Hybrid Deterministic-Stochastic Methods for Data Fitting (1104.2373v4)

Abstract: Many structured data-fitting applications require the solution of an optimization problem involving a sum over a potentially large number of measurements. Incremental gradient algorithms offer inexpensive iterations by sampling a subset of the terms in the sum. These methods can make great progress initially, but often slow as they approach a solution. In contrast, full-gradient methods achieve steady convergence at the expense of evaluating the full objective and gradient on each iteration. We explore hybrid methods that exhibit the benefits of both approaches. Rate-of-convergence analysis shows that by controlling the sample size in an incremental gradient algorithm, it is possible to maintain the steady convergence rates of full-gradient methods. We detail a practical quasi-Newton implementation based on this approach. Numerical experiments illustrate its potential benefits.

• The paper introduces a hybrid approach that combines incremental and full-gradient techniques to balance rapid progress with stable convergence.
• It dynamically adjusts sample sizes to achieve linear convergence rates under strong convexity while reducing computational costs.
• Empirical tests, including logistic regression and CRFs, validate the method's effectiveness in optimizing large-scale data fitting problems.

Hybrid Deterministic-Stochastic Methods for Data Fitting: An Overview

In the domain of structured data-fitting applications, optimization often involves equations summed over a vast number of individual measurements. This paper introduces an innovative methodology utilizing hybrid deterministic-stochastic approaches to enhance both convergence rates and computational efficiency. The paper's primary focus is on combining incremental-gradient methods, known for their initial efficiency, with full-gradient methods, praised for steady convergence, to foster a balanced optimization approach.

Core Concepts

The paper discusses optimization challenges where the objective function is a summation of individual misfit functions $f_i(x)$, highlighting the computational burden when the number of measurements $M$ is large. Standard techniques include:

• Incremental Gradient Methods: These select a subset of terms at each iteration, offering quick but potentially unstable progress.
• Full Gradient Methods: These evaluate the complete gradient at every iteration, showcasing consistent but costly convergence.

The hybrid method presented in the paper strategically adjusts the sample size in the incremental approach, achieving steady convergence rates typical of full-gradient methods while saving on computation time.

Analytical Insights

The authors delve into convergence analysis, establishing theoretical bounds and convergence rates for both deterministic and stochastic noise approaches:

• Strong Convexity Assumption: The method assumes strong convexity with constant Lipschitz gradients, providing a framework to derive linear convergence rates.
• Controlled Gradient Error: By managing gradient errors $e_k$ through bounded sequences, the method ensures convergence without requiring exact gradient computations.

Numerical Experiments

Experimental evaluations span a variety of data-fitting scenarios including logistic regression, multinomial logistic regression, and both chain-structured and general conditional random fields (CRFs). These experiments illustrate the hybrid method's capability to outperform standard deterministic and stochastic methods by leveraging initial rapid progress and stable convergent behavior.

For instance, in binary logistic regression, the hybrid method initially progresses as swiftly as stochastic methods, but continues to exhibit stable convergence akin to deterministic methods in later iterations. Similar trends are observed in other scenarios, reaffirming the method's effectiveness across diverse applications.

Practical Implications

The hybrid approach resonates deeply with real-world applications where the sheer size of data renders complete gradient evaluations impractical. By integrating the best aspects of incremental and full-gradient methods, this method is poised to benefit areas like machine learning and large-scale data analysis, where data scalability and optimization efficiency are paramount.

Moreover, the method's versatility extends beyond theoretical boundaries. The authors showcase scalability through practical implementations, allowing researchers to tackle extensive data-fitting problems with greater adaptability and less computational exhaust.

Future Directions

While the hybrid deterministic-stochastic approach provides substantial advancements, future investigations could explore its application to a broader spectrum of optimization algorithms. Extending these techniques to non-convex settings or integrating advanced probabilistic models could pave the way for even more robust and adaptable optimization strategies.

In conclusion, the paper’s hybrid approach marks a significant step towards optimizing large-scale data-fitting problems by effectively managing sample sizes and gradient evaluations. Its promising numerical results and theoretical justifications provide a solid foundation for future research in this area.