Lasso Regression L1 - Learning Module

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Feature Selection Aspect

Lasso as Automatic Feature Selection

One of Lasso's most celebrated properties is automatic feature selection. By driving coefficients exactly to zero, Lasso doesn't just shrink the model—it identifies which features matter and discards the rest. This transforms a regression method into a variable selection tool.

But this power comes with nuances. Feature selection via Lasso is not a black box that always works. Understanding when and how Lasso selects features—and when it fails—is essential for practitioners who rely on Lasso for scientific inference or dimensionality reduction.

This page addresses the critical questions:

When does Lasso correctly identify the true relevant features?
How does correlation among features affect selection?
Can we trust the selected features for scientific interpretation?
What are the connections to classical variable selection methods?
How do we quantify uncertainty in Lasso's selections?

What You Will Learn

By the end of this page, you will understand Lasso as a feature selection method, its guarantees and limitations, the crucial distinction between prediction and selection, stability selection for robust variable identification, and post-selection inference.

Selection vs. Prediction: Two Different Goals

A fundamental distinction underlies much confusion about Lasso: prediction and feature selection are different objectives that may require different approaches.

Prediction Goal:

Find a model $\hat{f}$ that minimizes expected prediction error:

$$\mathbb{E}[(Y - \hat{f}(\mathbf{X}))^2]$$

We don't care which features are used, only that predictions are accurate. Including irrelevant features is fine if it doesn't hurt prediction.

Feature Selection Goal:

Identify the true support $S = {j : \beta_j^* \neq 0}$—the features that genuinely influence the outcome. We want:

$$\hat{S} = S$$

with high probability. False positives (including irrelevant features) and false negatives (missing relevant features) both matter.

Why the Distinction Matters:

Prediction vs. Selection Trade-offs
Aspect	For Prediction	For Selection
Correlated features	Interchangeable—keep any	Must identify the correct one
False inclusions	Mild penalty (bias)	Serious error (wrong science)
Optimal λ	Minimize CV prediction error	May need different λ
Noise variables	Tolerable if shrunk	Must exclude all
Validation	Test set MSE	Known-truth simulations

Lasso's Dual Role:

Lasso attempts to serve both goals simultaneously, which creates tension:

For prediction: Choose λ via cross-validation (minimize test error)
For selection: This λ may include noise variables or exclude weak signals

Example of Divergence:

Consider 100 features where:

Features 1-5 are true signals with coefficients $\beta^* = (3, -2, 1.5, -1, 0.5)$
Feature 6 is pure noise but happens to correlate with feature 1 ($r = 0.95$)

For prediction: Including feature 6 instead of feature 1 barely changes predictions—they're nearly interchangeable.

For selection: Lasso might select feature 6 (noise) and exclude feature 1 (true signal). This is scientifically wrong despite good predictions.

Don't Confuse the Goals

Good cross-validation prediction error does NOT guarantee correct feature selection. A model with wrong features can predict well. If your goal is scientific understanding (identifying true drivers), prediction accuracy is necessary but not sufficient.

Conditions for Correct Selection

When does Lasso successfully recover the true support? This requires conditions on both the design matrix $\mathbf{X}$ and the true coefficients $\boldsymbol{\beta}^*$.

The Three Key Conditions:

1. Irrepresentable Condition (IC):

Let $S$ be the true support and $S^c$ its complement. The irrepresentable condition requires:

$$|\mathbf{C}{S^cS}\mathbf{C}{SS}^{-1}\text{sign}(\boldsymbol{\beta}S^*)|\infty < 1$$

where $\mathbf{C} = \frac{1}{n}\mathbf{X}^T\mathbf{X}$ is the sample correlation matrix.

Interpretation: Irrelevant features cannot be perfectly "explained by" relevant features. If an irrelevant feature is highly correlated with relevant ones (weighted by their signs), Lasso may incorrectly select it.

2. Beta-Min Condition:

The smallest true non-zero coefficient must be large enough:

$$\min_{j \in S} |\beta_j^*| \geq c \cdot \sigma \sqrt{\frac{\log p}{n}}$$

for some constant $c > 0$. This ensures signals are detectable above the noise floor.

