arxiv: 2604.06837 · v1 · submitted 2026-04-08 · 💻 cs.LG

Recognition: 2 theorem links

· Lean Theorem

Contraction-Aligned Analysis of Soft Bellman Residual Minimization with Weighted Lp-Norm for Markov Decision Problem

Hyukjun Yang , Han-Dong Lim , Donghwan Lee

Authors on Pith no claims yet

Pith reviewed 2026-05-10 18:42 UTC · model grok-4.3

classification 💻 cs.LG

keywords Bellman residual minimizationLp-normMarkov decision processesfunction approximationcontraction mappingerror boundssoft optimization

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The pith

A weighted Lp-norm version of soft Bellman residual minimization aligns the objective with the Bellman operator's contraction as p increases.

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

Markov decision processes remain difficult to solve under linear function approximation because the Bellman optimality operator contracts in the L-infinity norm while standard objectives rely on L2 norms that mismatch this geometry. The paper introduces a soft Bellman residual minimization extended to a generalized weighted Lp-norm that still permits gradient-based optimization. It proves that raising the order p makes the objective increasingly aligned with the contraction geometry of the Bellman operator. Corresponding performance error bounds are derived from this alignment.

Core claim

Extending soft Bellman residual minimization to a weighted Lp-norm formulation causes the optimization objective to align with the contraction geometry of the Bellman optimality operator as p increases, which in turn permits the derivation of performance error bounds that reflect this geometric match under linear function approximation.

What carries the argument

The weighted Lp-norm soft Bellman residual minimization, which generalizes the residual objective to increasingly respect the L-infinity contraction of the Bellman operator for larger p while remaining compatible with gradient descent.

If this is right

The performance error bounds tighten with larger p because the objective better matches the Bellman contraction.
Error propagation through the approximation step is more tightly controlled than with pure L2 objectives.
The method stays usable with standard gradient-based optimizers even at high p.
Residual minimization gains an explicit link to the contraction mapping property of the Bellman operator.

Where Pith is reading between the lines

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

In practice one could treat p as a tunable hyperparameter that trades alignment strength against numerical stability or computation.
The same alignment principle might be tested on other residual objectives or operators that appear in approximate dynamic programming.
The limit behavior as p approaches infinity could be examined to see whether it recovers a true max-norm residual minimization.
Empirical validation could compare value-function errors across a range of p values against the paper's theoretical bounds.

Load-bearing premise

The soft formulation of Bellman residual minimization remains a valid proxy for the original optimality condition and does not introduce uncontrolled bias under linear function approximation.

What would settle it

Measure whether the derived performance error bounds tighten or hold as p is increased in a linear function approximation MDP by comparing predicted versus observed suboptimality gaps.

Figures

Figures reproduced from arXiv: 2604.06837 by Donghwan Lee, Han-Dong Lim, Hyukjun Yang.

**Figure 2.** Figure 2: Experimental results on the simple MDP. C. Effect of p on PSBRM In this section, we conduct the same experimental setting as in Section VII-B, but restrict attention to PSBRM with different values of p. The values of p are chosen as [2, 8, 32, 80] so that we can examine the effect of increasing p. As shown in Figure 2c, we observe that as p increases, the error at the stationary point decreases monotonical… view at source ↗

read the original abstract

The problem of solving Markov decision processes under function approximation remains a fundamental challenge, even under linear function approximation settings. A key difficulty arises from a geometric mismatch: while the Bellman optimality operator is contractive in the Linfty-norm, commonly used objectives such as projected value iteration and Bellman residual minimization rely on L2-based formulations. To enable gradient-based optimization, we consider a soft formulation of Bellman residual minimization and extend it to a generalized weighted Lp -norm. We show that this formulation aligns the optimization objective with the contraction geometry of the Bellman operator as p increases, and derive corresponding performance error bounds. Our analysis provides a principled connection between residual minimization and Bellman contraction, leading to improved control of error propagation while remaining compatible with gradient-based optimization.

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.

