Learning Rate Schedules - Learning Module

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Cosine Annealing: Elegant Schedules for Modern Deep Learning

The Geometric Elegance of Cosine Schedules

Among the various learning rate schedules, cosine annealing stands out for its mathematical elegance and remarkable empirical success. Unlike step decay's sharp transitions or exponential decay's fixed percentage reduction, cosine annealing traces a smooth, S-shaped curve that starts gently, accelerates in the middle, and gently approaches the minimum—mirroring the natural dynamics of optimization convergence.

The cosine schedule has become ubiquitous in modern deep learning, serving as the default choice for transformer training (BERT, GPT, Vision Transformers) and achieving state-of-the-art results across computer vision, NLP, and beyond. Its extension to warm restarts (SGDR: Stochastic Gradient Descent with Warm Restarts) further enhances its power by enabling exploration-exploitation cycles within a single training run.

This page provides a comprehensive treatment of cosine annealing: from the basic formulation through warm restart variants, implementation details, and the theoretical principles that make it work so well.

What You Will Master

By the end of this page, you will understand why cosine annealing's shape matches optimization dynamics, implement both single-cycle and warm restart variants, tune the schedule's hyperparameters for your specific training scenario, and recognize when cosine annealing outperforms alternatives.

Mathematical Formulation: The Cosine Curve

Basic Cosine Annealing:

The learning rate follows a half-period cosine function from the maximum to minimum:

$$\eta_t = \eta_{\min} + \frac{1}{2}(\eta_{\max} - \eta_{\min})\left(1 + \cos\left(\frac{\pi t}{T}\right)\right)$$

Where:

$\eta_t$ is the learning rate at epoch $t$
$\eta_{\max}$ is the initial (maximum) learning rate
$\eta_{\min}$ is the final (minimum) learning rate (often 0 or very small)
$T$ is the total number of training epochs
$t \in [0, T]$ is the current epoch

Key Properties of the Cosine Curve:

Smooth Start: At $t=0$, $\cos(0) = 1$, so $\eta_0 = \eta_{\max}$. The derivative is zero—meaning the schedule starts flat.
Smooth End: At $t=T$, $\cos(\pi) = -1$, so $\eta_T = \eta_{\min}$. Again, the derivative is zero—the schedule ends flat.
Steepest Decay at Midpoint: At $t = T/2$, the decay rate is maximal. This concentrates the transition in the middle of training.
Monotonically Decreasing: The schedule strictly decreases from $\eta_{\max}$ to $\eta_{\min}$ without oscillations.

Cosine Annealing Values at Key Points
Training Progress (t/T)	cos(πt/T)	Learning Rate	Decay Rate
0%	1.0	η_max	Minimal (flat)
25%	0.707	0.85 × η_max	Increasing
50%	0.0	0.5 × η_max	Maximum
75%	-0.707	0.15 × η_max	Decreasing
100%	-1.0	η_min	Minimal (flat)

Intuitive Interpretation:

The cosine schedule embodies a compelling intuition about optimization:

Early Training (First 25%): Gentle reduction preserves exploration. The model is still finding good basins of attraction—we don't want to prematurely commit.
Middle Training (25-75%): Accelerated reduction as the model transitions from exploration to exploitation. This is where major phase changes happen.
Late Training (Last 25%): Gentle approach to minimum. Like a landing aircraft, we want to touch down smoothly rather than crash.

This matches empirical observations: most of the 'work' of optimization happens in the middle phase, with the beginning and end being more about setup and refinement.

The Flat Ends Matter

The zero derivative at start and end isn't just aesthetic—it provides stability. At the start, a sudden LR drop would be jarring; the flat region lets training settle. At the end, the flat region lets the model converge precisely without the LR racing toward zero.

Warm Restarts: Cosine Annealing with Multiple Cycles

SGDR: Stochastic Gradient Descent with Warm Restarts extends basic cosine annealing by periodically resetting the learning rate to its maximum value, creating multiple cosine cycles within a single training run. This seemingly simple modification has profound effects on optimization dynamics.