3. Restricted Eigenvalue Condition:

The design matrix must be well-conditioned on sparse directions:

$$\left|\mathbf{X}_S \mathbf{v}\right|_2^2 \geq \kappa n |\mathbf{v}|_2^2$$

for all vectors $\mathbf{v}$ and sparse index sets $S$. This prevents near-collinearity among relevant features.

When Selection Succeeds vs. Fails

•Success: IC holds, signals are strong, features weakly correlated → Lasso recovers true support with high probability
•Failure (IC violation): Irrelevant features highly correlated with relevant → Lasso may include wrong features
•Failure (weak signals): True coefficients too small → Lasso misses them (sets to zero)
•Failure (collinearity): True features are highly correlated → Lasso arbitrarily selects among them

Theoretical Guarantee:

Theorem (Selection Consistency): Under the irrepresentable condition, beta-min condition, and restricted eigenvalue condition, with $\lambda \asymp \sigma\sqrt{\frac{\log p}{n}}$:

$$P(\hat{S}_{\text{lasso}} = S) \to 1 \quad \text{as } n \to \infty$$

Reality Check:

These conditions are sufficient but not necessary. Lasso sometimes works even when conditions are violated (lucky data). But it can also fail when conditions appear satisfied (bad luck or finite-sample effects).

More importantly, we rarely know if conditions hold in practice. We don't know the true support $S$. The irrepresentable condition depends on $\text{sign}(\boldsymbol{\beta}^*)$, which is unknown. We must work with assumptions, not certainties.

Practical Implications

High correlation between features is a red flag for selection reliability. If you know or suspect groups of correlated features, consider using Elastic Net or Group Lasso. If you need reliable selection, use stability selection (coming next) instead of trusting a single Lasso run.

The Correlation Problem

The most common practical failure mode for Lasso selection is correlated features. This deserves detailed examination.

The Issue:

When features $j$ and $k$ are highly correlated ($|\rho_{jk}| \approx 1$), and the true model uses feature $j$:

Features $j$ and $k$ carry nearly the same information
Lasso only needs one—it's redundant to include both
Which one Lasso chooses is unstable—small data perturbations swap selections
The "wrong" choice is not wrong for prediction, but wrong for interpretation

Demonstration:

Correlation Instability Demo
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import numpy as np
from sklearn.linear_model import Lasso
 
def demonstrate_correlation_instability(rho=0.95, n_trials=100):
    """
    Show how Lasso selection becomes unstable with correlated features.
    
    True model: y = 2*x1 + noise
    x2 is correlated with x1 (correlation = rho)
    """
    np.random.seed(42)
    n = 100
    
    selections = {'x1_only': 0, 'x2_only': 0, 'both': 0, 'neither': 0}
    
    for trial in range(n_trials):
        # Generate correlated features
        x1 = np.random.randn(n)
        x2 = rho * x1 + np.sqrt(1 - rho**2) * np.random.randn(n)
        X = np.column_stack([x1, x2])
        
        # True model uses only x1
        y = 2 * x1 + 0.5 * np.random.randn(n)
        
        # Fit Lasso
        lasso = Lasso(alpha=0.1, fit_intercept=True)
        lasso.fit(X, y)
        
        coef1_nonzero = abs(lasso.coef_[0]) > 0.01
        coef2_nonzero = abs(lasso.coef_[1]) > 0.01
        
        if coef1_nonzero and not coef2_nonzero:
            selections['x1_only'] += 1
        elif coef2_nonzero and not coef1_nonzero:
            selections['x2_only'] += 1
        elif coef1_nonzero and coef2_nonzero:
            selections['both'] += 1
        else:
            selections['neither'] += 1
    
    print(f"Selection outcomes over {n_trials} trials (correlation = {rho}):")
    print(f"  x1 only (correct): {selections['x1_only']}%")
    print(f"  x2 only (wrong):   {selections['x2_only']}%")
    print(f"  Both selected:     {selections['both']}%")
    print(f"  Neither:           {selections['neither']}%")
    
    return selections
 
 
# Low correlation: stable selection
print("Low correlation (ρ = 0.3):")
demonstrate_correlation_instability(rho=0.3)
 
print("\nHigh correlation (ρ = 0.95):")
demonstrate_correlation_instability(rho=0.95)
 
# With high correlation, Lasso may select the wrong feature!

The Grouping Effect Problem:

When features form correlated groups, Lasso tends to:

Select one "representative" from each group
Ignore other group members, even if truly relevant
The selected representative may not be the most important

Example:

Genes A, B, C in the same pathway are correlated
True model: disease risk depends on all three
Lasso: selects gene A, ignores B and C
Conclusion: only gene A matters (wrong!)

