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arxiv: 2604.17560 · v1 · submitted 2026-04-19 · 🧮 math.OC

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The Multi-Block DC Function Class: Theory, Algorithms, and Applications

Alp Yurtsever, Hoomaan Maskan, Pouria Fatemi, Suvrit Sra

Authors on Pith no claims yet

Pith reviewed 2026-05-10 05:35 UTC · model grok-4.3

classification 🧮 math.OC
keywords multi-block DCDC programmingnonconvex optimizationReLU networkstensor factorizationpolynomialsconvergence analysisstochastic algorithms
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The pith

The multi-block DC class allows polynomial-size decompositions for models that require exponential size under standard DC.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines the multi-block DC class of nonconvex functions, which decomposes the difference-of-convex structure separately across multiple parameter blocks. It shows that polynomials and tensor factorizations must use exponentially large decompositions in the ordinary DC class yet admit only polynomial size in the new class. Explicit constructions are given for deep ReLU networks, a case with no known counterpart in standard DC. Algorithms are developed with non-asymptotic convergence guarantees that apply in both batch and stochastic regimes. These elements together indicate that block-structured decompositions can make certain nonconvex problems more tractable than global DC approaches allow.

Core claim

The central claim is that the multi-block DC class strictly contains ordinary DC programming and is more powerful, because standard models such as polynomials and tensor factorizations admit only exponential-size DC decompositions while their multi-block versions remain polynomial, and because explicit multi-block formulations can be written for deep ReLU networks.

What carries the argument

The multi-block DC decomposition, which writes a function as a difference of two convex functions defined separately over each parameter block rather than a single global pair of convex functions.

If this is right

  • Polynomials and tensor factorizations admit polynomial-size multi-block DC decompositions.
  • Deep ReLU networks possess explicit multi-block DC formulations.
  • Optimization algorithms for multi-block DC functions achieve non-asymptotic convergence in both batch and stochastic settings.
  • The class applies directly to several modern nonconvex tasks in machine learning and optimization.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Practitioners could derive closed-form decompositions for additional structured nonconvex functions not covered in the paper.
  • The block separation may combine with existing block-coordinate methods to produce hybrid solvers with improved scaling.
  • Empirical timing comparisons on tensor completion or network training tasks would quantify whether the polynomial size translates to faster run times.
  • Similar block-structured relaxations might apply to other difference-based function classes beyond DC.

Load-bearing premise

Explicit polynomial-size multi-block DC decompositions can be constructed for polynomials, tensor factorizations, and deep ReLU networks without hidden exponential cost in the construction itself.

What would settle it

A specific polynomial or ReLU network for which every multi-block DC decomposition requires exponential size in the number of variables or layers would falsify the separation and constructivity claims.

Figures

Figures reproduced from arXiv: 2604.17560 by Alp Yurtsever, Hoomaan Maskan, Pouria Fatemi, Suvrit Sra.

Figure 1
Figure 1. Figure 1: Estimated smoothness constant of gi(· ; θ¯ i) in (3.2) plotted against the gradient norm during training. Additional details are provided in Section A.6.1. L-smoothness by allowing the Hessian norm to grow with the gradient magnitude rather than remain uniformly bounded. Motivated by these developments, and supported by empirical observations that a multi-block version of the generalized smoothness holds i… view at source ↗
Figure 2
Figure 2. Figure 2: Reconstruction error and sparsity of codes for the [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of SBdc with SGD in regression (left) and classification (middle, right) for 10 Monte-Carlo instances. The shaded bands specify the 68% confidence intervals. As depicted, SBdc has comparable performance to SGD on test loss and test accuracy. where C =  D ∈ Rm×l | ∥dj∥2 ≤ 1 ∀j [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: SDL loss evolution with the nonconvex ℓ1 − ℓQ regularizer (Eq. 5.5) on synthetic data. The left panel shows the loss trajectory of the joint full-batch gradient descent method with an adaptive step size, while the right panel shows the loss evolution of the Bdc algorithm. Each curve reports the mean and ±2 standard deviations over 3 independent runs. Implementation Details. Both formulations (with ℓ1 norm … view at source ↗
read the original abstract

