Stochastic Variance Reduction for Nonconvex Optimization (1603.06160v2)

Abstract: We study nonconvex finite-sum problems and analyze stochastic variance reduced gradient (SVRG) methods for them. SVRG and related methods have recently surged into prominence for convex optimization given their edge over stochastic gradient descent (SGD); but their theoretical analysis almost exclusively assumes convexity. In contrast, we prove non-asymptotic rates of convergence (to stationary points) of SVRG for nonconvex optimization, and show that it is provably faster than SGD and gradient descent. We also analyze a subclass of nonconvex problems on which SVRG attains linear convergence to the global optimum. We extend our analysis to mini-batch variants of SVRG, showing (theoretical) linear speedup due to mini-batching in parallel settings.

• The paper establishes non-asymptotic convergence rates for SVRG in nonconvex settings, achieving an O(n^(2/3)/ε) improvement over SGD.
• The paper demonstrates that SVRG attains linear convergence for gradient dominated nonconvex functions, extending results from convex optimization.
• The paper introduces mini-batch variants of SVRG, providing linear speedup in parallel computations and enhancing scalability for complex models.

Stochastic Variance Reduction for Nonconvex Optimization

The paper "Stochastic Variance Reduction for Nonconvex Optimization" offers an in-depth exploration into the application of stochastic variance reduced gradient (SVRG) methods specifically within the domain of nonconvex optimization. With the increasing prevalence of nonconvex problems in practical applications, especially in deep learning, the optimization community has shown an increasing interest in advancing technologies beyond the conventional stochastic gradient descent (SGD).

Overview and Contributions

The authors focus on nonconvex finite-sum problems, framed within an Incremental First-order Oracle (IFO) framework. They examine SVRG, a method initially celebrated for its advantages in convex optimization, and extend its theoretical analysis to the nonconvex case by moving beyond the traditional assumption of convexity. Their principal contributions are as follows:

1. Convergence Rates: The paper establishes non-asymptotic rates of convergence to stationary points for SVRG in nonconvex settings, proving it to be consistently faster than both SGD and gradient descent. Particularly, SVRG achieves a convergence rate characterized by $O(n^{2/3}/\epsilon)$, substantially improving upon the rate of $O(1/\epsilon^2)$ typically associated with SGD.
2. Linear Convergence for Specific Nonconvex Classes: For a subclass of nonconvex problems identified as gradient dominated functions, the authors demonstrate that SVRG can achieve linear convergence to the global optimum. This finding extends the known applicability of linear convergence from strongly convex scenarios to certain nonconvex instances.
3. Mini-batch SVRG: The analysis is further extended to mini-batch variants of SVRG. Theoretically, mini-batching yields a linear speedup in a parallelized setting, enhancing the algorithm's scalability and efficiency, a claim not previously supported by existing literature in nonconvex optimization settings.
4. Experimental Insights: While primarily analytical, the paper suggests preliminary experiments illustrating the potential of SVRG in practice, though detailed empirical validation is yet to be explored.

Implications and Future Directions

The results presented have significant implications for both theoretical understanding and practical applications. By providing a framework wherein SVRG demonstrates superior performance across various dimensions of nonconvex optimization, the paper invites further exploration into algorithmic enhancements and variants tailored for specific problem structures.

The advancements also hint toward broader applicability in machine learning models that inherently involve complex, nonconvex landscapes, such as neural networks. This could lead to more efficient training processes by leveraging the reduced variance and improved convergence properties of SVRG.

From a theoretical perspective, these insights challenge the established paradigms regarding the limitations of variance reduction techniques in nonconvex domains. Future research may build upon this foundation, refining these techniques to achieve even greater robustness and efficiency.

In conclusion, the paper makes a significant stride in extending the frontiers of stochastic optimization, particularly for nonconvex challenges, offering valuable theoretical guarantees and practical considerations that bear potential for considerable impact in both academic and industrial contexts.