Solutions:

Elastic Net: Adds L2 penalty that promotes grouping—correlated features get similar coefficients
Group Lasso: Explicitly groups features; entire groups are in or out together
Stability Selection: Runs Lasso many times on subsamples; keeps frequently selected features
Pre-aggregation: Average or PCA within known groups before Lasso

Elastic Net for Correlated Features

Elastic Net (α*L1 + (1-α)*L2) with α < 1 exhibits the 'grouping effect': correlated features tend to have similar coefficients, all entering or leaving together. This stabilizes selection among correlated groups, though it reduces sparsity.

Stability Selection

Stability selection (Meinshausen & Bühlmann, 2010) addresses the instability of Lasso selection by aggregating results across many subsamples. Features that are consistently selected across perturbations are deemed reliable.

The Algorithm:

For $b = 1, \ldots, B$ bootstrap/subsample iterations:
- Draw a random subsample of size $\lfloor n/2 \rfloor$
- Fit Lasso (or run along a path of $\lambda$ values)
- Record which features are selected (non-zero)
Compute selection probability for each feature: $$\hat{\Pi}j^{(\lambda)} = \frac{1}{B}\sum{b=1}^B \mathbf{1}{\hat{\beta}_{j}^{(b)}(\lambda) \neq 0}$$
Select features with selection probability above threshold $\pi_{\text{thr}}$: $$\hat{S}^{\text{stable}} = {j : \hat{\Pi}j^{(\lambda)} \geq \pi{\text{thr}}}$$

Typical choices: $\pi_{\text{thr}} = 0.6$ to $0.9$, $B = 100$ to $500$ subsamples.

Stability Selection
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import numpy as np
from sklearn.linear_model import LassoCV
 
def stability_selection(X, y, n_bootstrap=100, sample_fraction=0.5,
                        threshold=0.6):
    """
    Perform stability selection for robust feature identification.
    
    Parameters
    ----------
    X : ndarray of shape (n, p)
        Feature matrix
    y : ndarray of shape (n,)
        Response vector
    n_bootstrap : int
        Number of subsampling iterations
    sample_fraction : float
        Fraction of samples to use in each subsample
    threshold : float
        Selection probability threshold (0.5 to 1.0)
    
    Returns
    -------
    stable_features : list
        Indices of stably selected features
    selection_probabilities : ndarray
        Selection probability for each feature
    """
    n, p = X.shape
    subsample_size = int(n * sample_fraction)
    
    # Track selection across bootstraps
    selection_counts = np.zeros(p)
    
    for b in range(n_bootstrap):
        # Random subsample
        indices = np.random.choice(n, size=subsample_size, replace=False)
        X_sub = X[indices]
        y_sub = y[indices]
        
        # Fit Lasso with CV-selected lambda
        lasso = LassoCV(cv=5, random_state=b)
        lasso.fit(X_sub, y_sub)
        
        # Record selections
        selected = np.abs(lasso.coef_) > 1e-10
        selection_counts += selected
    
    # Compute selection probabilities
    selection_probabilities = selection_counts / n_bootstrap
    
    # Identify stable features
    stable_features = np.where(selection_probabilities >= threshold)[0]
    
    return stable_features, selection_probabilities
 
 
# Example usage
np.random.seed(42)
n, p = 200, 50
 
# Create features with correlation structure
X = np.random.randn(n, p)
# Make features 0-4 correlated with each other
for j in range(1, 5):
    X[:, j] = 0.8 * X[:, 0] + 0.2 * np.random.randn(n)
 
# True model: features 0, 10, 20 matter
beta_true = np.zeros(p)
beta_true[0] = 2.0
beta_true[10] = -1.5
beta_true[20] = 1.0
y = X @ beta_true + 0.5 * np.random.randn(n)
 
