ADADELTA: An Adaptive Learning Rate Method (1212.5701v1)

Abstract: We present a novel per-dimension learning rate method for gradient descent called ADADELTA. The method dynamically adapts over time using only first order information and has minimal computational overhead beyond vanilla stochastic gradient descent. The method requires no manual tuning of a learning rate and appears robust to noisy gradient information, different model architecture choices, various data modalities and selection of hyperparameters. We show promising results compared to other methods on the MNIST digit classification task using a single machine and on a large scale voice dataset in a distributed cluster environment.

• The paper presents ADADELTA, an adaptive per-dimension learning rate method that uses only first-order gradient information to remove the need for manual tuning.
• It introduces a windowed accumulation of squared gradients and a unit-correcting update rule to maintain effective learning rates over time.
• Experimental results on MNIST and speech recognition tasks show ADADELTA’s robustness, scalability, and improved convergence compared to traditional methods.

ADADELTA: An Adaptive Learning Rate Method

The paper, "ADADELTA: An Adaptive Learning Rate Method", authored by Matthew D. Zeiler, introduces a novel per-dimension learning rate adaptation technique for gradient descent, termed ADADELTA. The technique is designed to dynamically adjust over time using exclusively first-order information, minimizing computational overhead relative to vanilla stochastic gradient descent (SGD). Notably, the ADADELTA method eliminates the need for manual learning rate tuning and showcases resilience against noisy gradient information, varying model architectures, data modalities, and hyperparameter settings.

Context and Motivation

Gradient descent algorithms typically involve iterative parameter updates to minimize an objective function $f(x)$. The traditional SGD approach uses a fixed learning rate $\eta$, which must be manually tuned—a process that can be both time-consuming and sensitive to the choice of initial parameters. ADADELTA aims to simplify this process by introducing an adaptive learning rate computed on a per-dimension basis, leveraging only first-order gradient information.

Related Work

The paper situates ADADELTA within a broader context of optimization techniques, highlighting several key methods:

• Newton's Method: Involves the computation of second-order derivatives for optimal step sizes but is computationally prohibitive for large models due to the Hessian matrix.
• Learning Rate Annealing: Involves adjusting the learning rate based on heuristics, typically leading to additional hyperparameters.
• Momentum: Accelerates SGD by accumulating a velocity vector in gradient directions.
• ADAGRAD: Adapts the learning rate by accumulating squared gradients, but suffers from continuously decaying learning rates.

ADADELTA Method

ADADELTA addresses two primary drawbacks of ADAGRAD—continuous decay of learning rates and sensitivity to initial conditions—by introducing two key ideas:

1. Accumulation Over Window: Instead of an unbounded accumulation of squared gradients, ADADELTA restricts the accumulation to a fixed-size window, implemented as an exponentially decaying average. This ensures learning rates do not diminish to zero over time.
2. Unit Correction with Hessian Approximation: ADADELTA corrects for unit inconsistencies in parameter updates by incorporating a measure proportional to the inverse of the Hessian approximation. This is accomplished without explicitly computing second-order derivatives, leveraging past gradients and updates for an adaptive, dimension-specific learning rate.

The update rule for ADADELTA is: $$\Delta x_t = - \frac{\text{RMS}[\Delta x]_{t-1}}{\text{RMS}[g]_t} \; g_t$$ where $\text{RMS}[g]_t$ is the root mean square of the gradients and $\text{RMS}[\Delta x]_{t-1}$ is the accumulated past updates.

Experimental Results

The efficacy of ADADELTA is evaluated on two tasks: the MNIST handwritten digit classification and a large-scale voice dataset for speech recognition.

• MNIST Classification: ADADELTA demonstrated competitive performance, achieving a test set error rate of $2.00\%$, outperforming other methods over sustained training epochs. Momentum eventually converged to a marginally better solution but required sensitive learning rate adjustments.
• Sensitivity Analysis: ADADELTA showed robustness across a range of hyperparameters, contrasting with the sensitivity observed in SGD, Momentum, and ADAGRAD.
• Speech Recognition: In a distributed computing environment with 100 and 200 replicas, ADADELTA maintained superior convergence speed and accuracy, demonstrating its scalability and applicability to large-scale, noisy datasets.

Implications and Future Directions

Practically, ADADELTA simplifies the training process of deep neural networks by obviating manual learning rate tuning. Theoretically, it contributes to the understanding of adaptive methods leveraging first-order information. Future work could explore integrating explicit annealing schedules to further optimize the convergence rates of ADADELTA, potentially yielding enhanced performance in both small and large-scale applications.

In conclusion, ADADELTA presents a robust, computationally efficient alternative to traditional and contemporary gradient descent methods, facilitating more effective and accessible deep learning model training.

Acknowledgements

The author acknowledges the contributions of Geoff Hinton, Yoram Singer, Ke Yang, Marc'Aurelio Ranzato, and Jeff Dean for their valuable inputs and discussions related to this research.

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