8.3 Proximal point algorithms and their convergence

Proximal point algorithms tackle monotone inclusion problems by iteratively solving regularized subproblems. They're a powerful tool for various optimization and variational tasks, generalizing methods like proximal gradient and augmented Lagrangian approaches.

These algorithms converge under mild conditions, with stronger results for strongly monotone operators. The choice of regularization parameters affects the trade-off between progress and computational cost, influencing convergence rates and practical performance.

Proximal Point Algorithm for Monotone Inclusions

Formulation and Key Components

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• Iterative method for solving monotone inclusion problems of the form $0 \in T(x)$, where $T$ is a maximal monotone operator
• Generates a sequence $\{x_k\}$ by solving a regularized subproblem at each iteration: $x_{k+1} = (I + \lambda_k T)^{-1}(x_k)$
  • $\lambda_k > 0$ is a regularization parameter
  • $I$ is the identity mapping
• Resolvent operator $(I + \lambda_k T)^{-1}$ is single-valued and firmly nonexpansive ensures well-definedness of the algorithm
• Generalization of proximal gradient method and augmented Lagrangian method for convex optimization problems (ADMM, Douglas-Rachford splitting)

Monotone Inclusion Problems and Applications

• Monotone inclusion problems encompass a wide range of optimization and variational problems
  • Convex optimization: $T = \partial f$, where $f$ is a convex function and $\partial f$ is its subdifferential
  • Variational inequalities: $T = F + N_C$, where $F$ is a monotone mapping and $N_C$ is the normal cone of a convex set $C$
  • Saddle point problems: $T = (A^T, -A)$, where $A$ is a linear operator
• Applications in machine learning, signal processing, and computational physics (image denoising, matrix completion, phase retrieval)

Convergence of Proximal Point Algorithm

Convergence under Bounded Regularization Parameters

• Convergence established under the assumption that regularization parameters $\{\lambda_k\}$ are bounded away from zero
  • There exists $\lambda_{\min} > 0$ such that $\lambda_k \geq \lambda_{\min}$ for all $k$
• Proof relies on firm nonexpansiveness of resolvent operator implies sequence $\{x_k\}$ is Fejér monotone with respect to the set of solutions
  • Fejér monotonicity: $\|x_{k+1} - x^*\| \leq \|x_k - x^*\|$ for any solution $x^*$
• Fejér monotonicity property, combined with the existence of a solution, ensures weak convergence of $\{x_k\}$ to a solution

Strong Convergence for Strongly Monotone Operators

• In the case of strongly monotone operators, convergence can be strengthened to strong convergence
  • Strong monotonicity: $\langle T(x) - T(y), x - y \rangle \geq \mu \|x - y\|^2$ for some $\mu > 0$
• Stronger convergence result relies on the contraction property of the resolvent operator under strong monotonicity

Convergence Rate Analysis

Linear Convergence for Strongly Monotone Operators

• For strongly monotone operators, proximal point algorithm achieves a linear rate of convergence
  • $\|x_k - x^*\| \leq c^k \|x_0 - x^*\|$, where $x^*$ is a solution and $c \in (0, 1)$ depends on strong monotonicity constant and regularization parameters
• Linear convergence rate is a consequence of the contraction property of the resolvent operator

Sublinear Convergence for Maximal Monotone Operators

• For maximal monotone operators that are not strongly monotone, proximal point algorithm exhibits a sublinear rate of convergence
  • Typically of the order $O(1/k)$ in terms of the residual $\|x_k - x_{k+1}\|$
• Sublinear convergence rate can be derived using the Fejér monotonicity property and the boundedness of the sequence $\{x_k\}$

Improved Convergence Rates with Variable Regularization Parameters

• Rate of convergence can be improved by employing variable regularization parameters $\{\lambda_k\}$ that decrease to zero at a suitable rate
  • Example: $\lambda_k = O(1/k)$ can lead to improved sublinear convergence rates
• Theoretical results provide guidance on the choice of regularization parameters for faster convergence

Regularization Parameters in Proximal Point Algorithm

Trade-off between Progress and Computational Cost

• Regularization parameters $\{\lambda_k\}$ control the trade-off between progress made at each iteration and computational cost of solving regularized subproblems
  • Larger $\lambda_k$ values lead to more aggressive updates and potentially faster convergence but may increase difficulty of solving subproblems accurately
  • Smaller $\lambda_k$ values result in more conservative updates and slower convergence but subproblems become easier to solve

Theoretical Guidance and Adaptive Strategies

• Choice of regularization parameters can be guided by theoretical convergence results
  • Ensuring $\{\lambda_k\}$ is bounded away from zero for weak convergence
  • Decreasing $\{\lambda_k\}$ at a suitable rate for improved convergence rates
• In practice, regularization parameters are often chosen adaptively based on the progress of the algorithm and computational cost of solving subproblems
  • Line search techniques: Backtracking or Armijo-type conditions to ensure sufficient decrease in a suitable merit function
  • Trust-region methods: Adjust the size of the regularized subproblem based on the agreement between the model and the actual decrease in the objective function