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High-dimensional regression with noisy and missing data: Provable guarantees with nonconvexity
Po-Ling Loh, Martin J. Wainwright
TL;DR
The paper asks how to perform high-dimensional sparse linear regression when covariates are noisy, missing, or dependent, settings that make standard formulations and guarantees inadequate. It constructs corrected nonconvex estimators, analyzes their statistical error, and proves projected gradient descent reaches near-global solutions. The guarantees provide high-probability error control with classical scaling and geometric computational convergence under suitable conditions.
Problem
Standard prediction assumes fully observed, noiseless, independently sampled covariates, whereas many applications involve noisy, missing, or dependent covariates and nonconvex optimization.
Method
The paper uses corrected ℓ1-constrained or Lagrangian estimators for sparse regression and analyzes projected gradient descent on the resulting nonconvex programs.
Results
The estimators achieve statistical error with the same scaling as classical fully observed, independently sampled cases, while projected gradient descent converges geometrically to near-global minimizers.
Takeaways & Limitations
Noisy, missing, and dependent covariates can be handled with estimators that retain high-probability statistical guarantees and polynomial-time optimization under the paper’s conditions.
Takeaways & Limitations
The constrained estimator’s radius R = ∥β∗∥1 is restrictive because β∗ is unknown; the Lagrangian formulation is more appealing in this respect.
Abstract
from arXiv · showhide
Although the standard formulations of prediction problems involve fully-observed and noiseless data drawn in an i.i.d. manner, many applications involve noisy and/or missing data, possibly involving dependence, as well. We study these issues in the context of high-dimensional sparse linear regression, and propose novel estimators for the cases of noisy, missing and/or dependent data. Many standard approaches to noisy or missing data, such as those using the EM algorithm, lead to optimization problems that are inherently nonconvex, and it is difficult to establish theoretical guarantees on practical algorithms. While our approach also involves optimizing nonconvex programs, we are able to both analyze the statistical error associated with any global optimum, and more surprisingly, to prove that a simple algorithm based on projected gradient descent will converge in polynomial time to a small neighborhood of the set of all global minimizers. On the statistical side, we provide nonasymptotic bounds that hold with high probability for the cases of noisy, missing and/or dependent data. On the computational side, we prove that under the same types of conditions required for statistical consistency, the projected gradient descent algorithm is guaranteed to converge at a geometric rate to a near-global minimizer. We illustrate these theoretical predictions with simulations, showing close agreement with the predicted scalings.
1. Introduction.
The paper addresses sparse linear regression when covariates are noisy, missing, or dependent, settings where standard methods face nonconvex optimization and limited guarantees. It develops estimators with statistical guarantees and shows projected gradient descent can efficiently approach global optima.
- Motivation: Many applications violate the usual assumptions of fully observed, noiseless, independently sampled covariates through missingness, measurement noise, or dependence.Examples include political voting, surveys, and sensor networks.
- Motivation: Likelihood-based approaches such as EM can produce nonconvex optimization problems whose local-minimum guarantees do not ensure proximity to a global optimum.This makes practical theoretical guarantees difficult to establish.
- Contributions: The proposed estimators target high-dimensional sparse linear regression with noisy, missing, and dependent predictors while analyzing both statistical and optimization error.The framework also applies to sparse Gaussian graphical model selection with missing data.
- Contributions: Projected gradient descent converges in polynomial time to a point sufficiently close to any global optimum, despite the underlying nonconvexity.The proximity is described as being on the scale of the statistical error.
- Contributions: The statistical error has the same scaling as minimax rates for classical fully observed, independently sampled covariates, with high-probability bounds also covering dependent data.Simulations show close agreement with the predicted scalings.
2. Background and problem setup.
The paper formulates sparse regression from corrupted covariates using corrected M-estimators that may be nonconvex, then analyzes restricted-eigenvalue conditions and projected gradient methods. The resulting procedures retain statistical guarantees and converge efficiently under suitable conditions.
- Observation model: The observation model allows additive noise, missing entries, and multiplicative noise, including missingness as a Bernoulli special case.The analysis considers both i.i.d. covariates and dependent covariates generated by a stationary VAR process.
- Observation model: The high-dimensional framework permits p to exceed n, so sparsity is imposed by requiring β∗ to have at most k nonzero entries.The sparsity level k may grow with p and n.
- M-estimators: The estimators plug unbiased surrogate estimates of Σx and Σxβ∗ into an ℓ1-constrained quadratic program adapted to corrupted covariates.The ideal convex program motivates this plug-in construction even though the needed population quantities are unknown.
- M-estimators: Noise and missingness can make the surrogate matrix indefinite, causing the associated quadratic objective to be nonconvex and, for the Lagrangian form, potentially unbounded below.This behavior occurs because low-rank sample matrices are combined with subtracted noise or missingness corrections when n ≪ p.
