Two-Step Iterative Regression

• Two-Step Iterative Regression is a method that alternates between regression (parameter fitting) and auxiliary operations like projection, local smoothing, or trimming to enforce constraints.
• The framework achieves robust, accelerated convergence by decomposing optimization into subproblems that often admit closed-form solutions or efficient approximations.
• It is widely applied in constrained learning, matrix completion, mixture recovery, and neural network training, while requiring careful tuning of hyperparameters and initialization conditions.

Two-Step Iterative Regression refers to a family of algorithms based on alternating between two distinct update steps—typically (i) regression (parameter estimation) and (ii) auxiliary operations such as constraint projection, trimming, local smoothing, or structural adjustment. This two-step structure underlies a variety of regression methodologies for constrained learning, robust fitting, isotonic and nonparametric models, matrix completion, $\ell_p$-norm optimization, mixture recovery, and neural network training. Below, key paradigms are systematically described, focusing on mathematical formulation, algorithmic structure, theoretical properties, and empirical performance, with all statements referenced to the arXiv record.

1. General Formulation and Core Principles

At its most abstract, two-step iterative regression algorithms decompose the overall optimization or estimation problem into two alternating subproblems that can often be solved efficiently or admit closed-form solutions per step. A canonical constrained regression formulation is as follows (C. et al., 2022):

$$\min_{\theta} L(y, f(X, \theta)) \quad\text{subject to}\quad f(X, \theta) \in C$$

where $L$ is a loss function (e.g., MSE, MAE), $f(X, \theta)$ is a parametric model, and $C$ is a closed convex set encoding constraints on predictions. The two-step iterative framework alternates:

1. Constraint Enforcement (Projection/Adjustment): Given the current prediction $\hat y^{(i)}$, compute a feasible surrogate target $z^{(i)}$ by projection onto $C$, often using a blend or proximity operator:

$$z^{(i)} = \operatorname{argmin}_{z \in C} L(z, (1-\alpha) y + \alpha \hat y^{(i)})$$

2. Regression (Learning/Fitting): Update model parameters to fit $z^{(i)}$ using $L$, resulting in new predictions $\hat y^{(i+1)}$:

$$\hat y^{(i+1)} = f\big( X,\, \operatorname{argmin}_\theta L(z^{(i)}, f(X, \theta)) \big)$$

This composition $(P_{B, L} \circ P_{C, L} \circ h)$ is central to contraction-based convergence arguments (C. et al., 2022).

Similar alternating schemes appear in:

• Row subsampling for large linear systems (Li et al., 2012)
• Iterative local least-squares imputation (Liu et al., 2012)
• Iterative refinement for $\ell_p$-regression (Adil et al., 2019)
• Alternating isotone/antitone projections (Guyader et al., 2012)
• Back-and-forth neuron/parameter fitting in neural nets (Khadilkar, 2023)
• Subset trimming and refitting for robust estimation (Shen et al., 2019)
• Two-alternative quasi-estimation with model-based selection (Gordinsky, 2010)

2. Detailed Method Classes and Mathematical Algorithms

2.1 Constrained Alternating Regression

For regression under arbitrary constraints $C$, (C. et al., 2022) proposes:

$$\begin{align*} & \text{Step 1 (Target adjustment)}:\ &\quad z^{(i)} = \arg\min_{z\in C} L(z, h(\hat y^{(i)})), \quad h(\hat y^{(i)}) = (1-\alpha)y + \alpha \hat y^{(i)} \ & \text{Step 2 (Regression)}:\ &\quad \hat y^{(i+1)} = \arg\min_{\hat y \in B} L(\hat y, z^{(i)}) \end{align*}$$

where $B$ is the model range. For MSE loss, $K=1$ and the process is a Banach contraction if $\alpha < 1$; for L1 loss, $K=2$ with $\alpha < 1/4$. Convergence is guaranteed in the complete metric space $(B, \|\cdot\|)$ (C. et al., 2022).

2.2 Iterative Local Least Squares (ILLS) for Matrix Completion

(Liu et al., 2012) describes a two-step workflow for imputation in sparse rating matrices:

• Step 1: Probabilistic Spreading (ProbS) produces a dense rating matrix via topology-aware resource propagation.
• Step 2: Pointwise local least-squares estimation: for each missing entry, identify $K$ similar users via cosine similarity and reconstruct the missing value as a weighted linear fit over the local neighborhood.

