An optimal randomized incremental gradient method (1507.02000v3)

Abstract: In this paper, we consider a class of finite-sum convex optimization problems whose objective function is given by the summation of $m$ ($\ge 1$) smooth components together with some other relatively simple terms. We first introduce a deterministic primal-dual gradient (PDG) method that can achieve the optimal black-box iteration complexity for solving these composite optimization problems using a primal-dual termination criterion. Our major contribution is to develop a randomized primal-dual gradient (RPDG) method, which needs to compute the gradient of only one randomly selected smooth component at each iteration, but can possibly achieve better complexity than PDG in terms of the total number of gradient evaluations. More specifically, we show that the total number of gradient evaluations performed by RPDG can be ${\cal O} (\sqrt{m})$ times smaller, both in expectation and with high probability, than those performed by deterministic optimal first-order methods under favorable situations. We also show that the complexity of the RPDG method is not improvable by developing a new lower complexity bound for a general class of randomized methods for solving large-scale finite-sum convex optimization problems. Moreover, through the development of PDG and RPDG, we introduce a novel game-theoretic interpretation for these optimal methods for convex optimization.

• The paper presents the RPDG method, which reduces gradient evaluations by sampling one component per iteration.
• It demonstrates superior performance over deterministic methods by lowering complexity by a factor of O(√m) for finite-sum convex problems.
• A new lower complexity bound and a game-theoretic interpretation confirm its optimal efficiency for large-scale applications.

An Analysis of the Randomized Primal-Dual Gradient Method for Convex Optimization

The paper "An Optimal Randomized Incremental Gradient Method" by Guanghui Lan and Yi Zhou addresses the problem of efficiently solving finite-sum convex optimization problems using a novel randomized incremental approach. The authors introduce a significant advancement in the optimization landscape with the development of the Randomized Primal-Dual Gradient (RPDG) method. This algorithm is designed to tackle problems characterized by objectives that are the sum of multiple smooth convex components, supplemented by other relatively simple terms. Here, the RPDG method is rigorously analyzed and shown to outperform deterministic methods under certain conditions, offering improved complexity bounds and introducing a game-theoretic interpretation.

Key Contributions

1. Deterministic Primal-Dual Method Development: The authors first develop a deterministic primal-dual gradient (PDG) method. This method is designed to achieve optimal black-box iteration complexity for composite optimization problems using a primal-dual termination criterion. The PDG method sets the foundation for the randomized approach and provides a game-theoretic interpretation that helps elucidate the mechanics behind Nesterov's accelerated gradient methodology.
2. Randomized Primal-Dual Gradient Method: The major contribution is the RPDG method, which dramatically reduces the number of gradient evaluations necessary to achieve convergence. Unlike traditional methods that require gradients from all smooth components, the RPDG method samples a single component at each iteration, reducing evaluation complexity by a factor of $\mathcal{O} (\sqrt{m})$ relative to deterministic first-order methods, both in expectation and high probability.
3. Lower Complexity Bound: A key theoretical result derived is a new lower complexity bound for randomized methods, which establishes that the RPDG method is optimally efficient in its use of gradient evaluations due to its sampling strategy. This is a notable achievement, as it asserts the optimality of the RPDG method under specific circumstances where the problem dimension is sufficiently large.
4. Game-Theoretic Interpretation: Both PDG and RPDG methods are equipped with a novel game-theoretic interpretation, framing the optimization problem as an iterative game between a buyer and multiple suppliers. This interpretation provides valuable insights into the algorithm's dynamics, portraying the optimization process as an equilibrium-seeking interaction.

Practical and Theoretical Implications

• Efficiency in Large-Scale Applications: The RPDG method's ability to reduce gradient computations makes it invaluable for large-scale applications common in machine learning, statistics, and image processing. Its randomized approach ensures efficient convergence, which is particularly beneficial where computational resources are a constraint.
• Generalizability and Versatility: The authors extend their work to encompass non-strongly convex and structured nonsmooth problems. This adaptation demonstrates the method's robustness and flexibility, positioning it as a versatile tool across various types of optimization challenges.
• Foundational Insights for Algorithm Design: The game-theoretic perspective not only provides a deeper understanding of existing acceleration techniques but also sets the stage for designing future algorithms that inherently incorporate strategic element-based optimizations.

Speculations on Future Developments

Given the substantial advancements presented, future research might expand on adaptive scheme developments that eliminate the need for known Lipschitz constants and strong convexity parameters. Addressing such practical considerations could spur even broader adoption of these methods in real-world applications. Further, exploring connections with other stochastic optimization techniques and deeper integrations with machine learning frameworks could open additional avenues for impactful innovations.

Conclusion

Lan and Zhou's work on the RPDG method represents a meaningful stride in the domain of incremental gradient methods for convex optimization. With improved complexity bounds and a clear optimality assertion for large-scale problems, coupled with a strategic interpretative framework, this research offers both a practical tool for immediate use and a theoretical foundation that underscores future exploratory pathways in optimization algorithm studies.