The Core Idea:

Instead of a single cosine decay from $\eta_{\max}$ to $\eta_{\min}$, SGDR executes multiple cosine cycles. At the end of each cycle, the learning rate 'restarts' to $\eta_{\max}$ (or a reduced maximum), beginning a new exploration phase.

Mathematical Formulation:

For a training run with cycle length $T_i$ for cycle $i$:

$$\eta_t = \eta_{\min} + \frac{1}{2}(\eta_{\max} - \eta_{\min})\left(1 + \cos\left(\frac{\pi T_{cur}}{T_i}\right)\right)$$

Where $T_{cur}$ is the number of epochs since the last restart.

Cycle Length Strategies:

Fixed Cycles: All cycles have the same length $T_0$. Train for $n \times T_0$ epochs with $n$ complete cycles.
Doubling Cycles: Each cycle is twice as long as the previous: $T_i = T_0 \times 2^i$. This provides increasingly long refinement periods.
Linear Growth: Cycles increase linearly: $T_i = T_0 \times (i + 1)$. Balanced between fixed and doubling.

Warm Restart Cycle Strategies
Strategy	Cycle Lengths	Total Epochs (4 cycles)	Best For
Fixed (T₀=50)	50, 50, 50, 50	200	Ensemble diversification
Doubling (T₀=10)	10, 20, 40, 80	150	Progressive refinement
Linear (T₀=25)	25, 50, 75, 100	250	Balanced exploration

Why Warm Restarts Help:

Escape Local Minima: The learning rate spike at restart can jolt the model out of poor local optima, enabling exploration of better basins.
Snapshot Ensembles: Models at cycle endpoints (just before restart) can be saved and ensembled, providing multiple diversified models from one training run.
Implicit Regularization: The periodic exploration prevents overfitting to specific features of the training data.
Faster Convergence: Counterintuitively, multiple shorter cycles often converge faster than one long decay, especially for complex loss landscapes.

The Snapshot Ensemble Bonus:

Each cycle endpoint represents a model that has converged with different initialization (the state at restart). These models, sharing early training but diverging in later cycles, provide diverse predictions that average well:

$$\text{Ensemble}(x) = \frac{1}{n}\sum_{i=1}^n f_{\theta_i}(x)$$

where $\theta_i$ are the parameters saved at the end of cycle $i$.

Restart is Not Reinitialization

The 'restart' only affects the learning rate—model parameters continue from where they left off. This 'warm' restart preserves learned representations while enabling new exploration. Cold restart (reinitializing weights) would waste all previous training.

Theoretical Justification: Why Cosine Outperforms

The empirical success of cosine annealing invites theoretical explanation. Several complementary perspectives illuminate why this particular shape works so well.

Optimization Dynamics Correspondence:

Neural network training exhibits characteristic dynamics that cosine annealing naturally matches:

Early Phase (Rapid Feature Learning): Loss drops quickly as the model learns basic features. High, stable LR supports rapid progress.
Middle Phase (Refinement): The model refines feature representations and resolves competition between alternative solutions. Moderate, decreasing LR enables this transition.
Late Phase (Fine-Tuning): Small adjustments maximize performance. Very low LR prevents overshooting optimal point.

The cosine curve's S-shape naturally concentrates change in the middle phase where the most optimization 'action' occurs.

Comparison with Polynomial Decay:

Theoretical analysis of SGD convergence often suggests polynomial decay: $$\eta_t \propto t^{-\alpha}$$

for some $\alpha \in (0.5, 1)$. Cosine annealing approximates this behavior:

For much of the schedule, cosine decay rate is comparable to polynomial decay
The smooth start and end provide additional stability benefits
The fixed endpoint (reaching exactly $\eta_{\min}$ at epoch $T$) is practically useful

Advantages Over Other Schedules

•Vs. Step Decay: Smoother transitions, no discontinuity-induced instability, better for batch normalization.
•Vs. Exponential Decay: Slow start preserves exploration; exponential starts decaying immediately. Cosine has predictable endpoint.
•Vs. Linear Decay: Cosine's curved shape better matches optimization dynamics; linear is too aggressive early and too gentle late.
•Vs. Polynomial Decay: Similar theoretical properties but cosine is simpler to specify (just endpoints) and has the smooth-end property.

