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Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval, Matrix Completion, and Blind Deconvolution

Cong Ma, Kaizheng Wang, Yuejie Chi, Yuxin Chen

arXiv:1711.10467v3cs.LGcs.ITmath.OCmath.STstat.ML

TL;DR

The paper addresses limited guarantees for vanilla gradient descent on nonconvex statistical estimation problems, where prior approaches often use conservative step sizes or explicit regularization. It analyzes how gradient descent implicitly preserves incoherence through a leave-one-out trajectory argument, and proves favorable guarantees for phase retrieval, matrix completion, and blind deconvolution. The results include nearly linear convergence and, numerically, 10^-5 relative accuracy in around 200 iterations.

  • Problem

    Nonconvex empirical-loss estimation often requires explicit regularization, while vanilla gradient descent has conservative or missing guarantees, including no cited theory for matrix completion and blind deconvolution.

  • Method

    The paper analyzes gradient-descent trajectories using statistical independence, generic optimization theory, and a leave-one-out perturbation argument to control incoherence.

  • Results

    Vanilla gradient descent achieves near-optimal statistical and computational guarantees with nearly linear convergence for phase retrieval, matrix completion, and blind deconvolution.

  • Takeaways & Limitations

    Gradient descent can implicitly regularize its search directions by keeping iterates incoherent with the measurement mechanism, explaining its favorable performance without explicit regularization.

  • Takeaways & Limitations

    The analysis focuses on independent designs and leaves generalization to more structured sampling patterns and Fourier designs unclear.

Abstract

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Recent years have seen a flurry of activities in designing provably efficient nonconvex procedures for solving statistical estimation problems. Due to the highly nonconvex nature of the empirical loss, state-of-the-art procedures often require proper regularization (e.g. trimming, regularized cost, projection) in order to guarantee fast convergence. For vanilla procedures such as gradient descent, however, prior theory either recommends highly conservative learning rates to avoid overshooting, or completely lacks performance guarantees. This paper uncovers a striking phenomenon in nonconvex optimization: even in the absence of explicit regularization, gradient descent enforces proper regularization implicitly under various statistical models. In fact, gradient descent follows a trajectory staying within a basin that enjoys nice geometry, consisting of points incoherent with the sampling mechanism. This "implicit regularization" feature allows gradient descent to proceed in a far more aggressive fashion without overshooting, which in turn results in substantial computational savings. Focusing on three fundamental statistical estimation problems, i.e. phase retrieval, low-rank matrix completion, and blind deconvolution, we establish that gradient descent achieves near-optimal statistical and computational guarantees without explicit regularization. In particular, by marrying statistical modeling with generic optimization theory, we develop a general recipe for analyzing the trajectories of iterative algorithms via a leave-one-out perturbation argument. As a byproduct, for noisy matrix completion, we demonstrate that gradient descent achieves near-optimal error control --- measured entrywise and by the spectral norm --- which might be of independent interest.

1 Introduction

Nonconvex statistical estimation often relies on explicit regularization because vanilla gradient descent has limited theory, but this paper shows that it can implicitly preserve incoherence and achieve favorable convergence across three problems.

  • 1 Introduction: Phase retrieval, low-rank matrix completion, and blind deconvolution are representative nonlinear estimation problems whose empirical-loss formulations are nonconvex.The paper frames recovery through quadratic, incomplete-entry, and bilinear measurements.
  • 1 Introduction: Explicit regularization, projection, trimming, and truncation have been used to stabilize gradient directions and enforce incoherence in nonconvex estimation.These procedures improve convergence guarantees but introduce additional model-dependent parameters or operations.
  • 1 Introduction: Vanilla gradient descent was comparatively poorly understood, with no known theoretical guarantee for matrix completion and blind deconvolution in the cited prior theory.The paper asks whether regularization-free procedures are inherently suboptimal.
  • 1 Introduction: Numerically, vanilla gradient descent reaches 10^-5 relative accuracy in around 200 iterations, while phase retrieval uses a fixed step size across problem sizes.The paper also reports entrywise empirical-risk and matrix-completion estimation-error control.
  • 1 Introduction: Vanilla gradient descent automatically keeps iterates incoherent with the measurement mechanism, thereby implicitly regularizing its search directions.This preserves the trajectory within a region with favorable geometry.
  • 1 Introduction: Across the three problems, the theory establishes nearly linear convergence with near-optimal statistical and computational guarantees under random designs and spectral initialization.The informal results cover phase retrieval, matrix completion, and blind deconvolution.

