Conservative-Markdown Redirect-UCB Pricing

• The paper demonstrates that CMRUP achieves a regret bound of approximately O(T^(2/3)), matching known lower bounds under censored binary feedback and discontinuous demand.
• It employs randomized parameter estimation and conservative markdown probing to discretize the residual range and form a robust one-dimensional UCB method.
• The method is significant for handling non-smooth, jump discontinuities in demand by shifting exploration to a residual grid with a one-step redirection strategy.

Conservative-Markdown Redirect-UCB Pricing (CMRUP) is a polynomial-time algorithm for contextual dynamic pricing with linear valuations, bounded-support agnostic noise, and censored binary purchase feedback. It addresses the regime in which the induced demand curve may be non-Lipschitz and may contain arbitrary jumps and atoms, so standard smooth-demand interpolation arguments are invalid. The method combines randomized parameter estimation, conservative residual-grid probing, and confidence-based one-step redirection, and achieves $\widetilde O(T^{2/3})$ regret under stochastic well-conditioned contexts, matching the known lower bounds up to logarithmic factors (Xu et al., 7 May 2026).

1. Formal model and pricing objective

The pricing environment is sequential. At each round $t=1,\dots,T$, a context $x_t\in\mathbb{R}^d$ is observed, a price $p_t\in[0,B]$ is posted, and the only feedback is whether the buyer purchases:

$o_t=\mathbf 1\{p_t\le y_t\}.$

The buyer valuation is modeled as

$y_t = x_t^\top \theta^\star + \xi_t,$

where $\theta^\star\in\mathbb{R}^d$ is unknown and $\xi_t$ is i.i.d. noise from an unknown distribution. Because only the binary outcome is observed, the problem is a censored-feedback contextual pricing problem rather than a full-information regression problem (Xu et al., 7 May 2026).

The expected purchase probability is written in terms of the residual survival function

$S(w):=\Pr(\xi_t\ge w),$

so that

$$\mathbb E[o_t\mid x_t,p_t]=S\!\left(p_t-x_t^\top\theta^\star\right).$$

The expected revenue at context $t=1,\dots,T$0 and price $t=1,\dots,T$1 is

$t=1,\dots,T$2

Regret is defined against the context-dependent oracle price:

$t=1,\dots,T$3

The paper also introduces the residual representation

$t=1,\dots,T$4

with oracle value

$t=1,\dots,T$5

This decomposition isolates a linear contextual component $t=1,\dots,T$6 and a one-dimensional residual pricing problem over $t=1,\dots,T$7, which is the structural basis of CMRUP (Xu et al., 7 May 2026).

2. Statistical regime and the source of non-Lipschitz difficulty

The analysis assumes known constants $t=1,\dots,T$8 such that almost surely

$t=1,\dots,T$9

The noises are i.i.d., independent of the context sequence, and satisfy

$x_t\in\mathbb{R}^d$0

Contexts are i.i.d. from a distribution $x_t\in\mathbb{R}^d$1 with well-conditioned covariance:

$x_t\in\mathbb{R}^d$2

A buffered admissible-price assumption requires a $x_t\in\mathbb{R}^d$3 such that

$x_t\in\mathbb{R}^d$4

These conditions ensure that the algorithm’s conservative markdown probes are not clipped by the price bounds (Xu et al., 7 May 2026).

The distinctive feature of the model is that no density, Lipschitz, smoothness, log-concavity, or parametric assumption is imposed on $x_t\in\mathbb{R}^d$5. The demand curve may therefore have arbitrary jumps and atoms. In this regime, cross-context interpolation fails: an arbitrarily small perturbation in residual price can cross a discontinuity and change demand by a constant amount, so

$x_t\in\mathbb{R}^d$6

This invalidates the smoothness-based confidence arguments used by prior contextual pricing methods. The paper positions this as the reason the earlier agnostic contextual pricing method of Xu and Wang achieved only $x_t\in\mathbb{R}^d$7 regret, while CMRUP avoids such interpolation entirely by working on a one-dimensional residual grid (Xu et al., 7 May 2026).

3. Three-stage construction of CMRUP

The algorithm proceeds in three stages. Stage 1 performs randomized parameter estimation of the linear valuation component. For $x_t\in\mathbb{R}^d$8, prices are posted uniformly at random:

$x_t\in\mathbb{R}^d$9

With $p_t\in[0,B]$0, the uniform randomization yields

$p_t\in[0,B]$1

This converts censored binary observations into a linear regression signal,

$p_t\in[0,B]$2

and the estimator is

$p_t\in[0,B]$3

The paper sets

$p_t\in[0,B]$4

and defines the prediction radius

$p_t\in[0,B]$5

The corresponding good event is

$p_t\in[0,B]$6

On this event, the linear component is uniformly estimated over the horizon (Xu et al., 7 May 2026).