The Warm Restart Mechanism:

Warm restarts provide a theoretically-grounded mechanism for escaping local minima. When the learning rate spikes:

Increased Noise: SGD noise amplitude scales with learning rate, enabling escape from shallow local minima.
Basin Exploration: The high LR allows the model to traverse between different basins of attraction.
Diversity Injection: Each restart creates a divergence point, generating model diversity for ensembles.

The cosine shape of each cycle then guides convergence to a new local minimum, potentially better than the previous one.

The Information Theory Perspective

Some researchers interpret warm restarts through information-theoretic lens: each cycle introduces new 'information' (exploration) followed by 'compression' (convergence). This explore-compress cycle mirrors how humans learn complex skills through alternating broad exploration and focused practice.

Implementation: From Basic to Production-Ready

PyTorch provides native support for cosine annealing, but understanding the implementation details enables customization and debugging.

cosine_annealing_implementations.py
Python
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import numpy as np
import torch
from torch.optim.lr_scheduler import (
    CosineAnnealingLR, 
    CosineAnnealingWarmRestarts,
    LambdaLR
)
from typing import Optional, List
import math
 
# =====================================================
# Implementation 1: PyTorch Native CosineAnnealingLR
# =====================================================
def create_cosine_annealing(
    optimizer,
    T_max: int,
    eta_min: float = 0,
    last_epoch: int = -1
):
    """
    Standard single-cycle cosine annealing.
    
    Args:
        optimizer: Wrapped optimizer
        T_max: Maximum number of epochs (full cosine period)
        eta_min: Minimum learning rate
        last_epoch: For resuming training
    
    LR formula: η_t = η_min + 0.5*(η_max - η_min)*(1 + cos(π*t/T_max))
    """
    return CosineAnnealingLR(
        optimizer, 
        T_max=T_max, 
        eta_min=eta_min,
        last_epoch=last_epoch
    )
 
 
# =====================================================
# Implementation 2: PyTorch Native Warm Restarts
# =====================================================
def create_warm_restarts(
    optimizer,
    T_0: int,
    T_mult: int = 1,
    eta_min: float = 0,
    last_epoch: int = -1
):
    """
    Cosine annealing with warm restarts (SGDR).
    
    Args:
        optimizer: Wrapped optimizer
        T_0: Number of epochs for first cycle
        T_mult: Multiplier for subsequent cycles (T_i = T_0 * T_mult^i)
        eta_min: Minimum learning rate
        last_epoch: For resuming
    
    T_mult=1: All cycles same length T_0
    T_mult=2: Cycles double (10, 20, 40, 80, ...)
    """
    return CosineAnnealingWarmRestarts(
        optimizer,
        T_0=T_0,
        T_mult=T_mult,
        eta_min=eta_min,
        last_epoch=last_epoch
    )
 
 
# =====================================================
# Implementation 3: Cosine with Warmup
# =====================================================
class CosineAnnealingWithWarmup:
    """
    Cosine annealing with linear warmup period.
    