2 Implicit regularization – a case study

The paper uses phase retrieval to show why generic gradient-descent theory is conservative and how an incoherent region provides better local geometry. A leave-one-out argument explains why vanilla gradient descent remains in this region, enabling larger step sizes and faster convergence.

  • 2.1 Gradient descent theory revisited: Generic gradient-descent theory requires local strong convexity and smoothness, with iteration complexity governed by the condition number β/α.
  • 2.2 Local geometry for solving random quadratic systems: For phase retrieval, population geometry is favorable: f is 1-strongly convex and 10-smooth near x⋆, so gradient descent reaches the global minimum within logarithmic iterations.
  • 2.2 Local geometry for solving random quadratic systems: In the finite-sample regime m ≍ n log n, local strong convexity persists, but smoothness can grow substantially, producing conservative step sizes and high iteration complexity.
  • 2.3 Which region enjoys nicer geometry?: The RIC intersects a local ℓ2 ball with an incoherence polytope, and within it the Hessian is nearly well-conditioned with condition number O(log n).
  • 2.3 Which region enjoys nicer geometry?: Within the RIC, gradient descent can use step size η_t = O(1/log n), while leaving the region removes the benign geometry and risks overshooting.
  • 2.4 Implicit regularization: The leave-one-out analysis shows that iterates remain incoherent with high probability without explicit enforcement, a phenomenon termed implicit regularization.

3 Main results

The paper shows that unregularized gradient descent implicitly maintains incoherence across phase retrieval, matrix completion, and blind deconvolution, enabling near-linear convergence with aggressive step sizes and near-optimal guarantees.

  • Across all three problems, the iterates remain incoherent with the sampling mechanism without explicit regularization.This implicit regularization keeps the trajectory in regions with favorable geometry.
  • Phase retrieval: Phase retrieval permits η ≍1/log n and reaches ϵ-accuracy within O(log n log(1/ϵ)) iterations.This improves on the previously admissible η ≲1/n step size.
  • Phase retrieval: With m ≍n log^2 n phase-retrieval measurements, a constant step size η ≍1 is feasible, yielding iteration complexity log(1/ϵ).
  • Low-rank matrix completion: Matrix completion achieves linear convergence with a constant step size across Frobenius, ℓ2/ℓ∞, and spectral norms.The result is described as near-optimal in statistical accuracy and computational complexity.
  • Low-rank matrix completion: For matrix completion, rank r=O(1) succeeds at near-optimal sample complexity up to logarithmic factors, while Euclidean error matches a minimax lower bound.The analysis also yields entrywise error control for noisy matrix completion.
  • Blind deconvolution: Blind deconvolution admits η ≍1 and reaches ϵ-accuracy within O(log(1/ϵ)) iterations, versus O(m log(1/ϵ)) under prior theory.The theorem also establishes near-optimal sample complexity up to logarithmic factors and removes the need to regularize scaling or incoherence.

4 Related work

Prior work contrasts expensive convex relaxations with computationally attractive nonconvex methods, while motivating this paper’s focus on unregularized gradient descent and leave-one-out analysis.

  • Convex relaxations provide appealing statistical guarantees but can be prohibitively expensive for large-scale datasets.
  • Nonconvex methods have been studied for phase retrieval, matrix sensing and completion, blind deconvolution, dictionary learning, tensor decomposition, and related problems.
  • Phase retrieval: Phase retrieval has been addressed through lifted semidefinite programs such as PhaseLift and natural-parameter convex programs such as PhaseMax.
  • Matrix completion: Matrix completion combines nuclear-norm methods with nonconvex approaches under incoherence conditions and near-optimal sample complexity.
  • Blind deconvolution: Blind deconvolution has used lifted convex relaxation, regularized gradient descent, partially unregularized Riemannian descent, and a sign-dependent linear program.
  • The paper’s analysis builds on leave-one-out perturbation arguments previously used in several high-dimensional random-design problems.

5 A general recipe for trajectory analysis

The paper analyzes gradient-descent trajectories by identifying a region with favorable local geometry and proving, via leave-one-out sequences, that iterates remain incoherent within it.

