On the convergence properties of a $K$-step averaging stochastic gradient descent algorithm for nonconvex optimization (1708.01012v3)

Abstract: Despite their popularity, the practical performance of asynchronous stochastic gradient descent methods (ASGD) for solving large scale machine learning problems are not as good as theoretical results indicate. We adopt and analyze a synchronous K-step averaging stochastic gradient descent algorithm which we call K-AVG. We establish the convergence results of K-AVG for nonconvex objectives and explain why the K-step delay is necessary and leads to better performance than traditional parallel stochastic gradient descent which is a special case of K-AVG with $K=1$. We also show that K-AVG scales better than ASGD. Another advantage of K-AVG over ASGD is that it allows larger stepsizes. On a cluster of $128$ GPUs, K-AVG is faster than ASGD implementations and achieves better accuracies and faster convergence for \cifar dataset.

• The paper proves that K-AVG converges for nonconvex functions by reducing average squared gradient norms through delayed averaging.
• The study shows that K-AVG scales efficiently with more processors by allowing larger stepsizes and lowering variance compared to ASGD.
• Empirical results indicate that K-AVG delivers superior accuracy and up to sevenfold speed improvements on benchmarks like CIFAR-10.

On the Convergence Properties of a $K$-Step Averaging Stochastic Gradient Descent Algorithm for Nonconvex Optimization

The paper "On the Convergence Properties of a $K$-step Averaging Stochastic Gradient Descent Algorithm for Nonconvex Optimization" by Fan Zhou and Guojing Cong introduces and analyzes the K-step Averaging Stochastic Gradient Descent (K-AVG) algorithm. This method is designed to tackle large-scale machine learning tasks characterized by nonconvex optimization landscapes, commonly seen in contemporary deep learning frameworks.

Overview of K-AVG Algorithm

K-AVG is a synchronous variant of Stochastic Gradient Descent (SGD) that incorporates a $K$-step delay in parameter averaging. In contrast to traditional Parallel SGD, which synchronizes parameters after each update ($K=1$), K-AVG allows for several local updates across $K$ iterations before global synchronization. This modification aims to reduce communication costs by amortizing them over $K$ updates and allowing for more substantial stepsizes, purportedly leading to faster convergence.

Theoretical Contributions

The contributions of the paper are primarily theoretical, focusing on proving convergences and scalability of the K-AVG method. Key results include:

1. Convergence Analysis: The paper provides rigorous proofs demonstrating that K-AVG converges for nonconvex objective functions. Through Assumptions commonly used in optimization theory (e.g., Lipschitz continuity, bounded variance of stochastic gradients), the authors establish that K-AVG leads to a reduction in average squared gradient norms. This indicates that the algorithm effectively reduces the expected gradient magnitude, eventually converging to a stationary point.
2. Scalability with Processors: The scalability of K-AVG with the number of processors ($P$) is analyzed extensively. The findings suggest that K-AVG scales better than Asynchronous SGD (ASGD) due to improved variance reduction and allowance for larger stepsizes, making it more suitable for large-scale distributed implementations.
3. Optimal Step Delay ($K_{opt}$): Contrary to initial assumptions, the paper argues that the optimal delay of parameter averaging, $K_{opt}$, may not be $1$. A key takeaway is that the optimal frequency of averaging is context-dependent and varies based on problem-specific characteristics, such as the Lipschitz constant and initial guess distance.

Empirical Results

Empirical validations on benchmark datasets, such as CIFAR-10, demonstrate K-AVG’s superior performance over popular ASGD implementations like Downpour and Elastic Averaging SGD (EASGD). Notably, at a large scale (e.g., 128 GPUs), K-AVG provides better accuracy and up to sevenfold speed improvements compared to ASGD systems.

Future Directions and Practical Implications

The findings suggest several practical implications for distributed machine learning systems. K-AVG potentially provides a viable alternative to address the communication bottleneck inherent in large-scale SGD implementations, particularly in environments with high GPU-to-GPU communication bandwidths.

In terms of theoretical advancements, future research could further explore optimization processes that dynamically adjust $K$ and stepsizes based on real-time convergence criteria to further enhance scalability and convergence speed. Additionally, applying such methodologies to novel nonconvex problem domains beyond conventional image recognition tasks could also be a fruitful avenue.

By addressing these facets, K-AVG manifests itself as a method that not only theoretically guarantees convergence but also pragmatically enhances the efficacy of distributed deep learning systems.