    This is the standard schedule for transformer training:
    1. Linear warmup from near-zero to base LR
    2. Cosine decay from base LR to minimum
    """
    
    def __init__(
        self,
        optimizer,
        warmup_epochs: int,
        total_epochs: int,
        warmup_start_lr: float = 1e-7,
        eta_min: float = 0
    ):
        self.optimizer = optimizer
        self.warmup_epochs = warmup_epochs
        self.total_epochs = total_epochs
        self.warmup_start_lr = warmup_start_lr
        self.eta_min = eta_min
        
        # Store base LRs from optimizer
        self.base_lrs = [pg['lr'] for pg in optimizer.param_groups]
        
        self.current_epoch = 0
        self.cosine_epochs = total_epochs - warmup_epochs
        
    def get_lr(self) -> List[float]:
        if self.current_epoch < self.warmup_epochs:
            # Linear warmup
            alpha = self.current_epoch / self.warmup_epochs
            return [
                self.warmup_start_lr + alpha * (base_lr - self.warmup_start_lr)
                for base_lr in self.base_lrs
            ]
        else:
            # Cosine annealing
            cosine_epoch = self.current_epoch - self.warmup_epochs
            cosine_factor = 0.5 * (1 + math.cos(math.pi * cosine_epoch / self.cosine_epochs))
            return [
                self.eta_min + cosine_factor * (base_lr - self.eta_min)
                for base_lr in self.base_lrs
            ]
    
    def step(self, epoch: Optional[int] = None):
        if epoch is not None:
            self.current_epoch = epoch
        else:
            self.current_epoch += 1
        
        lrs = self.get_lr()
        for param_group, lr in zip(self.optimizer.param_groups, lrs):
            param_group['lr'] = lr
    
    def state_dict(self):
        return {
            'current_epoch': self.current_epoch,
            'warmup_epochs': self.warmup_epochs,
            'total_epochs': self.total_epochs,
            'base_lrs': self.base_lrs,
            'warmup_start_lr': self.warmup_start_lr,
            'eta_min': self.eta_min
        }
    
    def load_state_dict(self, state_dict):
        self.current_epoch = state_dict['current_epoch']
        self.warmup_epochs = state_dict['warmup_epochs']
        self.total_epochs = state_dict['total_epochs']
        self.base_lrs = state_dict['base_lrs']
        self.warmup_start_lr = state_dict['warmup_start_lr']
        self.eta_min = state_dict['eta_min']
        self.cosine_epochs = self.total_epochs - self.warmup_epochs
 
 
# =====================================================
# Implementation 4: Warm Restarts with Warmup
# =====================================================
class WarmRestartWithWarmup:
    """
    Full-featured cosine schedule with:
    - Initial linear warmup
    - Multiple cosine cycles with restarts
    - Configurable cycle length growth
    - Maximum LR decay across restarts
    
    This is production-ready for transformer training.
    """
    
    def __init__(
        self,
        optimizer,
        warmup_epochs: int,
        cycle_epochs: int,
        num_cycles: int,
        cycle_mult: float = 1.0,
        max_lr_decay: float = 1.0,
        warmup_start_lr: float = 1e-7,
        eta_min: float = 0
    ):
        """
        Args:
            optimizer: Wrapped optimizer
            warmup_epochs: Linear warmup duration
            cycle_epochs: First cycle duration (after warmup)
            num_cycles: Number of cosine cycles
            cycle_mult: Cycle length multiplier (1.0 = fixed, 2.0 = doubling)
            max_lr_decay: Factor to reduce max LR at each restart
            warmup_start_lr: LR at epoch 0
            eta_min: Minimum LR (floor)
        """
        self.optimizer = optimizer
        self.warmup_epochs = warmup_epochs
        self.cycle_epochs = cycle_epochs
        self.num_cycles = num_cycles
        self.cycle_mult = cycle_mult
        self.max_lr_decay = max_lr_decay
        self.warmup_start_lr = warmup_start_lr
        self.eta_min = eta_min
        
        self.base_lrs = [pg['lr'] for pg in optimizer.param_groups]
        self.current_epoch = 0
        