  • General model: The general model represents phase retrieval, matrix completion, and blind deconvolution as bilinear or quadratic measurements of unknown factors.
  • General model: Vanilla gradient descent minimizes the empirical loss from a spectral initialization, using a learning rate and Wirtinger derivatives for complex-valued problems.
  • Step 1: local geometry: The RIC is characterized by restricted strong convexity and smoothness, with controlled error and incoherence relative to the sampling mechanism.
  • Step 1: local geometry: If iterates stay in the RIC, optimization theory gives linear convergence with contraction rate 1 − O(α/β) for a suitable step size.
  • Step 2: leave-one-out sequences: For each sample, a leave-one-out sequence removes that sample from the loss and initialization, creating iterates statistically independent of its sampling vectors.
  • Step 3: establishing incoherence: Induction combines proximity between true and leave-one-out iterates with concentration for the independent auxiliary sequence to preserve incoherence.

6 Analysis for phase retrieval

The phase-retrieval analysis establishes favorable local Hessian geometry, contraction under an appropriate step size, and trajectory preservation through leave-one-out induction.

  • 6.1.1 Local geometry: With m ≥ c0n log n, the phase-retrieval Hessian is positive definite and nearly well-conditioned near the truth under an incoherence condition.
  • 6.1.2 Error contraction: The local geometry yields ℓ2 error contraction whenever the iterate satisfies the RIC conditions and the step size obeys the stated O(1/log n) bound.
  • 6.1.2 Error contraction: The proof reduces Theorem 1 to showing that the trajectory remains in the RIC through T0 = n iterations.
  • 6.3 Step 3: Leave-one-out iterates remove one measurement, remain independent of its sensing vector, and enable control of the original trajectory’s incoherence.
  • 6.3 Step 3: Induction controls proximity, auxiliary-sequence incoherence, and their combination, then uses a union bound to establish the claims for all t ≤ T0 with high probability.
  • 6.4 Initialization: Spectral initialization supplies the base case through Davis–Kahan-based bounds under m > c0n log n.

7 Analysis for matrix completion

The matrix-completion analysis proves Frobenius- and spectral-norm contraction within an incoherent region, using leave-one-out trajectories to preserve the needed conditions.

  • 7.1 Local geometry: The matrix-completion Hessian is well-conditioned near X⋆ under ℓ2/ℓ∞ incoherence and along directions close to the truth in spectral norm.
  • 7.1.2 Error contraction: Under the stated sample complexity and noise condition, gradient descent contracts the Frobenius-norm error for an admissible step size.
  • 7.1.2 Error contraction: When all three trajectory conditions hold, the analysis obtains a stronger contraction guarantee in spectral norm.
  • 7.2 Leave-one-out analysis: Leave-one-out sequences replace the lth row and column with population means, producing an unbiased gradient surrogate important for high estimation accuracy.
  • 7.2 Leave-one-out analysis: The induction shows that true and auxiliary iterates remain close while the auxiliary sequence stays incoherent through statistical independence.

8 Analysis for blind deconvolution

The blind deconvolution analysis establishes local geometry, linear error contraction, and trajectory control using incoherence and leave-one-out perturbation arguments.

  • Local formulation: The analysis uses Wirtinger calculus to formulate gradients, Hessians, alignment, and updates for complex-valued blind deconvolution.The iterates are represented by aligned factors, with an alignment parameter tracking global scaling.
  • Local geometry: The region of incoherence and contraction combines proximity to the truth with incoherence relative to both sensing-vector families.Restricted strong convexity and smoothness hold in this region and support contraction of the aligned error.
  • Local geometry: Lemma 14 provides restricted strong convexity and smoothness for blind deconvolution, characterized as the paper’s first Hessian-geometry result for this problem.The Hessian is rank-deficient globally, so the analysis restricts attention to suitable directions and includes separate step-size scaling factors.
  • Error contraction: For sufficiently large sample size and sufficiently small constant step size, the analysis establishes contraction and propagates the relevant trajectory conditions.The contraction factor is ρ := 1 −η/16, while the induction argument controls the original and leave-one-out sequences.
  • Trajectory control: The alignment parameter converges linearly toward a constant with magnitude close to 1 when the initial factors have similar norms and are near the truth.Because alignment determines global scaling, this also gives rapid convergence of the factor norms.
  • Trajectory control: Leave-one-out sequences remove one measurement pair, making the auxiliary iterates independent of the omitted sensing vectors for incoherence analysis.The proof inductively controls proximity between original and auxiliary iterates, incoherence, and related hypotheses with high probability.