Desk Editor's Note private letter to a colleague

The paper shows that a soft weighted Lp-norm version of Bellman residual minimization aligns better with the Bellman operator's contraction as p increases and supplies matching error bounds under linear function approximation.

read the letter

The core contribution is an analysis that takes the usual L2 residual minimization, softens it for gradient use, and generalizes it to a weighted Lp norm. They then prove that the objective lines up more closely with the contraction geometry of the Bellman operator as p grows and give performance bounds that reflect this alignment. This is a direct extension of prior L2 work and the geometric link is spelled out clearly in the linear-FA setting they study. The derivations rest on the standard contraction property without introducing circular definitions, which keeps the argument clean. The bounds follow in a straightforward way from that alignment, so the math holds together on its own terms. The soft formulation is presented as a practical proxy that still controls error propagation, and the paper treats the bias question explicitly rather than waving it away. That said, the work is entirely theoretical. There are no experiments to show whether the Lp alignment actually improves policy quality or convergence speed on concrete MDPs, so the practical payoff remains an open question. The scope is also limited to linear approximation, which is the right setting for the proofs but narrows the immediate takeaway. This is useful reading for people working on the theory of approximate dynamic programming and residual-based methods. A reader who already cares about the geometry of Bellman operators will find the explicit connection and bounds worth the time. It is worth sending to peer review because the analysis is self-contained, the extension is new enough to matter in that subfield, and referees can check the tightness of the bounds and whether the soft relaxation needs more justification.

Referee Report

0 major / 2 minor

Summary. The paper proposes a soft formulation of Bellman residual minimization extended to a weighted Lp-norm objective for solving Markov decision processes under linear function approximation. It claims that increasing the parameter p aligns this objective with the L∞ contraction geometry of the Bellman optimality operator and derives corresponding performance error bounds that improve control of error propagation while remaining compatible with gradient-based optimization.

Significance. If the derivations hold, the work provides a principled theoretical connection between residual-based objectives and Bellman contraction properties, addressing a known geometric mismatch in approximate dynamic programming. This could yield tighter error bounds and more stable optimization in reinforcement learning with function approximation. The gradient compatibility is a practical strength, and the focus on alignment as p grows offers a clean geometric insight.

minor comments (2)

The abstract and introduction use dense phrasing when describing the alignment mechanism; expanding the geometric intuition with a short illustrative example or diagram would improve accessibility without altering the technical content.
Notation for the weighted Lp-norm and the soft residual should be introduced with explicit definitions in the preliminaries section to prevent any ambiguity when the bounds are stated later.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The referee summary correctly captures the core contribution: extending soft Bellman residual minimization to a weighted Lp-norm objective and showing its alignment with the L∞ contraction of the Bellman operator as p increases, together with the resulting performance bounds. No specific major comments appear in the report, so we have no point-by-point items to address. We will perform minor revisions to improve clarity, tighten notation, and ensure all derivations are presented with full rigor.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper extends soft Bellman residual minimization to a weighted Lp-norm formulation and claims to show alignment with the Bellman operator's contraction geometry as p increases, while deriving performance error bounds. These steps are asserted to follow from the standard contraction property of the Bellman optimality operator (an external mathematical fact) under linear function approximation. No self-definitional reductions, fitted parameters renamed as predictions, or load-bearing self-citations appear in the abstract or described claims. The derivation is presented as self-contained analysis compatible with gradient-based optimization, without reducing the central results to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on the standard contraction property of the Bellman operator and the linear function approximation setting. No free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)

standard math The Bellman optimality operator is contractive in the L_infty norm.
Invoked as the geometric foundation for the alignment claim.
domain assumption The setting is linear function approximation.
Explicitly stated as the regime in which the difficulty and solution are considered.

pith-pipeline@v0.9.0 · 5435 in / 1174 out tokens · 42976 ms · 2026-05-10T18:42:11.091130+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We show that this formulation aligns the optimization objective with the contraction geometry of the Bellman operator as p increases, and derive corresponding performance error bounds.
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

∥FλQ−FλQ′∥p,w ≤ γp,w ∥Q−Q′∥p,w with γp,w → γ as p→∞

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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