        # Precompute cycle boundaries
        self.cycle_ends = [warmup_epochs]
        for i in range(num_cycles):
            cycle_len = cycle_epochs * (cycle_mult ** i)
            self.cycle_ends.append(self.cycle_ends[-1] + cycle_len)
        
        self.total_epochs = int(self.cycle_ends[-1])
        print(f"WarmRestartWithWarmup: {num_cycles} cycles, "
              f"total {self.total_epochs} epochs")
        print(f"  Cycle ends: {[int(e) for e in self.cycle_ends]}")
    
    def _get_cycle_info(self, epoch: int):
        """Determine current cycle and position within it."""
        for cycle_idx in range(len(self.cycle_ends) - 1):
            if epoch < self.cycle_ends[cycle_idx + 1]:
                cycle_start = self.cycle_ends[cycle_idx]
                cycle_end = self.cycle_ends[cycle_idx + 1]
                cycle_progress = (epoch - cycle_start) / (cycle_end - cycle_start)
                return cycle_idx, cycle_progress
        
        # Beyond planned cycles: stay at minimum
        return self.num_cycles - 1, 1.0
    
    def get_lr(self) -> List[float]:
        if self.current_epoch < self.warmup_epochs:
            # Warmup phase
            alpha = self.current_epoch / self.warmup_epochs
            return [
                self.warmup_start_lr + alpha * (base_lr - self.warmup_start_lr)
                for base_lr in self.base_lrs
            ]
        
        cycle_idx, progress = self._get_cycle_info(self.current_epoch)
        
        # Compute max LR for this cycle (may decay across restarts)
        cycle_max_lr_factor = self.max_lr_decay ** cycle_idx
        
        # Cosine factor
        cosine_factor = 0.5 * (1 + math.cos(math.pi * progress))
        
        return [
            self.eta_min + cosine_factor * 
            (base_lr * cycle_max_lr_factor - self.eta_min)
            for base_lr in self.base_lrs
        ]
    
    def step(self, epoch: Optional[int] = None):
        if epoch is not None:
            self.current_epoch = epoch
        else:
            self.current_epoch += 1
        
        lrs = self.get_lr()
        for param_group, lr in zip(self.optimizer.param_groups, lrs):
            param_group['lr'] = lr
 
 
# =====================================================
# Implementation 5: Lambda-Based for Maximum Flexibility
# =====================================================
def create_cosine_with_lambda(
    optimizer,
    warmup_epochs: int,
    total_epochs: int,
    min_lr_ratio: float = 0.0,
    num_cycles: float = 0.5  # 0.5 = half period (standard), 1.0 = full period
):
    """
    Use LambdaLR for customized cosine schedules.
    
    Allows fractional cycles and other customizations.
    """
    
    def lr_lambda(epoch: int) -> float:
        if epoch < warmup_epochs:
            return epoch / warmup_epochs
        
        progress = (epoch - warmup_epochs) / (total_epochs - warmup_epochs)
        cosine_value = math.cos(math.pi * num_cycles * 2 * progress)
        # Map [-1, 1] to [min_lr_ratio, 1]
        return min_lr_ratio + (1 + cosine_value) * (1 - min_lr_ratio) / 2
    
    return LambdaLR(optimizer, lr_lambda)
 
 
# =====================================================
# Visualization and Analysis
# =====================================================
def visualize_cosine_schedules(total_epochs: int = 200):
    """Compare different cosine schedule variants."""
    import matplotlib.pyplot as plt
    
    epochs = np.arange(total_epochs)
    base_lr = 0.1
    
    # Basic cosine
    basic = base_lr * 0.5 * (1 + np.cos(np.pi * epochs / total_epochs))
    
    # With warmup (10%)
    warmup_length = int(total_epochs * 0.1)
    with_warmup = np.where(
        epochs < warmup_length,
        base_lr * epochs / warmup_length,
        base_lr * 0.5 * (1 + np.cos(np.pi * (epochs - warmup_length) / (total_epochs - warmup_length)))
    )
    
    # Warm restarts (4 cycles)
    cycle_length = total_epochs // 4
    warm_restart = base_lr * 0.5 * (1 + np.cos(np.pi * (epochs % cycle_length) / cycle_length))
    
