The paper discusses computational challenges in convex bandit and zeroth-order optimization, focusing on the manipulation of convex sets.
Various representations of convex sets and their computational implications are analyzed, including polytope, convex hull, separation oracle, and linear optimization oracle.
Methods for improving computational efficiency, such as projection algorithms, finding self-concordant barriers, and approximating MVEEs, are proposed.
Open problems in the field are identified, emphasizing the need for algorithms that balance efficiency and accuracy, adapt to different constraints, and scale with problem complexity.
Computational efficiency in convex bandit and zeroth-order optimization problems heavily depends on the representation and manipulation of convex sets. This paper discusses various computational challenges encountered when dealing with convex sets in optimization problems and proposes several methods to overcome these hurdles. The core of the discussion revolves around efficient ways to perform operations such as projections, finding self-concordant barriers, and approximating minimum volume enclosing ellipsoids (MVEEs). These operations are crucial in implementing algorithms for both convex bandit optimization and zeroth-order optimization.
Convex sets can be represented in several ways, each offering different computational advantages and challenges:
Polytope representation allows defining a convex set as a collection of linear inequalities. Operations such as projection and finding a separation oracle are computationally feasible but can be complex depending on the size of the inequality system.
Convex hull representation involves defining a convex set through a set of points. While this representation simplifies certain computations, it becomes challenging when the point set is large.
Separation oracle provides a mechanism to determine if a point lies inside a convex set and, if not, to find a hyperplane that separates the point from the set. This approach is versatile and can be applied to various convex set representations but may require iterative procedures to find an accurate separation.
Linear optimization oracle offers a direct way to perform optimization over convex sets by solving a linear program defined by the convex set constraints. This method is efficient but lacks flexibility in handling non-linear constraints.
Projection onto Convex Sets: Efficient projection algorithms are key in many optimization procedures. For polytope and convex hull representations, various Newton method-based and Frank-Wolfe algorithm-based techniques provide efficient solutions. When only separation oracles are available, the ellipsoid method serves as an effective though computationally intensive option.
Finding Self-Concordant Barriers: Identifying appropriate self-concordant barriers is crucial for interior-point methods in optimization. While logarithmic and volumetric barriers provide practical solutions for polytopes, identifying efficient barriers for more complex convex sets remains challenging.
Approximating Minimum Volume Enclosing Ellipsoids (MVEEs): Approximating MVEEs is vital for many optimization algorithms. The paper discusses procedures applicable to both polytope representations and separation oracles, ensuring computational efficiency while maintaining a balance between approximation fidelity and operational complexity.
The computation in convex optimization, particularly in bandit and zeroth-order settings, presents various open problems:
Balancing Efficiency and Accuracy: Finding the most computationally efficient algorithms that do not compromise the accuracy of optimization solutions, especially in high-dimensional spaces.
Adaptive Algorithms: Developing algorithms that can adapt to different convex set representations and optimization constraints without significant manual tuning or prior knowledge.
Scalability: Ensuring that computational methods remain efficient as the dimensionality of the problem space and the complexity of the convex sets increase.
Effective computation in convex bandit and zeroth-order optimization is contingent upon the precise representation of convex sets and the efficiency of operations such as projection, barrier identification, and MVEE approximation. Future research directions include improving algorithmic adaptability, enhancing computational efficiency, and solving open problems related to convex set manipulation.