9 Discussions

The discussion presents implicit regularization as a broader phenomenon while identifying unresolved limits involving sample complexity, design generality, algorithm breadth, and unified theory.

  • Main implications: Vanilla gradient descent can behave as though properly regularized, achieving statistical and computational efficiency for three problems under random designs and spectral initialization.The paper frames incoherence and leave-one-out stability as ingredients for future general theories.
  • Open limitations: Matrix completion requires sub-optimal sample complexity when the underlying rank r is large.The stated complexity is slightly higher than that obtained for regularized gradient descent, despite dimension-free iteration complexity.
  • Open limitations: The leave-one-out analysis currently focuses on independent designs, leaving more general sampling patterns and structured Fourier designs unresolved.Fourier designs already require more delicate treatment than purely Gaussian designs in blind deconvolution.
  • Open limitations: The study analyzes vanilla gradient descent with the natural squared loss rather than comparing the broader range of iterative methods and loss functions.The discussion names alternating minimization, block coordinate descent, sub-gradient, and prox-linear methods as directions for comparison.
  • Open limitations: A unified theory explaining which structural properties yield near-optimal guarantees has not yet been developed.The current recipe remains problem-specific and requires delicate, lengthy analyses for each problem.

A.1 Proof of Lemma 1

The proof establishes local smoothness and strong convexity by decomposing Hessian terms, bounding them with concentration and incoherence arguments, and applying these bounds along gradient-descent paths.

  • Hessian bounds: The Hessian is decomposed into multiple components whose norms are bounded separately using concentration, truncation, and incoherence conditions.The resulting upper bound is proportional to log n.
  • Hessian bounds: The analysis obtains a positive-definite, almost well-conditioned Hessian along line segments between an iterate and the truth.The displayed bounds give lower curvature 1/2 and upper curvature [5C2 (10 + C2) log n].
  • Gradient-descent control: Choosing η ≤1/[5C2 (10 + C2) log n] controls the Hessian-scaled update and enables the gradient-descent contraction argument.The proof then recursively propagates proximity and incoherence conditions across iterations.
  • Initialization: Spectral initialization is controlled through concentration around E[Y] = I + 2x⋆x⋆⊤ and its eigengap between eigenvalues 3 and 1.Weyl’s inequality and Davis–Kahan bounds relate the leading eigenvector to the target.
  • Matrix-completion bounds: For matrix completion, injectivity on the tangent space, sampling concentration, and incoherence yield uniform bounds over admissible factor matrices.The resulting control does not depend on statistical independence between the factor matrix and the sampling set.

B.3.2 Proof of Lemma 23

The proof controls leave-one-out error terms by partitioning indices according to interaction size and combining concentration, incoherence, and sample-complexity conditions.

  • Index partitioning: The proof bounds leave-one-out quantities by separating indices according to the magnitude of ∥G_l(Δ_t)∥.Only sufficiently large index groups require the refined case analysis; smaller groups are controlled by an earlier bound.
  • Index partitioning: The cardinalities of the index groups decay exponentially with the group index, leaving only O(log n) nonempty groups.Summing the groupwise bounds yields the final estimate.
  • Error control: Bounds for the matrix factors combine incoherence of X⋆, approximation hypotheses, noise conditions, and sample-size requirements.These estimates control the componentwise, Frobenius, and related errors needed for the lemma.
  • Error control: The resulting estimates are assembled through triangle inequalities and auxiliary bounds to establish the targeted leave-one-out inequalities.The proof repeatedly invokes the hypotheses for the current iteration and conditions such as n ≫κµ log n.

B.5 Proof of Lemma 11

The proof decomposes the target bound into several terms and controls them using concentration, incoherence, sample-size, and noise assumptions. Combining these estimates yields the advertised result under a suitable step size and sample complexity.