Stage 2 discretizes the residual interval $p_t\in[0,B]$7 by

$p_t\in[0,B]$8

For each cell $p_t\in[0,B]$9, the conservative markdown probe is

$o_t=\mathbf 1\{p_t\le y_t\}.$0

The deliberate $o_t=\mathbf 1\{p_t\le y_t\}.$1 shift is the conservative geometry of the method. On $o_t=\mathbf 1\{p_t\le y_t\}.$2,

$o_t=\mathbf 1\{p_t\le y_t\}.$3

which yields the purchase-probability sandwich

$o_t=\mathbf 1\{p_t\le y_t\}.$4

The adjacent jump size is defined by

$o_t=\mathbf 1\{p_t\le y_t\}.$5

This is the paper’s robust observation model for discontinuous demand: each probe identifies an interval of survival values rather than relying on pointwise smoothness (Xu et al., 7 May 2026).

Stage 2 then includes a warmup period of length

$o_t=\mathbf 1\{p_t\le y_t\}.$6

during which the grid index is sampled uniformly,

$o_t=\mathbf 1\{p_t\le y_t\}.$7

and the corresponding conservative markdown probe is played. Stage 3 is the adaptive pricing phase. For each cell $o_t=\mathbf 1\{p_t\le y_t\}.$8, define the pre-round count $o_t=\mathbf 1\{p_t\le y_t\}.$9, empirical mean $y_t = x_t^\top \theta^\star + \xi_t,$0, and confidence radius

$y_t = x_t^\top \theta^\star + \xi_t,$1

The optimistic score is

$y_t = x_t^\top \theta^\star + \xi_t,$2

and the selected index is

$y_t = x_t^\top \theta^\star + \xi_t,$3

The pricing rule then applies confidence-based one-step redirection: if $y_t = x_t^\top \theta^\star + \xi_t,$4, the algorithm directly probes $y_t = x_t^\top \theta^\star + \xi_t,$5; if $y_t = x_t^\top \theta^\star + \xi_t,$6 and $y_t = x_t^\top \theta^\star + \xi_t,$7, it redirects to $y_t = x_t^\top \theta^\star + \xi_t,$8; if $y_t = x_t^\top \theta^\star + \xi_t,$9, it plays $\theta^\star\in\mathbb{R}^d$0 (Xu et al., 7 May 2026).

4. Confidence structure, redirection, and regret accounting

The confidence analysis is uniform over Stage-3 rounds and cells:

$\theta^\star\in\mathbb{R}^d$1

where $\theta^\star\in\mathbb{R}^d$2 is the predictable average mean of actual plays of probe $\theta^\star\in\mathbb{R}^d$3. Combined with the markdown sandwich, this gives

$\theta^\star\in\mathbb{R}^d$4

From this, the optimistic score satisfies

$\theta^\star\in\mathbb{R}^d$5

The role of UCB is therefore localized to the residual grid: it is a confidence device over discretized residual cells rather than a smooth-demand or per-context interpolation argument (Xu et al., 7 May 2026).

The redirect rule is introduced to control the cost of discontinuities. If the algorithm were to keep probing a cell adjacent to a jump, it could repeatedly pay the adjacent-jump penalty $\theta^\star\in\mathbb{R}^d$6. Once the confidence radius at an index becomes small enough, direct probing is no longer necessary. Redirecting one step left preserves optimism, avoids probing the discontinuity directly, and incurs only $\theta^\star\in\mathbb{R}^d$7 extra price loss. This is the meaning of “Redirect-UCB” in the algorithm name: exploration is confidence-based, but execution is redirected when confidence has crossed the grid scale (Xu et al., 7 May 2026).

The paper formalizes one-round regret in two modes. In direct-probe mode,

$\theta^\star\in\mathbb{R}^d$8

Thus the instantaneous loss contains a grid approximation term, an adjacent-jump term, and a confidence-radius term. In redirect mode,

$\theta^\star\in\mathbb{R}^d$9

where $\xi_t$0 depends only on $\xi_t$1. Because redirection is used only when $\xi_t$2, this constant-level loss is absorbed into the $\xi_t$3 analysis (Xu et al., 7 May 2026).

