- L1 penalty: P = λ · ||w||₁, where ||w||₁ = Σ |wᵢ|.
- L2 penalty: P = λ · ||w||₂², where ||w||₂² = Σ wᵢ².
- Elastic penalty: P = λ · ( α||w||₁ + (1−α)||w||₂² ).
- Choose a regularization type: L1, L2, or elastic mix.
- Enter λ. Increase it to apply stronger shrinkage.
- For elastic mix, set α to balance L1 vs L2.
- Paste your coefficients as comma-separated values.
- Enter the base loss to compute the total objective.
- Press Calculate. Download CSV or PDF if needed.
| Type | λ | α | Coefficients | Base loss | Penalty | Total |
|---|---|---|---|---|---|---|
| Elastic | 0.1 | 0.5 | 1.2, -0.8, 0.3, 2 | 1.75 | 0.5235 | 2.2735 |
| L1 | 0.25 | — | 0.6, 0, -1.1 | 0.92 | 0.425 | 1.345 |
| L2 | 0.05 | — | 2.4, -0.4, 0.9 | 2.1 | 0.3365 | 2.4365 |
Lambda as a control knob for generalization
The regularization strength λ scales the penalty added to the base loss. When λ = 0, the objective equals the unregularized loss. As λ increases, the optimizer favors smaller coefficients, reducing variance and improving stability under noise. In practice, scan λ on a log grid such as 1e-4, 1e-3, 1e-2, 1e-1, 1, and 10, then validate.
Understanding L1 penalty behavior
L1 uses ||w||1 = Σ|wi|, so every coefficient contributes linearly. This geometry tends to create sparse solutions where some weights become exactly zero, which is useful for feature selection. The calculator reports the L1 norm and λ·||w||1 so you can compare how sparsity pressure changes across candidate vectors.
Understanding L2 penalty behavior
L2 uses ||w||2² = Σwi², which penalizes large weights more aggressively than small ones. It rarely forces exact zeros, but it shrinks correlated features together and improves numerical conditioning. The calculator shows both ||w||2 and ||w||2², letting you see how the squared term dominates when a few coefficients are large.
Elastic mix as a practical compromise
Elastic combines L1 and L2² with α in [0,1]: λ(α||w||1 + (1−α)||w||2²). Higher α leans toward sparsity; lower α leans toward smooth shrinkage. A common starting point is α = 0.5, then adjust based on whether you prefer fewer active features or more stable coefficients.
Interpreting totals and reporting outputs
The total objective J = base_loss + penalty is the value you would minimize during training or compare across settings. Keep base loss and penalty units consistent, and standardize features so λ has a comparable effect across coefficients. Bias handling matters: if the intercept is a baseline shift, you may exclude it from the penalty to avoid moving the mean. Track penalty share versus base loss; when penalty dominates, the model may underfit. Pair results with validation curves, reporting training loss and validation error across λ. For vector inputs, the norms grow with dimension, so compare settings using the same feature set. If you change the number of coefficients, re-tune λ. When comparing two candidate weight vectors, prefer the one with lower total objective at the same λ, then confirm on held-out data in practice.
1) What does this calculator return?
It returns L1, L2, and L2-squared norms, the selected penalty value, and the total objective J = base_loss + penalty for your inputs.
2) When should I choose L1?
Choose L1 when you want sparsity, simpler models, or feature selection. It can push some coefficients exactly to zero as λ increases.
3) When should I choose L2?
Choose L2 when you want smooth shrinkage, better conditioning, and stability with correlated features. It typically reduces magnitudes without forcing exact zeros.
4) How do I set alpha for elastic mix?
Start at α = 0.5. Increase α for more sparsity (more L1) or decrease α for more stability (more L2²), then select using validation results.
5) Should I regularize the bias term?
Often no, because the intercept represents a baseline shift. Regularizing it can move the mean prediction unnecessarily. Include it only if your method requires it.
6) Why does feature scaling matter?
Without scaling, coefficients reflect feature units, so the same λ penalizes features unevenly. Standardizing features makes λ comparable and improves fair shrinkage across weights.