- Statistical guarantees: Restricted-eigenvalue analysis establishes low-error behavior for sparse solutions and shows that corrected matrices can satisfy the needed lower-RE condition with high probability.The paper develops similar scaling to classical Lasso analyses for its corrected choices of bΓ.
- Optimization: Projected gradient descent is guaranteed to approach global-optimum precision under conditions that hold for the considered statistical models, despite negative eigenvalues in the surrogate matrix.The iterates converge extremely close to any global optimum in both ℓ1- and ℓ2-norms.
3. Main results and consequences.
The paper establishes statistical guarantees for global optima under surrogate deviation and lower-RE conditions, then shows projected and composite gradient methods efficiently approach those optima despite nonconvexity. Simulations support the predicted consistency and sample-size scaling, while the constrained estimator requires an unknown radius.
- Statistical error: Theorem 1 bounds the ℓ2- and ℓ1-statistical errors of global optima when the surrogates satisfy deviation and lower-RE conditions.The result applies to both the Lagrangian and constrained programs, with their respective parameter conditions.
- Statistical error: Under the stated conditions, all global optima lie within an ℓ2-ball whose radius shrinks to zero when k log p/n = o(1).Theorem 1 also prevents widely separated global optima under this scaling.
- Statistical error: For additive-noise simulations, the ℓ2-error decreases to zero with increasing sample size, demonstrating consistency across p ∈ {128,256,512}.The observed curves shift right as p increases and align after rescaling sample size by n/(k log p).
- Statistical error: Rescaling the sample size by n/(k log p) produces the predicted stacking behavior across dimensions in the additive-noise simulations.The rescaled curves roughly align for different values of p.
- Statistical error: The constrained estimator's radius R = ∥β∗∥1 is restrictive because β∗ is unknown, whereas the Lagrangian estimator only requires b0 > ∥β∗∥2.The paper presents the Lagrangian formulation as more appealing under this scope boundary.
- Optimization error: Theorem 2 bounds optimization error for projected and composite gradient descent and guarantees geometric convergence toward a small neighborhood of global optima.The contraction coefficient γ ∈ (0,1) is independent of (n,p,k), and the remaining error is controlled by statistical-error terms.
3.2. Some consequences.
The paper derives high-probability statistical and optimization guarantees for sparse regression with additive noise, missingness, and dependent covariates. These guarantees extend across i.i.d. and VAR settings, including cases where corruption parameters are estimated.
- General guarantees: Under n ≿ k log p, the corollaries provide high-probability guarantees for noisy, missing, and dependent-data settings.The probability is at least 1 − c1 exp(−c2 log p), with universal positive constants c1 and c2.
- Additive noise: For additive noise, the statistical error depends on an inverse signal-to-noise prefactor that grows with σw/σx and σε/σx.When covariate noise vanishes, the bound agrees with known results for uncorrupted covariates.
- Estimated corruption parameters: Unknown noise covariance and unknown missingness parameters can be estimated from auxiliary or observed-data information without changing the stated corollary guarantees.For dependent data, the paper says these extensions follow by the same arguments as in the i.i.d. case.
- Missing data: Missing-data guarantees assume ρmax < 1 and produce bounds whose effective variance depends on the missingness fraction and the conditioning of Σx.The same corollary remains valid when missingness parameters are estimated empirically.
- Dependent data: For Gaussian VAR covariates, the same theorem pair applies to additive-noise and missing-data cases despite dependence across rows.The scaling and error form remain similar across Corollaries 2–4, with different effective variances.
3.3. Application to graphical model inverse covariance estimation.
The paper extends sparse inverse-covariance estimation to corrupted observations by combining regression-based precision-matrix recovery with symmetry projection. The resulting procedure has high-probability error guarantees under sparse columns and lower-RE conditions.
- Matrix assembly: The estimated regression vectors and scalar coefficients are combined into an initial precision estimate, then projected onto symmetric matrices by an ℓ1 minimization.The final minimization is a linear program and can be solved with standard methods.
- Regression construction: The method estimates each precision-matrix column through a regression of Zj on Z−j, using corruption-specific surrogate estimators.The approach covers additive-noise and missing-data observations, including settings with i.i.d. or VAR covariates.
- Guarantees: When the precision matrix has k-sparse columns and finite nonzero condition number, Corollary 5 gives an error bound under deviation and uniform lower-RE conditions.The result parallels Theorem 1 and applies the preceding corollaries to the relevant data scenarios.
- Guarantees: The resulting bound holds with probability at least 1 − c1 exp(−c2 log p) when n ≿ k log p, with scenario-specific values of ϕ and α1.The paper states that the needed deviation and lower-RE conditions hold across the considered settings.
