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Certified Learning under Distribution Shift: Sound Verification and Identifiable Structure

Chandrasekhar Gokavarapu, Sudhakar Gadde, Y. Rajasekhar, S. R. Bhargava

TL;DR

A unified framework is developed in which risk under distribution shift is certified by explicit inequalities, verification of learned models is sound for nontrivial sizes, and interpretability is enforced through identifiability conditions rather than post hoc explanations.

Abstract

Proposition. Let $f$ be a predictor trained on a distribution $P$ and evaluated on a shifted distribution $Q$. Under verifiable regularity and complexity constraints, the excess risk under shift admits an explicit upper bound determined by a computable shift metric and model parameters. We develop a unified framework in which (i) risk under distribution shift is certified by explicit inequalities, (ii) verification of learned models is sound for nontrivial sizes, and (iii) interpretability is enforced through identifiability conditions rather than post hoc explanations. All claims are stated with explicit assumptions. Failure modes are isolated. Non-certifiable regimes are characterized.

Certified Learning under Distribution Shift: Sound Verification and Identifiable Structure

TL;DR

A unified framework is developed in which risk under distribution shift is certified by explicit inequalities, verification of learned models is sound for nontrivial sizes, and interpretability is enforced through identifiability conditions rather than post hoc explanations.

Abstract

Proposition. Let be a predictor trained on a distribution and evaluated on a shifted distribution . Under verifiable regularity and complexity constraints, the excess risk under shift admits an explicit upper bound determined by a computable shift metric and model parameters. We develop a unified framework in which (i) risk under distribution shift is certified by explicit inequalities, (ii) verification of learned models is sound for nontrivial sizes, and (iii) interpretability is enforced through identifiability conditions rather than post hoc explanations. All claims are stated with explicit assumptions. Failure modes are isolated. Non-certifiable regimes are characterized.
Paper Structure (28 sections, 21 theorems, 61 equations)

This paper contains 28 sections, 21 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. The three gaps as one problem
  3. Assumptions and verifiability
  4. Main contributions
  5. A failure of common logic
  6. Plan
  7. Preliminaries
  8. Probability, risks, and losses
  9. Complexity control
  10. Verification model
  11. Core result I
  12. Assumptions and verifiable quantities
  13. Risk under shift: certified bound
  14. Core result II
  15. Certificate form and soundness
  16. ...and 13 more sections

Key Result

Proposition 1.1

Let $P$ be a training distribution on $\mathcal{X}\times\mathcal{Y}$, and let $Q$ be a test distribution. Let $\ell\colon \mathbb{R}\times\mathcal{Y}\to[0,1]$ be a loss and $f\colon\mathcal{X}\to\mathbb{R}$ a predictor. A mathematically checkable theory of learning under shift should deliver an expl where $\mathcal{R}_\mu(f):=\mathbb{E}_{(X,Y)\sim \mu}\,\ell(f(X),Y)$, the "shift" term is a metric

Theorems & Definitions (75)

  • Proposition 1.1: A certifiable target
  • Theorem 1.6: Shift risk as a robust program with computable dual bound
  • Theorem 1.7: Sound verification as a dual certificate; explicit scaling law
  • Theorem 1.8: Identifiability-implies-certifiability for an interpretable class
  • Proposition 1.9: A negative result: when certification cannot scale
  • Definition 2.1: 1-Wasserstein metric
  • Definition 2.2: Lipschitz constant
  • Lemma 2.3: Kantorovich--Rubinstein duality
  • proof
  • Lemma 2.4: Worst-case expectation under a $W_1$-ball
  • ...and 65 more