We present the Multi-Block DC (BDC) class, a rich class of structured nonconvex functions that admit a DC ("difference-of-convex") decomposition across parameter blocks. This multi-block class not only subsumes the usual DC programming, but also turns out to be provably more powerful. Specifically, we demonstrate how standard models (e.g., polynomials and tensor factorization) must have DC decompositions of exponential size, while their BDC formulation is polynomial. This separation in complexity also underscores another key aspect: unlike DC formulations, obtaining BDC formulations for problems is vastly easier and constructive. We illustrate this aspect by presenting explicit BDC formulations for modern tasks such as deep ReLU networks, a result with no known equivalent in the DC class. Moreover, we complement the theory by developing algorithms with non-asymptotic convergence theory, including both batch and stochastic settings, and demonstrate the broad applicability of our method through several applications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper introduces the Multi-Block DC (BDC) function class, a structured generalization of standard DC functions in which the difference-of-convex decomposition is performed across multiple parameter blocks. It claims to establish a complexity separation showing that standard models (polynomials, tensor factorization) require exponential-size DC decompositions under the paper's representation measure while admitting polynomial-size BDC formulations, supplies explicit constructive BDC decompositions for deep ReLU networks (with no known DC equivalent), and derives non-asymptotic convergence guarantees for both batch and stochastic algorithms together with illustrative applications.

Significance. If the explicit constructions and separation results hold, the BDC class would provide a strictly more powerful yet tractable framework for nonconvex optimization than classical DC programming. The constructive polynomial-size formulations for modern models such as deep ReLU networks, combined with matching exponential lower bounds on ordinary DC size and non-asymptotic convergence theory for batch and stochastic variants, constitute a clear strength that could enable new algorithmic approaches in machine learning and tensor optimization.

minor comments (3)
  1. [Complexity analysis] The precise definition of the representation measure used for the DC versus BDC size comparison should be stated once in a dedicated subsection and cross-referenced in the complexity theorems to make the separation fully self-contained.
  2. [Algorithmic development] In the algorithmic sections, the dependence of the convergence rates on the number of blocks should be made explicit (e.g., in the statement of the main convergence theorem) so that readers can immediately see the scaling with respect to the multi-block structure.
  3. [Applications and examples] A short table summarizing the BDC decompositions for the running examples (polynomials, tensor factorization, ReLU networks) would improve readability and allow quick comparison of the polynomial versus exponential sizes.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report recognizes the key contributions of the Multi-Block DC class, including the complexity separation from standard DC programming, constructive polynomial-size formulations for deep ReLU networks, and non-asymptotic convergence results for batch and stochastic algorithms. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivations are constructive and independent

full rationale

The paper defines the BDC class and supplies explicit, polynomial-size BDC decompositions for polynomials, tensor factorizations, and deep ReLU networks, paired with separate exponential lower bounds on ordinary DC size under the stated representation measure. These constructions and bounds are presented as direct, verifiable results rather than reductions to fitted parameters or self-referential equations. Algorithmic convergence guarantees follow from the multi-block structure without invoking self-citations as load-bearing premises or smuggling ansatzes. The central separation claim therefore rests on independent proofs and explicit formulations, not on any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claims rest on the definition of the new BDC class and on standard convex-analysis facts; no free parameters or invented physical entities are mentioned.

axioms (1)
  • standard math Standard properties of convex functions and DC decompositions from prior literature
    The paper builds directly on the existing theory of DC programming.
invented entities (1)
  • Multi-Block DC (BDC) function class no independent evidence
    purpose: To admit structured DC decompositions across parameter blocks and achieve polynomial complexity for models that are exponential in ordinary DC
    The class is defined and analyzed in the paper; no independent empirical or formal verification outside the manuscript is provided.

pith-pipeline@v0.9.0 · 5467 in / 1368 out tokens · 41495 ms · 2026-05-10T05:35:19.594823+00:00 · methodology

discussion (0)

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Reference graph

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