4. Simulations.
Simulations examine sparse regression and inverse-covariance estimation under additive noise, missingness, and dependence. Across these settings, rescaling the sample size or error as predicted by theory produces aligned or approximately constant curves.
- Sparse regression: For missing-data regression, rescaled error curves are roughly constant, supporting the predicted dependence on sample size, sparsity, dimension, and missingness.Each point in the reported experiment averages 200 trials.
- Noise scaling: For additive-noise regression, the rescaled error versus σw is roughly constant, agreeing with the predicted form of ϕ(Q,σε).The experiment uses Σx = I, σε = 0.5, p = 256, and k ≈ log p.
- Graphical models: For graphical models, chain-structured inverse-covariance errors align when plotted against rescaled sample size, with qualitatively similar results for star and Erdős–Renyi graphs.The experiments include additive-noise and missing-data observations, and each point averages 100 trials.
5. Proofs.
The proofs establish statistical error bounds from a basic inequality and lower-RE curvature, while the paper directs technical corollary proofs to supplementary material. The proof strategy bounds the stochastic deviation term before applying curvature.
- Proof scope: The paper proves its two main theorems in this section and places the more technical corollary proofs in the supplementary appendix.The corollaries require additional verification that the deterministic theorem conditions hold in specific statistical models.
- Statistical error proof: The loss framework covers both unregularized and ℓ1-regularized estimators, with optimality yielding the basic inequality L(bβ) ≤ L(β∗).The regularized argument uses k-sparsity to control ∥β∗∥1.
- Statistical error proof: The proof defines the error vector bν := bβ − β∗, upper-bounds the inequality’s right-hand side, and then uses the lower-RE condition to control bν.Hölder’s inequality bounds the key inner product by ∥bν∥1∥bγ − bΓβ∗∥∞.
- Statistical error proof: Combining the lower bound from the lower-RE condition with the earlier upper bound yields the stated error claim.The argument proceeds through the basic inequality and an intermediate inequality for the error vector.
Upper bound on right-hand side.
The derivation upper-bounds the right-hand side of inequality (5.1) by combining sparsity, triangle-inequality, feasibility, and regularization arguments.
- The argument combines these upper bounds with intermediate inequalities to conclude the desired right-hand-side control.
- Sparsity of β∗ and the triangle inequality are used to upper-bound the right-hand side of inequality (5.1).
- The resulting bound holds for any nonnegative choice of λn.
- For the constrained estimator, feasibility yields an ℓ1 relationship between the error on and off the support of β∗.The proof derives ∥bνSc∥1 ≤ ∥bνS∥1 and then ∥bν∥1 ≤ 2∥bνS∥1.
- For the regularized estimator, the selected λn controls term (5.5) by at most 3λn.
Lower bound on left-hand side.
The derivation establishes lower and contraction bounds for projected-gradient and Lagrangian analyses using restricted-eigenvalue conditions and sparsity-related assumptions.
- The derivation concludes the stated inequalities by applying the restricted-eigenvalue conditions and rearranging the resulting bounds.
- The ℓ2-error analysis applies restricted strong convexity and smoothness arguments through lower- and upper-RE conditions.The proof adapts a theorem originally stated for convex loss functions.
- The proof computes the tolerance parameter ε2 and contraction coefficient needed to invoke the convergence results.
- The assumption n ≿ k log p ensures the projected-gradient contraction coefficient lies in (0,1).
- The constrained-program analysis uses feasibility, optimality, and the triangle inequality to control the ℓ1 error of iterates.
- For the Lagrangian formulation, the proof restricts the comparison subspace to vectors supported on β∗'s support.
- Under n ≿ k log p, the associated geometric quantities remain controlled and the compound tolerance parameter satisfies ε2 = O(k log p.
6. Discussion.
The discussion summarizes guarantees for corrupted sparse regression and identifies broader corruption, dependence, distributional, loss-function, and applicability settings as future directions.
- Projected gradient descent converges to within statistical precision of the optimum despite the generally nonconvex objective.The formulation uses an ℓ1-constrained minimization problem for sparse linear regression with additive noise or missing data.
- High-probability ℓ1- and ℓ2-error bounds hold for i.i.d. sub-Gaussian data and Gaussian vector autoregressive data.
- Sparse inverse covariance estimation with corruptions achieves spectral norm rates of the same order as existing rates for uncorrupted, i.i.d. data.
- Future work includes more general covariate dependencies and corruption types, including broader multiplicative-noise models.
- Open settings include unknown noise covariance without replicates, non-sub-Gaussian data, model mismatch, harder loss corrections, and broader nonconvex applications.
Supplementary material for: High-dimensional regression with noisy and missing data: Provable guarantees with nonconvexity
Technical details for the remaining proofs are provided in supplementary material because of space constraints.
- Technical details of the remaining proofs are relegated to the supplement because of space constraints.