# Single Lasso run
from sklearn.linear_model import LassoCV
single_lasso = LassoCV(cv=5).fit(X, y)
single_selected = np.where(np.abs(single_lasso.coef_) > 1e-10)[0]
print(f"Single Lasso selected: {single_selected}")
print(f"True features: [0, 10, 20]")
 
# Stability selection
stable_features, probs = stability_selection(X, y, n_bootstrap=100, threshold=0.7)
print(f"\nStability selection (threshold=0.7): {stable_features}")
print(f"Selection probabilities for true features:")
print(f"  Feature 0: {probs[0]:.2f}")
print(f"  Feature 10: {probs[10]:.2f}")
print(f"  Feature 20: {probs[20]:.2f}")

Theoretical Guarantees:

Stability selection provides control over false positives. Under mild conditions:

$$\mathbb{E}[|\hat{S}^{\text{stable}} \cap S^c|] \leq \frac{q^2}{(2\pi_{\text{thr}} - 1)p}$$

where $q$ is the expected number of selected variables per subsample and $S^c$ is the set of irrelevant variables.

Benefits:

Robustness: Unstable selections (due to correlation) are filtered out
False positive control: Mathematical bound on expected false discoveries
Works with any selector: Can use Lasso, Elastic Net, Random Forest, etc.
Interpretable output: Selection probabilities indicate confidence

Limitations:

Computational cost: $O(B)$ times slower than single Lasso
Power loss: Threshold may exclude true but weakly selected features
Doesn't fix fundamental IC violations: If wrong feature is always selected, stability selection won't help

Complementary Selection Plot

The 'stability path' plot shows selection probability vs. regularization strength for each feature. Stable features maintain high selection probability across λ values. Unstable features show erratic selection patterns. This plot is diagnostic gold for understanding selection reliability.

Post-Selection Inference

A subtle but critical issue arises when using Lasso for scientific inference: the same data used for selection cannot naively be used for inference.

The Problem:

Suppose Lasso selects features $\hat{S} = {1, 5, 12}$. We want to:

Test hypotheses: Is $\beta_1 = 0$?
Construct confidence intervals: $\beta_1 \in [?, ?]$
Report p-values: What's the significance of feature 1?

Naive approach: Fit OLS on selected features, use standard inference.

Why this fails: The selection event $\hat{S} = {1, 5, 12}$ was chosen because features 1, 5, 12 had the strongest apparent associations. Standard inference assumes the model is fixed before seeing data. Selecting based on data and then testing on the same data inflates significance.

The Selection Bias:

Consider: we select feature 1 because $|\hat{\beta}_1^{\text{lasso}}| > 0$. This already tells us the data supports a non-zero coefficient. Testing $H_0: \beta_1 = 0$ after this selection is circular—the selection guarantees the test rejects!

Simulation Evidence:

Post-Selection Bias Demonstration
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import numpy as np
from scipy import stats
from sklearn.linear_model import LassoCV
 
def demonstrate_selection_bias(n_sims=1000):
    """
    Show how naive post-selection inference is biased.
    
    True model: y = noise (no true signals)
    Yet naive p-values after Lasso selection are often significant!
    """
    np.random.seed(42)
    n, p = 100, 50
    
    naive_pvalues = []
    
    for sim in range(n_sims):
        # Generate null data (no true signal)
        X = np.random.randn(n, p)
        y = np.random.randn(n)  # Pure noise!
        
        # Lasso selection
        lasso = LassoCV(cv=5).fit(X, y)
        selected = np.where(np.abs(lasso.coef_) > 1e-10)[0]
        
        if len(selected) == 0:
            continue
        
        # Naive OLS on selected features
        X_sel = X[:, selected]
        beta_ols = np.linalg.lstsq(X_sel, y, rcond=None)[0]
        
        # Naive standard errors
        residuals = y - X_sel @ beta_ols
        sigma_hat = np.sqrt(np.sum(residuals**2) / (n - len(selected)))
        var_beta = sigma_hat**2 * np.diag(np.linalg.inv(X_sel.T @ X_sel))
        se_beta = np.sqrt(var_beta)
        
