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Title: Hyperparameter Tuning and Implicit Regularization in Minibatch SGD

Paper content:
This paper makes two contributions towards understanding how the hyperparameters of stochastic gradient descent affect the final training loss and test accuracy of neural networks.
First, we argue that stochastic gradient descent exhibits two regimes with different behaviours; a noise dominated regime which typically arises for small or moderate batch sizes, and a curvature dominated regime which typically arises when the batch size is large.
In the noise dominated regime, the optimal learning rate increases as the batch size rises, and the training loss and test accuracy are independent of batch size under a constant epoch budget.
In the curvature dominated regime, the optimal learning rate is independent of batch size, and the training loss and test accuracy degrade as the batch size rises.
We support these claims with experiments on a range of architectures including ResNets, LSTMs and autoencoders.
We always perform a grid search over learning rates at all batch sizes.
Second, we demonstrate that small or moderately large batch sizes continue to outperform very large batches on the test set, even when both models are trained for the same number of steps and reach similar training losses.
Furthermore, when training Wide-ResNets on CIFAR-10 with a constant batch size of 64, the optimal learning rate to maximize the test accuracy only decays by a factor of 2 when the epoch budget is increased by a factor of 128, while the optimal learning rate to minimize the training loss decays by a factor of 16.
These results confirm that the noise in stochastic gradients can introduce beneficial implicit regularization.
Stochastic gradient descent (SGD) is the most popular optimization algorithm in deep learning, but it remains poorly understood.
A number of papers propose simple scaling rules that predict how changing the learning rate and batch size will influence the final performance of popular network architectures (Hoffer et al., 2017; Goyal et al., 2017; Jastrzębski et al., 2017) .
Some of these scaling rules are contradictory, and Shallue et al. (2018) argue that none of these simple prescriptions work reliably across multiple architectures.
Some papers claim SGD with Momentum significantly outperforms SGD without Momentum (Sutskever et al., 2013) , but others observe little difference between both algorithms in practice (Kidambi et al., 2018; Zhang et al., 2019) .
We hope to clarify this debate.
We argue that minibatch stochastic gradient descent exhibits two regimes with different behaviours: a noise dominated regime and a curvature dominated regime (Ma et al., 2017b; McCandlish et al., 2018; Liu & Belkin, 2018) .
The noise dominated regime typically arises for small or moderate batch sizes, while the curvature dominated regime typically arises when the batch size is large.
The curvature dominated regime may also arise if the epoch budget is small or the loss is poorly conditioned (McCandlish et al., 2018) .
Our extensive experiments demonstrate that, 1.
In the noise dominated regime, the final training loss and test accuracy are independent of batch size under a constant epoch budget, and the optimal learning rate increases as the batch size rises.
In the curvature dominated regime, the optimal learning rate is independent of batch size, and the training loss and test accuracy degrade with increasing batch size.
The critical learning rate which separates the two regimes varies between architectures.
2. If specific assumptions are satisfied, then the optimal learning rate is proportional to batch size in the noise dominated regime.
These assumptions hold for most tasks.
However we observe a square root scaling rule when performing language modelling with an LSTM.
This is not surprising, since consecutive gradients in a language model are not independent.
3. SGD with Momentum and learning rate warmup do not outperform vanilla SGD in the noise dominated regime, but they can outperform vanilla SGD in the curvature dominated regime.
There is also an active debate regarding the role of stochastic gradients in promoting generalization.
It has been suspected for a long time that stochastic gradients sometimes generalize better than full batch gradient descent (Heskes & Kappen, 1993; LeCun et al., 2012) .
This topic was revived by Keskar et al. (2016) , who showed that the test accuracy often falls if one holds the learning rate constant and increases the batch size, even if one continues training until the training loss ceases to fall.
Many authors have studied this effect (Jastrzębski et al., 2017; Chaudhari & Soatto, 2018) , but to our knowledge no paper has demonstrated a clear generalization gap between small and large batch training under a constant step budget on a challenging benchmark while simultaneously tuning the learning rate.
