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arxiv: 1907.04669 · v1 · pith:3OBIAI2Xnew · submitted 2019-07-08 · 💻 cs.LG · stat.ML

Optimal Explanations of Linear Models

Dimitris Bertsimas , Arthur Delarue , Patrick Jaillet , Sebastien Martin This is my paper

Pith reviewed 2026-05-25 01:20 UTC · model grok-4.3

classification 💻 cs.LG stat.ML
keywords linear modelsmodel interpretabilityexplanationsoptimizationcoordinate updatesaccuracy-interpretability tradeoffmachine learning
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The pith

Linear models can be explained by decomposing them into a sequence of simpler models through successive coordinate updates on their coefficients.

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

The paper sets out a general optimization framework that turns any linear model into an ordered chain of models, each one more complex than the last. The chain is built by repeatedly updating a single coefficient at a time until the original model is recovered. A reader would care because the method converts the loose idea of interpretability into a concrete, solvable problem that comes with exact algorithms and fast heuristics. It also supplies a tunable family of metrics for how easy a model is to explain while keeping its predictions intact.

Core claim

The central claim is that explanations for linear models are obtained by solving an optimization problem whose solution is a sequence of coordinate updates; this sequence yields a parametrized family of interpretability metrics and makes the accuracy-interpretability tradeoff explicit and computable.

What carries the argument

The decomposition of a linear model into a sequence of models of increasing complexity obtained by coordinate updates on the coefficients, computed by exact algorithms or scalable heuristics.

If this is right

  • The framework produces a family of interpretability metrics that generalize typical proxies used in practice.
  • Exact and heuristic algorithms exist to compute the optimal decomposition for any given linear model.
  • The approach quantifies the tradeoff between how interpretable a model is and how much predictive accuracy it retains.
  • The resulting explanations can be used to support complex decisions by revealing the model's reasoning step by step.

Where Pith is reading between the lines

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

  • The same decomposition idea could be tested on generalized linear models to see whether the coordinate-update sequence still yields useful explanations.
  • Organizations could adopt the derived metrics as a standardized way to audit the explainability of deployed linear models.
  • If the sequences align with human judgments on benchmark tasks, they might reduce vulnerability to adversarial attacks by making the model's logic transparent at each step.

Load-bearing premise

That the sequence of models produced by coordinate updates on the coefficients forms a meaningful explanation of the original model's reasoning rather than an arbitrary reordering of terms.

What would settle it

Apply the method to a small linear regression on a dataset whose intuitive feature ordering is already known and check whether the generated sequence deviates from that ordering on more than a small fraction of runs.

Figures

Figures reproduced from arXiv: 1907.04669 by Arthur Delarue, Dimitris Bertsimas, Patrick Jaillet, Sebastien Martin.

Figure 1
Figure 1. Figure 1: Tradeoff between the cost of the first and second models for a coordinate path of length 2 on problem (1). [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Pareto front between interpretability loss [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Example of a Pareto-efficient coordinate path. On the left we see the benefits of each coefficient modification. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of interpretable path and sequence of models of increasing sparsity selected by LASSO. Ftemp is [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
read the original abstract

When predictive models are used to support complex and important decisions, the ability to explain a model's reasoning can increase trust, expose hidden biases, and reduce vulnerability to adversarial attacks. However, attempts at interpreting models are often ad hoc and application-specific, and the concept of interpretability itself is not well-defined. We propose a general optimization framework to create explanations for linear models. Our methodology decomposes a linear model into a sequence of models of increasing complexity using coordinate updates on the coefficients. Computing this decomposition optimally is a difficult optimization problem for which we propose exact algorithms and scalable heuristics. By solving this problem, we can derive a parametrized family of interpretability metrics for linear models that generalizes typical proxies, and study the tradeoff between interpretability and predictive accuracy.

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

1 major / 2 minor

Summary. The manuscript proposes a general optimization framework for explaining linear models. It decomposes a given linear model into a sequence of models of increasing complexity by performing coordinate updates on the coefficients. Exact algorithms and scalable heuristics are developed to solve the resulting optimization problem. From the optimal decomposition the authors derive a parametrized family of interpretability metrics that generalize common proxies and examine the resulting interpretability-accuracy tradeoff.

Significance. If the coordinate-update decomposition can be shown to produce sequences that capture model reasoning in a non-arbitrary way, the framework would supply a principled, optimization-based route to interpretability metrics for linear models together with concrete algorithms. The algorithmic component and the explicit parametrization of metrics are concrete strengths.

major comments (1)
  1. [Abstract and §3] Abstract and §3 (methodology): the central claim that the coordinate-update sequence constitutes an 'explanation' of the original model's reasoning rests on the unstated assumption that the ordering induced by successive coordinate updates aligns with semantic reasoning steps. No linking criterion, invariance property, or validation is supplied showing why this ordering is explanatory rather than an arbitrary reordering of coefficients; this assumption is load-bearing for the claim that the framework produces explanations rather than merely parametrized orderings.
minor comments (2)
  1. Notation for the parametrized interpretability metrics is introduced without an explicit list of the free parameters or a table showing how common proxies arise as special cases.
  2. The abstract states that the decomposition is computed 'optimally,' yet the distinction between the exact algorithm and the scalable heuristic is not quantified in terms of approximation guarantees or empirical gap on the objective.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive review. We respond to the major comment below.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (methodology): the central claim that the coordinate-update sequence constitutes an 'explanation' of the original model's reasoning rests on the unstated assumption that the ordering induced by successive coordinate updates aligns with semantic reasoning steps. No linking criterion, invariance property, or validation is supplied showing why this ordering is explanatory rather than an arbitrary reordering of coefficients; this assumption is load-bearing for the claim that the framework produces explanations rather than merely parametrized orderings.

    Authors: The framework defines an explanation as the sequence obtained by solving the stated optimization problem, which yields a decomposition that is optimal with respect to an objective trading off fidelity to the original model against a complexity penalty. This optimality supplies a non-arbitrary criterion for the induced ordering. The parametrized interpretability metrics are then derived directly from properties of any such optimal sequence and are shown to generalize standard proxies. The manuscript does not claim or require that the ordering matches semantic reasoning steps of a human interpreter; its contribution is the optimization-based construction itself and the resulting metrics for the interpretability-accuracy tradeoff. We will revise the abstract and §3 to state this grounding explicitly and to remove any phrasing that could suggest semantic alignment. revision: yes

Circularity Check

0 steps flagged

No circularity: forward proposal of optimization framework with independent derivation of metrics

full rationale

The paper introduces a new optimization-based decomposition of linear models via coordinate updates on coefficients and then derives a parametrized family of interpretability metrics from that construction. No step reduces a claimed prediction, uniqueness result, or metric to a fitted quantity or self-citation defined inside the paper; the central claim is a methodological proposal whose validity rests on external interpretability criteria rather than internal redefinition. The derivation chain is self-contained against the stated inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; free parameters, axioms, and invented entities cannot be enumerated from the given text.

pith-pipeline@v0.9.0 · 5655 in / 867 out tokens · 16913 ms · 2026-05-25T01:20:56.000979+00:00 · methodology

discussion (0)

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