        # Naive t-test for first selected feature
        t_stat = beta_ols[0] / se_beta[0]
        pvalue = 2 * (1 - stats.t.cdf(abs(t_stat), df=n-len(selected)))
        naive_pvalues.append(pvalue)
    
    # Under the null, p-values should be uniform
    # But selection bias makes small p-values too common!
    
    naive_pvalues = np.array(naive_pvalues)
    print("Under the NULL (no true signals):")
    print(f"  Simulations with selection: {len(naive_pvalues)}")
    print(f"  P-values < 0.05: {np.mean(naive_pvalues < 0.05):.1%} (should be ~5%)")
    print(f"  P-values < 0.01: {np.mean(naive_pvalues < 0.01):.1%} (should be ~1%)")
    print(f"  Mean p-value: {np.mean(naive_pvalues):.3f} (should be ~0.5)")
    
    return naive_pvalues
 
 
pvals = demonstrate_selection_bias()
print("\nConclusion: Naive inference after selection is SEVERELY biased!")

Solutions for Valid Post-Selection Inference:

1. Data Splitting:

Split data into selection set and inference set
Use selection set for Lasso
Use inference set for hypothesis testing on selected features
Pro: Simple, valid. Con: Less power (smaller sample sizes)

2. Selective Inference (Conditional Inference):

Condition on the selection event ${\hat{S} = S}$
Derive the distribution of $\hat{\beta}$ conditional on Lasso selecting $S$
Compute valid p-values and confidence intervals
Pro: Uses full data. Con: Complex, conservative

3. Debiased Lasso:

Construct a "debiased" estimator that corrects for Lasso shrinkage
Valid asymptotically for high-dimensional settings
Pro: Allows inference on all coefficients. Con: Requires careful implementation

4. Bootstrap Methods:

Residual bootstrap or pairs bootstrap
Construct confidence intervals via bootstrap distribution
Pro: Flexible. Con: May not be valid for selection

The Double-Dipping Fallacy

Using the same data for selection and inference is 'double dipping.' Papers reporting significant p-values for features selected by Lasso on the same data are often invalid. Always split data or use valid post-selection inference methods if making scientific claims.

Lasso vs. Other Feature Selection Methods

Lasso is one of many feature selection approaches. Understanding alternatives helps choose the right tool.

The Feature Selection Landscape:

Comparison of Feature Selection Methods
Method	Type	Pros	Cons
Lasso	Embedded (in model)	Efficient, interpretable coefficients	Unstable with correlations
Elastic Net	Embedded	Handles correlations better	Less sparse than Lasso
Forward Stepwise	Wrapper	Well-understood, fast	Greedy, can't recover from errors
Best Subset	Wrapper	Optimal if computable	NP-hard for large p
Random Forest Importance	Embedded/Filter	Captures nonlinearities	Not for coefficient interpretation
Univariate Filtering	Filter	Very fast, simple	Ignores feature interactions
SCAD/MCP	Embedded	Less biased than Lasso	Non-convex, harder to optimize

When to Prefer Lasso:

Linear models: Lasso is designed for linear/GLM settings
Sparse truth: When you believe few features truly matter
High dimensions: Lasso scales well to $p > n$
Interpretability: Coefficients have clear meanings
Computational efficiency: Coordinate descent is very fast

When to Consider Alternatives:

Highly correlated features: Elastic Net or Group Lasso
Nonlinear relationships: Random Forest, Gradient Boosting
Strong signals: Best Subset (if computable) has less bias
Mixed types: Combination methods (e.g., filter then Lasso)
Strict false positive control: Knockoffs, stability selection

Non-Convex Penalties (SCAD, MCP):

SCAD (Smoothly Clipped Absolute Deviation) and MCP (Minimax Concave Penalty) reduce Lasso's bias on large coefficients:

$$\text{Lasso: } |\beta|$$ $$\text{SCAD: } \begin{cases} |\beta| & |\beta| \leq \lambda \ \text{(quadratic transition)} & \lambda < |\beta| < a\lambda \ \frac{(a+1)\lambda}{2} & |\beta| \geq a\lambda \end{cases}$$

SCAD/MCP still produce sparsity but don't shrink large coefficients. The cost: non-convex optimization (local minima possible).

