Stochastic Cubic Regularization for Fast Nonconvex Optimization (1711.02838v2)

Abstract: This paper proposes a stochastic variant of a classic algorithm---the cubic-regularized Newton method [Nesterov and Polyak 2006]. The proposed algorithm efficiently escapes saddle points and finds approximate local minima for general smooth, nonconvex functions in only $\mathcal{\tilde{O}}(\epsilon^{-3.5})$ stochastic gradient and stochastic Hessian-vector product evaluations. The latter can be computed as efficiently as stochastic gradients. This improves upon the $\mathcal{\tilde{O}}(\epsilon^{-4})$ rate of stochastic gradient descent. Our rate matches the best-known result for finding local minima without requiring any delicate acceleration or variance-reduction techniques.

• The paper introduces stochastic cubic regularization, an algorithm for nonconvex optimization that efficiently escapes saddle points and converges with a better iteration complexity than SGD.
• The method uses stochastic gradients and Hessian-vector products, minimizing a cubic submodel to manage stochastic noise without computing the full Hessian.
• This approach offers practical benefits for large-scale machine learning by improving computational efficiency and opens avenues for future hybrid optimization research.

Optimizing Nonconvex Functions with Stochastic Cubic Regularization

The paper "Stochastic Cubic Regularization for Fast Nonconvex Optimization" introduces a novel approach to nonconvex optimization by leveraging a stochastic variant of the cubic-regularized Newton method. The authors propose an algorithm that efficiently escapes saddle points and converges to approximate local minima, a substantial improvement over established methods like Stochastic Gradient Descent (SGD). Herein, we analyze the theoretical advancements and practical implications of this research, placed within the broader context of nonconvex optimization strategies.

Key Contributions

The core contribution of this paper lies in adapting the cubic-regularized Newton method to the stochastic setting. The conventional cubic method, originally introduced by Nesterov and Polyak, is renowned for its ability to exploit second-order information to escape saddle points swiftly. By incorporating a stochastic variant, the presented algorithm achieves a computational rate of approximately $\mathcal{\tilde{O}}(\epsilon^{-3.5})$ in terms of stochastic gradient and Hessian-vector product evaluations. This rate outperforms the $\mathcal{\tilde{O}}(\epsilon^{-4})$ iteration complexity of SGD, marking a significant leap in optimizing smooth, nonconvex functions without reliance on complex acceleration or variance-reduction techniques.

Theoretical Framework

Fundamentally, the algorithm addresses the following nonconvex optimization problem under the stochastic approximation framework: $$\min_{\mathbf{x} \in \mathbb{R}^d} f(\mathbf{x}) = \mathbb{E}_{\xi \sim \mathcal{D}} [f(\mathbf{x}; \xi)]$$. Here, $\xi$ is a random variable sampled from a distribution $\mathcal{D}$, covering a wide range of statistical and machine learning applications such as deep neural network training.

The proposed method capitalizes on two critical stochastic components:

• Stochastic gradients: $\nabla f(\mathbf{x}; \xi)$
• Stochastic Hessian-vector products: $\nabla^2 f(\mathbf{x}; \xi) \cdot \mathbf{v}$

These are incorporated into an iterative approach using a cubic submodel to approximate local behavior, effectively managing the stochastic nature of the problem.

Methodological Innovations

The algorithm employs a tailored Cubic-Subsolver, a first-order method such as gradient descent, to approximately minimize the cubic submodel at each iteration. Rather than demanding the computation of a full Hessian, which is computationally prohibitive, the approach leverages Hessian-vector products—a significant reduction in complexity. Additionally, the theoretical setup introduces rigorous non-asymptotic analysis to support its $\epsilon$-approximation guarantees.

Practical Implications and Future Directions

The stochastic cubic regularization method proposed offers notable benefits for high-dimensional and large-scale optimization problems common in machine learning. By reducing the required computation per iteration while maintaining rapid convergence to local minima, it enables more efficient training of neural networks and similar models.

However, the impact of stochastic cubic regularization is not limited to computational efficiency. It opens doors to further research in optimizing nonconvex problems, where improved handling of noise in gradient and Hessian computations can yield better generalization and convergence properties. Future works could investigate potential hybrid approaches that combine second-order optimization with adaptive gradient-based techniques to further enhance optimization performance in the context of noisy data.

Conclusion

The introduction of a stochastic cubic-regularized Newton method represents an advancement in nonconvex optimization, addressing critical limitations of existing methods like SGD while enhancing computational efficiency. This paper's methodological and theoretical strides create a robust foundation for future exploration and innovation in stochastic optimization, particularly within the field of machine learning and beyond.