constraints are kept intact, the relationship min f(x) = max - f(x) can be utilized for converting the concave to convex. A concave function f(x) can be converted to a convex function equal to -f(x). There are a lot of tricks to convert a problem to a convex function, and this reduces the computation time required to solve the non-convex problems.

Convex Optimization and Machine Learning

Optimization is a crucial step in practical machine learning algorithms. In machine learning, an objective function and constraints are defined over a set of data to which the algorithms have to find an optimum solution. In machine learning, the problems are not necessarily convex. However, converting them to convex functions as convex optimization is simpler, computationally less intensive, and faster. 

A  fundamental convex optimization problem in machine learning can be expressed as:

min i=1mfi(x)+R(x)

where x ∈ Rn  , R, and fi(x) are convex functions and  ≥ 0 is a fixed parameter. Depending on the function fi(x), the fundamental optimization equation transforms into various convex optimization problems such as the linear programming problem, least-square problem, or ridge regression problem, which can be easily solved using convex optimization.

The problems in machine learning can be with or without constraints. In a constrained optimization problem, you can see whether the generated solution is feasible by using a convex optimization technique. In convex optimization, the solution lies within the convex region and this helps to rule out infeasible solutions. In the process of solving the unconstrained problem with convex optimization, the local search algorithm finalizes the optimal solution from a list of prospective solutions. 

As machine learning gains popularity, convex optimization does too. If you are looking to train a system using machine learning algorithms, convex optimization will save a lot of computation time. The feasibility check of the solution is also much easier using convex optimization. Use convex optimization to quickly and easily convexify your optimization problems in machine learning. 

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