SPIDER: Near-Optimal Non-Convex Optimization via Stochastic Path Integrated Differential Estimator (1807.01695v2)

Abstract: In this paper, we propose a new technique named \textit{Stochastic Path-Integrated Differential EstimatoR} (SPIDER), which can be used to track many deterministic quantities of interest with significantly reduced computational cost. We apply SPIDER to two tasks, namely the stochastic first-order and zeroth-order methods. For stochastic first-order method, combining SPIDER with normalized gradient descent, we propose two new algorithms, namely SPIDER-SFO and SPIDER-SFO\textsuperscript{+}, that solve non-convex stochastic optimization problems using stochastic gradients only. We provide sharp error-bound results on their convergence rates. In special, we prove that the SPIDER-SFO and SPIDER-SFO\textsuperscript{+} algorithms achieve a record-breaking gradient computation cost of $\mathcal{O}\left( \min( n^{1/2} \epsilon^{-2}, \epsilon^{-3} ) \right)$ for finding an $\epsilon$-approximate first-order and $\tilde{\mathcal{O}}\left( \min( n^{1/2} \epsilon^{-2}+\epsilon^{-2.5}, \epsilon^{-3} ) \right)$ for finding an $(\epsilon, \mathcal{O}(\epsilon^{0.5}))$-approximate second-order stationary point, respectively. In addition, we prove that SPIDER-SFO nearly matches the algorithmic lower bound for finding approximate first-order stationary points under the gradient Lipschitz assumption in the finite-sum setting. For stochastic zeroth-order method, we prove a cost of $\mathcal{O}( d \min( n^{1/2} \epsilon^{-2}, \epsilon^{-3}) )$ which outperforms all existing results.

• The paper introduces SPIDER, a technique that nearly attains optimal gradient computation complexity for first-order stochastic non-convex optimization.
• It extends the estimator to zeroth-order methods, significantly lowering computational costs in scenarios lacking direct gradient access.
• Rigorous theoretical analysis and high-probability convergence guarantees underscore SPIDER's practical and theoretical impact.

Near-Optimal Non-Convex Optimization via SPIDER

In the field of non-convex optimization, the paper presents an innovative contribution through the introduction of the Stochastic Path-Integrated Differential Estimator (SPIDER), applied to both first-order and zeroth-order stochastic optimization methods. The SPIDER approach is designed to efficiently track deterministic quantities, significantly lowering computational costs. The paper is primarily focused on enhancing convergence rates for finding approximate first-order and second-order stationary points. This summary explores the intricate details of the methods proposed, the theoretical guarantees provided, and the implications of these findings.

Key Contributions

1. Stochastic First-Order Optimization: The authors develop two algorithms, SPIDER-SFO and SPIDER-SFO⁺, which employ the SPIDER technique in conjunction with normalized gradient descent to tackle non-convex stochastic optimization problems. The SPIDER-SFO algorithm achieves a noteworthy gradient computation complexity of $$\mathcal{O}(\min(n^{1/2} \epsilon^{-2}, \epsilon^{-3}))$$, aligning closely with the algorithmic lower bound for finite-sum settings. This result markedly improves the previous benchmarks in convergence rates for finding first-order stationary points.
2. Stochastic Zeroth-Order Optimization: For zeroth-order methods, the SPIDER technique establishes a gradient computation cost of $$\mathcal{O}(d \min(n^{1/2} \epsilon^{-2}, \epsilon^{-3}))$$, surpassing existing methods. This approach is particularly advantageous for scenarios where gradients cannot be directly computed.
3. Theoretical Insights: The paper presents rigorous theoretical analyses to support the claims regarding convergence rates. The authors utilize a variety of mathematical tools, including martingale theory and Lipschitz continuity arguments, to establish the efficiency and optimality of the proposed methods.
4. High Probability Convergence: Beyond expectations, the work further provides convergence guarantees in high probability using concentration inequalities. This establishes a more robust understanding of SPIDER's performance in practical applications.

Numerical and Theoretical Implications

The SPIDER techniques offer substantial enhancements in computational efficiency for stochastic optimization, promising both practical and theoretical advancements. The application of SPIDER to stochastic first-order and zeroth-order methods presents a pathway for future research in various optimization domains, potentially influencing areas such as machine learning model training and large-scale data analysis.

Future Directions

The paper opens several avenues for future exploration:

• Extension to Other Optimization Problems: There is potential for extending SPIDER to other forms of optimization, including those with additional constraints or specific structure.
• Implementation in Real-World Applications: Further empirical validation in practical scenarios, such as training complex neural networks, could solidify SPIDER's utility.
• Exploration of Lower Bounds: Continued examination of theoretical lower bounds in non-convex optimization could offer deeper insights into the limits of stochastic gradient methods.

Overall, the proposed SPIDER methodology constitutes a substantial advancement in the efficiency of solving non-convex optimization problems. By reducing gradient computation costs and maintaining optimal convergence rates, this research paves the way for more effective and efficient optimization algorithms in both theoretical and applied settings.