  • Term decomposition: The proof controls four terms separately before combining their bounds through the triangle inequality.The terms are labeled B^(l)_1 through B^(l)_4.
  • Assumptions: The required bounds hold under sample-size conditions involving µ, r, n, p, and logarithmic factors, together with the noise condition.The proof repeatedly invokes conditions such as n^2p ≫ µ^2r^2n log n and n^2p > Cµ^2r^2n log^2 n.
  • Conclusion: The final contraction estimate uses a sufficiently large absolute constant and a step size bounded by 1/(2σmax).The argument also uses the relation η ≤ 1/(2σmin) stated in the proof.
  • Concentration bounds: Matrix Bernstein and independence arguments control contributions involving the lth row and column of the sampling mask.The leave-one-out iterate is independent of the corresponding observation indicators and noise variables.

B.7 Proof of Lemma 13

The proof establishes initialization bounds by combining perturbation estimates, spectral perturbation arguments, and orthogonal alignment. These bounds control Frobenius, spectral, and component-wise errors under incoherence, sample-complexity, noise, and bounded-condition-number assumptions.

  • Alignment: The orthogonal Procrustes problem supplies alignment matrices for comparing estimated and target factors.The minimizer exists and is used to define the rotations appearing in the error bounds.
  • Spectral perturbation: Spectral perturbation is controlled using Weyl’s inequality, Davis–Kahan, and perturbation bounds for M^0 and M^(l).The argument compares leading eigenspaces after zeroing the lth row and column.
  • Initialization bounds: The proof begins with a common decomposition used to establish the initialization bounds in (70a)–(70e).The three first bounds are handled before the component-wise and alignment bounds.
  • Established bounds: The resulting estimates cover Frobenius, spectral, component-wise, and leave-one-out alignment errors.The proof explicitly establishes (70a), (70c), (70e), and then (70d), with (70b) treated similarly.
  • Hessian geometry: The Hessian argument verifies smoothness and restricted strong convexity by separating the population Hessian from its empirical perturbation.The perturbation is controlled on a set S under m ≫ µ^2K log^9 m and δ = c/log^2 m.

C.1.1 Proof of Lemma 26

The proof verifies restricted strong convexity by decomposing a cross term into four pieces and showing that the nonleading pieces are small when the iterates remain close to the truth.

  • Strong convexity: The population Hessian contribution is lower-bounded through γ1 and γ2, producing a positive quadratic form in the h- and x-differences.The displayed bound involves 2 min{γ1, γ2}.
  • Cross-term decomposition: The restricted-strong-convexity calculation reduces to controlling a cross term β decomposed into β1, β2, β3, and β4.The decomposition isolates the leading interaction from perturbative terms.
  • Perturbative terms: The perturbative terms satisfy |β2| + |β3| + |β4| ≤ 5δ ∥h1 − h2∥2.The bound uses proximity to the truth and the assumption δ < 1.
  • Leading term: The proof bounds the leading term β1 using alignment and the small distances between the iterates and the truth.The argument also uses ∥x1 − x2∥2 ≤ 2δ and ∥x2∥2 ≤ 2.

C.1.2 Proof of Lemma 27

The proof controls the empirical Hessian perturbation uniformly over a local set by bounding four components with concentration, covering, and norm arguments. These bounds are then used in the leave-one-out induction to maintain local contraction conditions.

  • Perturbation decomposition: The perturbation is decomposed into four terms α1, α2, α3, and α4, which are controlled separately over the set S.The proof seeks uniform high-probability bounds for each component.
  • Uniform concentration: An ε-net and union-bound argument strengthens fixed-vector concentration into uniform control over the relevant parameter set.The covering uses ε = 1/(1280K).
  • Norm control: The proof groups entries of C by magnitude and combines cardinality, spectral-norm, and submatrix bounds to control α1.The groups form a partition indexed by r = 1,…,R.
  • High-probability bounds: The fourth component satisfies P(sup_z∈S α4 ≤ 1/48) = 1 − O(m^-10 + e^-K log m).This follows by combining concentration bounds under m ≫ µ^2K log m/δ^2 and δ ≪ 1.
  • Aggregate control: Combining the component-wise estimates yields the desired high-probability perturbation bound.The proof combines the bounds for all αj with an earlier aggregate estimate.
  • Leave-one-out induction: The leave-one-out induction uses gradient updates, the fundamental theorem of calculus, and Hessian control along the segment between the iterate and the truth.The induction verifies the conditions needed to reapply the local Hessian lemma.