Hybrid Approaches

Consider combining methods: (1) Use univariate filtering to remove obviously irrelevant features, (2) Apply Lasso to the reduced set, (3) Use stability selection for robustness, (4) Apply debiased Lasso for inference. Each step addresses different challenges.

Practical Guidelines for Feature Selection

Based on theory and experience, here are practical recommendations for using Lasso for feature selection.

Step-by-Step Workflow:

Recommended Workflow

•Explore correlations: Compute the feature correlation matrix. Identify groups of highly correlated features. If many correlations exceed $|r| > 0.7$, expect selection instability.
•Consider domain grouping: If features naturally form groups (e.g., genes in pathways, related survey items), consider Group Lasso or pre-aggregation.
•Run stability selection: Don't trust a single Lasso run. Use 100+ subsamples with threshold 0.6-0.8 for reliable selection.
•Examine the stability path: Plot selection probability vs. λ for each feature. Stable features show consistent selection; unstable features oscillate.
•Split data for inference: If reporting p-values or confidence intervals, use a held-out dataset not used for selection.
•Report uncertainty: Present selection probabilities, not just Yes/No. Acknowledge that selection has uncertainty.
•Validate externally: The ultimate test is replication. Do selected features matter in an independent dataset?

What to Report:

For scientific publications using Lasso selection:

Selection method details: λ selection criteria, algorithm used, software/version
Correlation structure: Acknowledge and show correlations among features
Stability analysis: Selection probabilities from stability selection
Inference approach: How p-values/CIs were computed (data splitting, selective inference, etc.)
Sensitivity analysis: How results change with different λ or thresholds
Limitations: Explicitly state what selection cannot tell you

Common Mistakes to Avoid:

Common Mistakes

•Trusting single runs: Selection from one Lasso fit is unreliable. Always assess stability.
•Ignoring correlations: High correlation invalidates selection interpretations.
•Double-dipping inference: Same data for selection and hypothesis testing is invalid.
•Over-interpreting zeros: A zero coefficient doesn't prove irrelevance; it may indicate redundancy.
•Ignoring sample size: In small samples, selection variance is high; be humble about findings.
•Choosing λ for selection by prediction CV: Optimal prediction λ may include noise variables or miss weak signals.

The Humility Principle

Feature selection is inherently uncertain. No method—Lasso included—magically identifies 'the true model.' Present results as 'features selected by Lasso under these conditions' rather than 'the features that matter.' Scientific humility protects against over-interpretation.

Summary: Lasso as Feature Selector

We've explored Lasso's role as an automatic feature selection method. Here are the key insights:

Key Takeaways

•Selection ≠ Prediction: Good prediction doesn't guarantee correct feature identification. These are distinct goals.
•Theoretical Conditions: Irrepresentable condition, beta-min, and restricted eigenvalue conditions govern selection success. Often unverifiable in practice.
•Correlation Instability: Highly correlated features cause unstable, arbitrary selection. This is Lasso's Achilles' heel.
•Stability Selection: Aggregate selections across subsamples for robust identification. Selection probabilities quantify confidence.
•Post-Selection Inference: Naive inference after selection is biased. Use data splitting or specialized methods for valid p-values.
•Method Comparisons: Lasso is one tool among many. Consider Elastic Net for correlations, nonlinear methods for complex relationships.
•Practical Humility: Selection is uncertain. Report stability, acknowledge limitations, validate externally.

The Big Picture:

Lasso's automatic feature selection is powerful but imperfect. It works best when:

Features are weakly correlated
True signals are moderately strong
You use stability selection for robustness
You separate selection from inference

Used thoughtfully, Lasso is an invaluable tool for dimensionality reduction and generating hypotheses. Used naively, it can mislead. The difference is understanding its assumptions and limitations—which you now do.

Module Complete:

You've now mastered Lasso regression comprehensively: the L1 formulation, sparsity mechanics, geometric interpretation, optimization algorithms, and feature selection aspects. This knowledge equips you to apply Lasso effectively and understand its behavior in high-dimensional machine learning problems.

Module Complete

Congratulations! You have completed the comprehensive study of Lasso Regression (L1 Regularization). You understand the mathematical formulation, why L1 produces sparsity, the geometric interpretation, solution algorithms, and the crucial feature selection aspects including stability and post-selection inference.