Applying a learning rate scheduler to gently decrease the learning rate over the course of training can lead to improved model accuracy and less overfitting compared to using a constant learning rate. While the exact cause is debated, one theoretical explanation suggests that taking smaller step sizes forces the model parameters to remain closer to zero, resulting in a simpler model, although this does not completely explain the phenomenon.

Effect of Learning Rate Scheduling on Overfitting

A polynomial learning rate decay schedule gradually reduces the learning rate over the course of training according to a polynomial function. It is considered one of the common policy choices for dynamically adjusting the learning rate.

Polynomial Learning Rate Decay

A piecewise constant learning rate schedule, often referred to as a multi-factor scheduler, maintains a steady learning rate for predefined intervals of training and abruptly drops the rate at specific milestones. Mathematically, given a set of milestone times $$s$$ (such as $$s = \{5, 10, 20\}$$), the learning rate is updated according to the rule $$\eta_{t+1} \leftarrow \eta_t \cdot \alpha$$ whenever the current step $$t \in s$$, where $$\alpha$$ is the designated decay factor.

Piecewise Constant Learning Rate Schedule

A cosine learning rate schedule, proposed by Loshchilov and Hutter (2016), dynamically adjusts the learning rate by following the shape of a cosine curve. It relies on the observation that the learning rate should not decrease too drastically at the beginning of training, and that the solution should be refined at the end using a very small learning rate. For learning rates in the range $$t \in [0, T]$$, this results in a schedule with the functional form:

$$\eta_t = \eta_T + \frac{\eta_0 - \eta_T}{2} \left(1 + \cos(\pi t/T)\right)$$

Here, $$\eta_0$$ is the initial learning rate and $$\eta_T$$ is the target rate at the maximum update step $$T$$. For steps $$t > T$$, the learning rate is simply pinned to $$\eta_T$$ without increasing it again.

Cosine Learning Rate Schedule

To address the dilemma of choosing an initial learning rate that is either too small (causing slow progress) or too large (causing divergence), a simple strategy called optimizer warmup is used. During a warmup period, the learning rate gradually increases—typically linearly—from a small value to its initial maximum, after which it cools down until the end of the optimization process.

Optimizer Warmup

A factor learning rate scheduler is an alternative to polynomial decay that utilizes a multiplicative reduction strategy. The learning rate is updated at each step using the equation $$\eta_{t+1} \leftarrow \eta_t \cdot \alpha$$, where $$\alpha \in (0, 1)$$. To prevent the learning rate from decaying beyond a reasonable lower bound, the update equation is typically modified to $$\eta_{t+1} \leftarrow \mathop{\mathrm{max}}(\eta_{\mathrm{min}}, \eta_t \cdot \alpha)$$.

Factor Learning Rate Scheduler

A basic learning rate adjustment can be performed explicitly at each step by modifying the optimizer's parameter groups directly. The following Python code demonstrates how to set a new learning rate manually in a PyTorch optimizer:

```python
lr = 0.1
trainer.param_groups[0]["lr"] = lr
print(f'learning rate is now {trainer.param_groups[0]["lr"]:.2f}')
```

Explicit Learning Rate Adjustment Implementation

To empirically study learning rate scheduling, a computationally efficient yet nontrivial toy problem can be constructed. A common setup involves training a modernized LeNet architecture on the Fashion-MNIST dataset. This modernized LeNet updates the classic design by replacing sigmoid activations with ReLU activations and substituting AveragePooling operations with MaxPooling.

Learning Rate Scheduler Toy Problem

A simple learning rate scheduler can be defined to decay the learning rate proportionally to the inverse square root of the number of updates. Specifically, at step $$t$$, the learning rate is set to $$\eta = \eta_0 (t + 1)^{-\frac{1}{2}}$$, where $$\eta_0$$ is the initial learning rate. This formulation ensures that the step size gently and continuously decreases as training progresses.

Square Root Learning Rate Scheduler

A learning rate scheduler is a mechanism used to dynamically adjust the learning rate during the training of a model. Rather than keeping the learning rate constant, the scheduler acts as a function that takes the number of optimization updates (such as iterations or epochs) and outputs the appropriate learning rate value for the next step.

Claude

The learning rate $$\eta$$ is a positive scalar chosen by the algorithm designer that controls the size of each parameter update step in gradient descent. It directly scales the gradient to determine how far the parameters move in the negative gradient direction at each iteration. Setting $$\eta$$ appropriately is critical: a value that is too small results in very slow updates, requiring many more iterations to approach the optimum, while a value that is too large can cause the update step $$\eta f'(x)$$ to become so large that the first-order Taylor approximation breaks down, potentially causing the iterates to overshoot the minimum and diverge rather than converge.

Learning Rate

Dive into Deep Learning

When the learning rate $$\eta$$ is chosen to be too small, each gradient descent update moves the parameter $$x$$ only a tiny distance toward the optimum. This results in extremely slow progress, with the algorithm requiring a large number of iterations to reach a satisfactory solution. For instance, applying gradient descent to the quadratic $$f(x) = x^2$$ with $$\eta = 0.05$$ and starting from $$x = 10$$, the parameter value is still approximately $$3.49$$ after $$10$$ iterations—far from the optimal solution at $$x = 0$$. While a small learning rate ensures that the first-order Taylor approximation remains valid and the function value decreases at every step, the practical cost is an unacceptably slow convergence rate.

Effect of a Small Learning Rate on Gradient Descent

When the learning rate $$\eta$$ is set too high, the magnitude of the gradient step $$|\eta f'(x)|$$ can become large enough to invalidate the first-order Taylor expansion used to justify gradient descent. Specifically, the higher-order remainder term $$\mathcal{O}(\eta^2 (f'(x))^2)$$ is no longer negligible, so there is no guarantee that the function value will decrease after each update. In practice, the iterates overshoot the optimal solution and can diverge. For example, applying gradient descent to $$f(x) = x^2$$ with $$\eta = 1.1$$ starting from $$x = 5$$ causes $$x$$ to overshoot the minimum at $$x = 0$$ and gradually diverge, reaching approximately $$61.92$$ after $$10$$ iterations instead of converging.

Effect of an Excessive Learning Rate on Gradient Descent

Learning Rate Scheduler

When training advanced neural network designs, initializing the parameters is sometimes insufficient to guarantee stable optimization. This creates an optimization dilemma: choosing a sufficiently small initial learning rate prevents early divergence but results in extremely slow progress, whereas choosing a large initial learning rate leads to immediate divergence.

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