    # Warm restarts with doubling
    def doubling_restarts(epoch, base=50):
        # Find which cycle we're in
        cumsum = 0
        cycle = 0
        while cumsum + base * (2 ** cycle) <= epoch:
            cumsum += base * (2 ** cycle)
            cycle += 1
        cycle_len = base * (2 ** cycle)
        progress = (epoch - cumsum) / cycle_len
        return 0.5 * (1 + np.cos(np.pi * progress))
    
    doubling = np.array([base_lr * doubling_restarts(e) for e in epochs])
    
    fig, ax = plt.subplots(figsize=(12, 6))
    ax.plot(epochs, basic, label='Basic Cosine', linewidth=2)
    ax.plot(epochs, with_warmup, label='Cosine + Warmup', linewidth=2)
    ax.plot(epochs, warm_restart, label='Warm Restarts (fixed)', linewidth=2)
    ax.plot(epochs, doubling, label='Warm Restarts (doubling)', linewidth=2)
    
    ax.set_xlabel('Epoch')
    ax.set_ylabel('Learning Rate')
    ax.set_title('Cosine Annealing Variants Comparison')
    ax.legend()
    ax.grid(True, alpha=0.3)
    
    return fig

Per-Epoch vs Per-Step Scheduling

CosineAnnealingLR expects T_max in epochs, with scheduler.step() called once per epoch. If you call step() per batch/step, you'll complete the schedule in T_max steps, not epochs—a common source of bugs. For per-step scheduling, compute T_max = epochs × steps_per_epoch.

Hyperparameter Selection: Tuning Cosine Schedules

Cosine annealing has fewer hyperparameters than some alternatives, but each choice significantly impacts training dynamics.

T_max (Total Epochs/Cycle Length):

The most important parameter. T_max controls the rate of learning rate decay:

Too small: LR decays too quickly, causing premature convergence or incomplete exploration.
Too large: LR stays high too long, causing late-stage oscillation; you're wasting epochs with unnecessarily high LR.
Just right: Training progresses smoothly through exploration → refinement → convergence phases.

Practical Guidelines:

For basic cosine, set T_max = total_epochs (the schedule naturally fits the training duration)
For warm restarts, experiment with T_0 = total_epochs / num_desired_cycles
Doubling cycles (T_mult=2) often work well: early cycles for exploration, later long cycles for refinement

eta_min (Minimum Learning Rate):

Controls the LR floor. Options:

eta_min = 0: Full decay to zero. Simple, often works fine. Model converges to a locally stable point.
eta_min = 1e-6 to 1e-4: Small but non-zero floor. Prevents complete stalling, allows small adjustments at the end.
eta_min = 0.01 × eta_max: 1% of maximum. Preserves some learning capability for late refinement.

Cosine Annealing Configuration Guidelines
Scenario	T_max / T_0	T_mult	eta_min	Warmup
ImageNet training (90 epochs)	90	—	0	0
Transformer pretraining (long)	Total epochs	—	1e-5	10% of total
SGDR ensemble (100 epochs)	25	1	0	5 epochs
Progressive refinement (150 epochs)	10	2	1e-6	5 epochs
Fine-tuning pretrained (20 epochs)	20	—	1e-6	2 epochs

Warmup Duration:

For cosine schedules with warmup:

Typical range: 5-10% of total training
Large batch training: May need up to 10-20% warmup (large batches require stability)
Small models/datasets: May work with minimal or no warmup
Transformer training: Often use 1,000-10,000 steps of warmup regardless of total length

Cycle Multiplier (T_mult) for Warm Restarts:

T_mult = 1: Fixed cycle length. Good for snapshot ensembles where you want equally-weighted models.
T_mult = 2: Doubling cycles. Natural progression from exploration to refinement.
T_mult = 1.5: Moderate growth. Compromise between fixed and doubling.

Max LR Decay Across Restarts:

Some implementations reduce the maximum learning rate at each restart: $$\eta_{\max}^{(i)} = \eta_{\max}^{(0)} \cdot \gamma^i$$

This provides additional annealing across cycles, ensuring later cycles are more about refinement than exploration.

