Adaptive Backtracking Line Search (2408.13150v2)

Abstract: Backtracking line search is foundational in numerical optimization. The basic idea is to adjust the step-size of an algorithm by a constant factor until some chosen criterion (e.g. Armijo, Descent Lemma) is satisfied. We propose a novel way to adjust step-sizes, replacing the constant factor used in regular backtracking with one that takes into account the degree to which the chosen criterion is violated, with no additional computational burden. This light-weight adjustment leads to significantly faster optimization, which we confirm by performing a variety of experiments on over fifteen real world datasets. For convex problems, we prove adaptive backtracking requires no more adjustments to produce a feasible step-size than regular backtracking does. For nonconvex smooth problems, we prove adaptive backtracking enjoys the same guarantees of regular backtracking. Furthermore, we prove adaptive backtracking preserves the convergence rates of gradient descent and its accelerated variant.

• The paper introduces adaptive backtracking factors that adjust step sizes based on the violation degree of line search criteria.
• It rigorously validates the method through theoretical proofs and empirical experiments, demonstrating fewer function evaluations in convex and nonconvex problems.
• Empirical results across datasets, including logistic regression and matrix factorization, confirm robust performance improvements over conventional backtracking.

Adaptive Backtracking For Faster Optimization

"Adaptive Backtracking For Faster Optimization" presents a novel approach to optimizing the line search mechanism in numerical optimization algorithms. The authors, Joao V. Cavalcanti, Laurent Lessard, and Ashia C. Wilson, propose an adaptive backtracking strategy that incorporates the degree of violation of the line search criterion to adjust step sizes dynamically, rather than using a constant factor.

Overview

Backtracking line search is a cornerstone in numerical optimization to determine appropriate step sizes in iteration-based optimization algorithms like Gradient Descent (GD), Accelerated Gradient Descent (AGD), and others. The current prevalent method involves scaling the step size by a constant factor until certain criteria, such as the Armijo condition or the Descent Lemma, are satisfied. This paper introduces an adaptive mechanism that modifies this procedure by accounting for how much the criterion is violated.

Contributions

The contributions of this paper are multi-faceted:

1. Adaptive Backtracking Factors: The authors introduce adaptive backtracking factors as a general alternative to constant factors in the adjustment scheme.
2. Armijo Condition and Descent Lemma: They present specific implementations of adaptive backtracking for the Armijo condition and the descent lemma, respectively.
3. Theoretical Guarantees: Detailed proofs are provided showing that for convex problems, their adaptive approach requires fewer function evaluations compared to regular backtracking.
4. Empirical Validation: They verify the efficacy of their approach through extensive experiments across multiple real-world datasets.

Methodology

The adaptive backtracking approach proposed in this paper replaces the constant factor typically used in backtracking with an online variable factor tailored to the degree of violation of the line search criteria. The primary components discussed include:

1. Armijo Condition: The authors propose a specific adaptive factor:

   $$\hat\rho(v(\alpha_{k})) = \max \left(\epsilon, \rho \frac{1 - c}{1 - c \cdot v(\alpha_{k})} \right),$$

   where $v(\alpha_{k})$ indicates the degree to which the Armijo condition is violated.

2. Descent Lemma: For the descent lemma, they propose:

   $\hat{\rho}(v(\alpha_{k})) = \rho v(\alpha_{k}),$

   where $v(\alpha_{k})$ measures the criterion violation similarly.

The motivation is to aggressively reduce the step size when the criteria are violated while ensuring sufficiently large step sizes for fast convergence.

Theoretical Results

The paper establishes several key theoretical results:

• Convex Problems: For convex optimization, adaptive backtracking not only requires fewer function evaluations but maintains global theoretical guarantees similar to traditional backtracking. The step size adjustments are shown to be a non-increasing function of the adjustment factor $\rho$.
• Nonconvex Problems: For nonconvex smooth problems, the adaptive backtracking maintains the same level of guarantees as regular backtracking, effectively bound the number of function evaluations by $\log_{\rho}(\bar{\alpha}/\alpha_{0}) + 1$ adjustments, where $\bar{\alpha}$ is an upper step size bound based on the problem's Lipschitz constant.

Empirical Validation

The authors conduct empirical validation using over fifteen datasets, covering both convex and nonconvex problem configurations. Key results from their experiments highlight:

• Logistic Regression: Adaptive backtracking leads to faster convergence across various datasets using GD, AGD, and Adagrad.
• Linear Inverse Problems: For FISTA, adaptive backtracking showed consistent improvements in terms of fewer function evaluations and faster convergence.
• Rosenbrock Function: On this nonconvex problem, the adaptive backtracking method showed substantial improvements in both final solutions and convergence speed.
• Matrix Factorization: Experiments on the MovieLens dataset with varying matrix ranks confirmed the robustness and effectiveness of adaptive backtracking strategies.

Implications and Future Work

The adaptive backtracking approach proposed in this paper has several practical and theoretical implications:

• Practical Implications: Optimization algorithms incorporating adaptive backtracking are empirically faster and can be more robust to initial step size settings, which is extremely valuable in complex, real-world optimization scenarios.
• Theoretical Extension: Future research could further explore how adaptive backtracking can be generalized to other line search criteria and optimization methods. Additionally, testing in large-scale machine learning frameworks, such as deep learning optimization routines, could provide deeper insights into practical utility.

Conclusion

"Adaptive Backtracking For Faster Optimization" makes significant strides in refining the optimization process by proposing a more nuanced adjustment of step sizes during line search. Through rigorous theoretical proof and extensive empirical validation, Cavalcanti et al. demonstrate that their adaptive approach can achieve faster and more efficient optimization. This work opens several avenues for further exploration in adaptive optimization techniques, potentially influencing a broad spectrum of applications in machine learning and beyond.