This phenomenon has also been questioned by a number of authors.
Shallue et al. (2018) argued that one can reduce the generalization gap between small and large batch sizes if one introduces additional regularization (we note that this is consistent with the claim that stochastic gradients can enhance generalization).
Zhang et al. (2019) suggested that a noisy quadratic model is sufficient to describe the performance of neural networks on both the training set and the test set.
In this work, we verify that small or moderately large batch sizes substantially outperform very large batches on the test set in some cases, even when compared under a constant step budget.
However the batch size at which the test accuracy begins to degrade can be larger than previously thought.
We find that the test accuracy of a 16-4 Wide-ResNet (Zagoruyko & Komodakis, 2016 ) trained on CIFAR-10 for 9725 updates falls from 94.7% at a batch size of 4096 to 92.8% at a batch size of 16384.
When performing language modelling with an LSTM on the Penn TreeBank dataset for 16560 updates (Zaremba et al., 2014) , the test perplexity rises from 81.7 to 92.2 when the batch size rises from 64 to 256.
We observe no degradation in the final training loss as the batch size rises in either model.
These surprising results motivated us to study how the optimal learning rate depends on the epoch budget for a fixed batch size.
As expected, the optimal test accuracy is maximized for a finite epoch budget, consistent with the well known phenomenon of early stopping (Prechelt, 1998) .
Meanwhile the training loss falls monotonically as the epoch budget increases, consistent with classical optimization theory.
More surprisingly, the learning rate that maximizes the final test accuracy decays very slowly as the epoch budget increases, while the learning rate that minimizes the training loss decays rapidly.
1 These results provide further evidence that the noise in stochastic gradients can enhance generalization in some cases, and they suggest novel hyper-parameter tuning strategies that may reduce the cost of identifying the optimal learning rate and optimal epoch budget.
We describe the noise dominated and curvature dominated regimes of SGD with and without Momentum in section 2.
We focus on the analogy between SGD and stochastic differential equations (Gardiner et al., 1985; Welling & Teh, 2011; Mandt et al., 2017; Li et al., 2017) , but our primary contributions are empirical and many of our conclusions can be derived from different assumptions (Ma et al., 2017b; Zhang et al., 2019) .
In section 3, we provide an empirical study of the relationship between the optimal learning rate and the batch size under a constant epoch budget, which verifies the existence of the two regimes in practice.
In section 4, we study the relationship between the optimal learning rate and the batch size under a constant step budget, which confirms that stochastic gradients can introduce implicit regularization enhancing the test set accuracy.
Finally in section 5, we fix the batch size and consider the relationship between the optimal learning rate and the epoch budget.
The contributions of this work are twofold.
First, we verified that SGD exhibits two regimes with different behaviours.
In the noise dominated regime which arises when the batch size is small, the test accuracy is independent of batch size under a constant epoch budget, the optimal learning rate increases as the batch size rises, and acceleration techniques do not outperform vanilla SGD.
Meanwhile in the curvature dominated regime which arises when the batch size is large, the optimal learning rate is independent of batch size, acceleration techniques outperform vanilla SGD, and the test accuracy degrades with batch size.
If certain assumptions are satisfied, the optimal learning rate in the noise dominated regime is proportional to batch size.
These assumptions hold for most tasks.
Second, we confirm that a gap in test accuracy between small or moderately large batch sizes and very large batches persists even when one trains under a constant step budget.
When training a 16-4 Wide-ResNet on CIFAR-10 for 9765 updates, the test accuracy drops from 94.7% at a batch size of 4096 to 92.5% at a batch size of 16384.
We also find that the optimal learning rate which maximizes the test accuracy of Wide-ResNets depends very weakly on the epoch budget while the learning rate which minimizes the training loss falls rapidly as the epoch budget increases.
These results confirm that stochastic gradients introduce implicit regularization which enhances generalization, and they provide novel insights which could be used to reduce the cost of identifying the optimal learning rate.

assistant: text: Smaller batch sizes can outperform very large batches on the test set under constant step budgets and with properly tuned learning rate schedules.