The Warmup Ratio Rule

For most scenarios, 5-10% warmup followed by cosine decay works well. The intuition: warmup lets batch norm and optimizer state stabilize; cosine then provides smooth, principled decay. When in doubt, start with 5% warmup and adjust based on early training stability.

Comparison and Best Practices

Cosine annealing has become the default choice for modern deep learning, but it's not universally optimal. Understanding when to use it—and when alternatives might be better—requires examining specific scenarios.

When Cosine Annealing Excels:

Ideal Scenarios for Cosine Annealing

•Transformer Training: BERT, GPT, ViT, and most modern transformers use cosine with warmup as the standard schedule.
•Long Training Runs: The smooth curve naturally fits extended training without discrete phase decisions.
•Ensemble Generation: Warm restarts provide model diversity for free during single training runs.
•Unknown Optimal Schedule: When you don't have published baselines, cosine is a reliable default.
•Reproducibility: The schedule is fully determined by few parameters, making reproduction easy.

When Alternatives Might Be Better:

Following Established Recipes: If a seminal paper (e.g., original ResNet) uses step decay with specific milestones, reproducing their results requires their schedule.
Validation-Based Reduction: When you need adaptive scheduling based on training dynamics, ReduceLROnPlateau might be more appropriate.
Very Short Training: For very few epochs (< 20), the overhead of cosine's gentle start/end phases might not be worthwhile.
Debugging Training Issues: Step decay's discrete phases are easier to analyze when diagnosing problems.

Schedule Selection Decision Matrix
Criterion	Favor Cosine	Favor Step	Favor Exponential
Training length	100+ epochs	30-100 epochs	Long with stability needs
Published baseline	Transformer papers	CNN papers (ResNet)	Few formal baselines
Stability priority	High	Moderate	Very high
Interpretability need	Moderate	High	Low
Ensemble goal	Strong (warm restarts)	Weak	Weak

Production Best Practices

•Always include warmup for large models or large batch sizes. The cost is minimal; the stability benefit is significant.
•Log learning rate every epoch to diagnose schedule issues and correlate with loss curves.
•Save model at cycle endpoints for warm restart runs to enable snapshot ensembles.
•Validate schedule before training by running the scheduler through all epochs and plotting the curve.
•Include scheduler state in checkpoints to enable correct resume from any point.

The Modern Default

If you're starting a new project without specific schedule requirements, cosine annealing with 5-10% linear warmup is the recommended default. It works well across architectures, has theoretical grounding, and produces reliable results with minimal tuning.

Summary: Cosine Annealing Mastery

Cosine annealing represents the current state-of-the-art in learning rate scheduling, combining mathematical elegance with practical effectiveness. Its S-shaped curve naturally matches optimization dynamics, and the warm restart variant enables sophisticated exploration-exploitation strategies.

Key Takeaways

•Core Formula: $\eta_t = \eta_{\min} + \frac{1}{2}(\eta_{\max} - \eta_{\min})(1 + \cos(\pi t/T))$ — smooth decay with flat start and end.
•Warm Restarts: Periodic LR resets enable escape from local minima and snapshot ensemble generation.
•Warmup Integration: 5-10% linear warmup followed by cosine decay is the modern standard for transformer training.
•Natural Dynamics Match: The S-curve concentrates change in the middle of training, matching optimization phase transitions.
•Cycle Length Strategies: Fixed cycles for ensembles; doubling cycles for progressive refinement.
•Modern Default: When uncertain, cosine with warmup is the recommended starting point.

What's Next:

The next page explores warmup strategies in greater depth, examining why large models and large batches require careful initialization of the learning rate and how to design effective warmup schedules for your specific training scenario.

Page Complete

You now understand cosine annealing from mathematical foundations through advanced warm restart variants. This scheduling approach, combined with warmup strategies covered next, forms the backbone of modern deep learning training pipelines.