Alternating these steps for a few iterations reduces NRMSE and boosts AUC, particularly in moderately dense data regimes.

2.3 Iterative Row Sampling for Tall Linear Systems

(Li et al., 2012) presents a two-step iterative row sampling approach to produce spectral sparsifiers for least-squares regression:

1. Reduction: Collapse blocks of rows using random Gaussian maps, yielding a much smaller "sketch" matrix.
2. Recovery: Estimate statistical leverage scores on the sketch, propagate them as upper bounds to the previous level, and use these for importance sampling to select rows.

Iterating this reduction/recovery structure logarithmically many times yields a core matrix $B$ such that solving $\min_x \|B x - b'\|_2$ approximates the full problem within $(1 \pm \epsilon)$.

2.4 Iterative Least Trimmed Squares (ILTS) for Mixture Models

ILTS in (Shen et al., 2019) alternates:

• Subset selection: Identify the $\tau n$ samples with lowest current residuals (trimming).
• Model refit: Compute least-squares fit on this subset, yielding a new parameter vector.

This process, under appropriate separation and corruption assumptions, achieves linear (geometric) convergence locally and, with global initialization, recovers all mixture components efficiently.

2.5 Iterative Refinement for $\ell_p$-Norm Regression

(Adil et al., 2019) introduces a two-step scheme for general $p$-norm regression:

1. Smooth quadratic subproblem: Approximate the $\ell_p$-norm by a smooth (Huber-type) surrogate, then solve a constrained quadratic approximation via a KKT system in each iteration.
2. Approximate solve with maintained inverse: Update the solution by solving the KKT system using fast data structures for incremental inverse maintenance.

Convergence requires only $O_p(\log n)$ iterations; total cost is dominated by the per-iteration solve.

2.6 Alternating Projections in Isotonic/Antitonic Regression

(Guyader et al., 2012) formalizes iterative isotone regression (IIR):

• Alternate projection onto the cone of non-decreasing functions (isotonic regression) and the cone of non-increasing functions (antitonic regression), iteratively refining a Jordan decomposition.

This process is equivalent to Von Neumann’s algorithm for projections onto convex sets, converging to the data vector itself, interpolating the noise unless regularized by early stopping.

2.7 Two-Stage Quasi-Estimation

(Gordinsky, 2010) defines a two-stage procedure based on constructing two alternative estimates by adjusting the OLS solution with scaled residuals in the maximal-risk direction. A single auxiliary information bit (e.g., sign constraint, prior, external measurement) is used in the second stage to choose the lower-risk estimate. This yields significant risk reduction compared to OLS, is robust to distributional misspecification, and requires minimal extra information.

3. Convergence Guarantees and Theoretical Properties

Contraction, monotonicity, or geometric convergence is analyzable for many two-step procedures:

• Constrained Alternating Projections: With Lipschitz continuity and appropriate choice of $\alpha$, the iterative map is a Banach contraction. For $\ell_2$ loss ($K=1$), any $\alpha<1$ suffices for guaranteed convergence to the unique fixed point (C. et al., 2022).
• ILTS: Local linear convergence with contraction factor determined by data regularity, separation, and corruption fraction (Shen et al., 2019). Proper initialization suffices for global identification in the mixture setting.
• Iterative Refinement for $\ell_p$: Each step reduces the optimality gap by a fixed proportion; attaining a $1/\operatorname{poly}(n)$ solution in $O(\log n)$ steps (Adil et al., 2019).
• Geometric Isotonic Regression: Alternating projections between convex cones converge to the intersection (or, with additive cones, interpolate the data).
• ILLS and Matrix Sparsification: Empirically converge within a small number of iterations (typically 4–6), with theoretical underpinning from random-projection and leverage score analysis (Liu et al., 2012, Li et al., 2012).