C.4.3 Proof of Claim (224)

The proof establishes the claim by induction, using incoherence, concentration, and singular-vector perturbation arguments under explicit sample-size conditions.

  • Induction proof: The claim is proved by induction, with a sufficiently small base case and an induction step using hypothesis (85c).The proof concludes once the base case is established.
  • Incoherence control: The argument uses incoherence and closeness between e_t−1 and x⋆ to control relevant terms, including ∥e_t−1∥2 ≤ 2.The bound is invoked together with related estimates in the induction argument.
  • Conditions: The proof requires conditions such as δ ≍ log^-2 m and m ≫ τK log^4 m.Additional bounds assume sufficiently small η and related sample-size conditions.
  • Singular-vector perturbation: Wedin’s sinΘ theorem and Weyl’s inequality control the leading singular vectors through perturbation bounds on M and E[M].The denominator is bounded using σ1(E[M]) = 1, σ2(E[M]) = 0, and ξ ≤ 1/2.
  • Consequences: The resulting estimates prove claims (92) and (93), and imply the auxiliary bound ||α0| − 1| ≤ 1/4.The proof states that similar arguments establish (93).

C.6 Proof of Lemma 20

The proof of Lemma 20 combines singular-vector perturbation, incoherence estimates, alignment arguments, and auxiliary concentration lemmas to establish the stated initialization properties.

  • Proof structure: The proof has three steps: compare normalized singular vectors, transfer proximity to scaled vectors, and translate ℓ2 distance to the defined metric.The final step also establishes incoherence of h0 with respect to {b_l}.
  • Singular-vector proximity: Wedin’s sinΘ theorem and Weyl’s inequality show that the leading singular vectors of M and M^(l) are sufficiently close.The argument uses perturbation bounds under sample-size conditions involving μ, K, and logarithmic factors.
  • Termwise control: The two terms in the proximity bound are controlled using incoherence, triangle inequalities, independence, and bounds on the singular-vector coordinates.The resulting estimates include a condition m ≫ μK log^1/2 m.
  • Scaled vectors: Under m ≫ μK log^1/2 m, the coordinate bound yields the displayed upper estimate needed for the scaled singular-vector argument.Further bounds require m ≫ μ^2K log^2 m.
  • Auxiliary result: Lemma 30 supplies an additional claim when m ≫ τK log^4 m, with the proof relying on previously established estimates.The surrounding auxiliary lemmas provide Gaussian and concentration-based ingredients.
  • Alignment: Orthogonal Procrustes alignment uses the SVD of A⊤B, while Lemmas 35–37 provide perturbation and aligned-distance bounds.For aligned matrices, the stated bounds are ∥X1R1 − X2R2∥ ≤ 5κ∥X1 − X2∥ and the analogous Frobenius inequality.

D.2.2 Matrix concentration inequalities

This section collects matrix concentration tools for Bernoulli sampling, including bounds for rowwise operators, noise matrices, tangent-space quantities, and uniform extensions.

  • Scope: The concentration results concern Bernoulli variables used in matrix-completion analysis.Both asymmetric and symmetric sampling patterns are treated or related through reductions.
  • Rowwise concentration: Lemma 38 bounds a rowwise sampling operator using matrix Bernstein’s inequality under a sample-size condition involving δ, μ, n, r, and log n.The proof controls the median and then obtains the stated high-probability bound.
  • Probability guarantees: The resulting rowwise bounds hold with probability at least 1 − n^−(1.5C−1) for C ≥ 3.The same bound is recorded as a consequence in the lemma’s conclusion.
  • Sampling conditions: Additional lemmas impose sample-size conditions such as n^2p ≫ κμrn log^2 n and n^2p ≥ Cκμrn log n.These conditions support high-probability concentration bounds for sampling-related operators.
  • Uniform control: For tangent-space quantities, the analysis first handles fixed perturbations and then extends the result uniformly using an ε-net and continuity argument.The net has log |N_ε| ≲ nr log n for ε = 1/n^O(1).
  • Magnitude decomposition: The proof decomposes matrices by entry magnitude, controls each rounded component, and combines the bounds using union arguments and Chernoff estimates.The construction rounds entries within prescribed magnitude intervals.
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