4. Representative Algorithms and Pseudocode

Several two-step iterative regression methods are formulated in explicit, modular pseudocode. For example, the constrained regression framework from (C. et al., 2022):

Input: y ∈ ℝⁿ, model f(·, θ), loss L, constraint set C, α ∈ [0, 1/K²), β ≥ 0, max-iterations N Initialize: ŷ¹ ← argmin_θ L(y, f(X, θ)) for i = 1 to N−1 do if ŷⁱ ∉ C: h ← (1 − α)·y + α·ŷⁱ zⁱ ← argmin_{z ∈ C} L(z, h) else: zⁱ ← argmin_{z∈C} L(z, y) subject to L(z, ŷⁱ) ≤ β ŷ^{i+1} ← argmin_θ L(zⁱ, f(X, θ)) Output: ŷ^N

Other procedures, such as ILLS for neural nets (Khadilkar, 2023), IIR for isotonic regression (Guyader et al., 2012), and iterative local least squares (Liu et al., 2012), present similarly structured alternating-step schemes with minor adjustments for the specific context (e.g., local neighborhoods, diagonal preconditioning, regularization terms, or data splitting).

5. Empirical Performance and Application Domains

A range of empirical evidence supports the practical merits of two-step iterative regression:

Algorithm	Application Domain	Core Metrics	Main Outcomes
(C. et al., 2022)	Constrained regression (fairness, monotonicity, structure)	R², constraint compliance (DIDI), std. dev.	Achieves better fairness-accuracy trade-offs and more stable convergence vs. baselines
(Liu et al., 2012)	Recommender systems	NRMSE, AUC, Precision, Recall	Converges in 4–6 iterations, sharp NRMSE drop, improved AUC/Recall
(Khadilkar, 2023)	Neural network regression	MSE, epochs to convergence	Outperforms Adam (by an order of magnitude in epochs), remains stable at higher learning rates
(Shen et al., 2019)	Mixed linear regression (corruptions, mixtures)	Recovery error	Achieves linear convergence, near-optimal sample complexity
(Gordinsky, 2010)	Robust/auxiliary info regression	Quadratic risk	Reduces risk by 60%, robust across error distributions

This table summarizes the main performance indicators and experimental observations for the primary two-step iterative regression frameworks.

6. Structural Generality and Connections

The two-step alternating structure, with each step designed either as a projection/proximal map or as an exact/approximate minimization over a restricted domain, is closely linked to:

• Alternating Projection Methods: As in Dykstra’s and Von Neumann-type algorithms for convex sets (Guyader et al., 2012).
• Majorization-Minimization (MM) and EM Algorithms: When the two steps can be understood as alternately improving tangent majorizers or optimizing over partitioned parameter spaces.
• Randomized Sketching and Sampling: Row sampling/re-sampling for randomized linear algebra, matrix sketching, and spectral sparsification (Li et al., 2012).
• Gradient-based and Newton-type Optimization: Iterative refinement with approximate system solvers and variable preconditioning (as in fast $\ell_p$-norm regression) (Adil et al., 2019).
• Block Coordinate Descent and Backfitting: Seen in additive and nonparametric modeling (Guyader et al., 2012).

These methods illustrate the breadth of applicability and the unifying role of two-step iterative regression across modern statistical and computational learning.

7. Limitations, Practical Issues, and Extensions

Two-step iterative regression methods generally require:

• Closed or efficiently computable projection/sub-step operations (often relying on convexity, linearity, or local quadraticity)
• Mild regularity, separation, or initialization conditions for global convergence or component identification (mixture estimation)
• Explicit or implicit hyperparameter tuning (e.g., choice of $\alpha$, neighborhood size $K$, number of trimmed samples, number of iterations)
• Contextualization of stopping rules to avoid overfitting or instability (especially in boosted or isotonic procedures (Guyader et al., 2012))
• For some frameworks (e.g., quasi-estimation (Gordinsky, 2010)), auxiliary information at the second stage is necessary for optimality, which may not always be available.

Extensions and open directions involve multi-step generalizations, stochastic and online variants, nonconvex constraints, high-dimensional regime adaptations, and integration with deep or structured prediction models.

---

References:

• "Iterative Supervised Learning for Regression with Constraints" (C. et al., 2022)
• "A two-step Recommendation Algorithm via Iterative Local Least Squares" (Liu et al., 2012)
• "Iterative Row Sampling" (Li et al., 2012)
• "Using Linear Regression for Iteratively Training Neural Networks" (Khadilkar, 2023)
• "Iterative Least Trimmed Squares for Mixed Linear Regression" (Shen et al., 2019)
• "Quasi-estimation as a Basis for Two-stage Solving of Regression Problem" (Gordinsky, 2010)
• "A Geometrical Approach to Iterative Isotone Regression" (Guyader et al., 2012)
• "Iterative Refinement for $\ell_p$-norm Regression" (Adil et al., 2019)