5. Regret guarantees and optimality

The main theorem states that under the bounded-support, stochastic-context, and buffered-price assumptions,

$\xi_t$4

even when the noise distribution has arbitrary atoms, jumps, and no Lipschitz or smoothness regularity. The proof balances four contributions: the cost of Stages 1 and 2, each $\xi_t$5; the grid-scale approximation term, roughly $\xi_t$6; the UCB accumulation

$\xi_t$7

and the jump cost

$\xi_t$8

With

$\xi_t$9

the dominant terms balance at $S(w):=\Pr(\xi_t\ge w),$0 (Xu et al., 7 May 2026).

The result is presented as optimal up to logarithmic factors. The paper explains that the classical non-contextual lower bound of Kleinberg and Leighton gives

$S(w):=\Pr(\xi_t\ge w),$1

regret for posted-price learning, that this lower bound can be embedded into the contextual linear-valuation model, and that Xu and Wang also show a $S(w):=\Pr(\xi_t\ge w),$2 lower bound for contextual pricing under Lipschitz demand. Under the paper’s assumptions—especially i.i.d. stochastic contexts, full-rank covariance, bounded-support noise, and a buffer away from clipping—the long-standing regret gap is therefore closed:

$S(w):=\Pr(\xi_t\ge w),$3

The paper also records the prior best upper bound in the agnostic bounded setting as $S(w):=\Pr(\xi_t\ge w),$4, so the improvement is specifically from $S(w):=\Pr(\xi_t\ge w),$5 to $S(w):=\Pr(\xi_t\ge w),$6 (Xu et al., 7 May 2026).

6. Relation to conservative bandits and markdown pricing

CMRUP belongs to a broader family of conservative pricing and bandit methods, but the term “conservative” is used in materially different senses across adjacent lines of work. In nonlinear conservative contextual bandits, the conservative requirement is an explicit baseline-relative performance constraint:

$S(w):=\Pr(\xi_t\ge w),$7

with algorithms $S(w):=\Pr(\xi_t\ge w),$8 and $S(w):=\Pr(\xi_t\ge w),$9 combining an online regression oracle, Inverse Gap Weighting exploration, and a safety check that reverts to the baseline when needed. In markdown pricing under an unknown parametric demand family, conservatism refers instead to monotone downward pricing under irreversible overshooting risk, leading to Cautious Myopic (CM) and Iterative Cautious Myopic (ICM), which choose the largest safe price consistent with confidence information and enforce non-increasing prices by truncation or cautious descent. CMRUP differs from both: it studies contextual pricing with censored feedback and agnostic non-Lipschitz demand, and its conservative mechanism is the combination of a downward residual probe offset and one-step redirection on a residual grid rather than an explicit baseline safety budget or a globally non-increasing price path (Deb et al., 2024, Jia et al., 2023, Xu et al., 7 May 2026).

Work	Conservative object	Mechanism
CMRUP (Xu et al., 7 May 2026)	Residual-grid probing near jumps	Randomized parameter estimation, conservative markdown probing, one-step redirection
Conservative contextual bandits beyond linear (Deb et al., 2024)	Cumulative loss relative to a baseline policy	Online regression oracle, IGW exploration, safety inequality
Markdown pricing under unknown parametric demand (Jia et al., 2023)	Non-increasing markdown path under overshooting risk	Confidence-based cautious descent via CM or ICM

A common misconception is to read CMRUP as a standard smooth-demand UCB algorithm. The paper states the opposite motivation: smooth interpolation is precisely what fails when $$\mathbb E[o_t\mid x_t,p_t]=S\!\left(p_t-x_t^\top\theta^\star\right).$$0 has arbitrary jumps and atoms. Another possible confusion is to equate its “markdown” component with the monotone markdown constraint studied in single-product markdown pricing. The terminology overlaps, but the mechanisms are different. In CMRUP, markdown refers to the conservative $$\mathbb E[o_t\mid x_t,p_t]=S\!\left(p_t-x_t^\top\theta^\star\right).$$1 shift used to ensure that each probe’s true residual lies inside a known grid cell, whereas the monotonicity constraint in parametric markdown pricing is a structural restriction requiring $$\mathbb E[o_t\mid x_t,p_t]=S\!\left(p_t-x_t^\top\theta^\star\right).$$2 almost surely for all $$\mathbb E[o_t\mid x_t,p_t]=S\!\left(p_t-x_t^\top\theta^\star\right).$$3 (Xu et al., 7 May 2026, Jia et al., 2023).

A plausible implication is that CMRUP can be viewed as a bridge between two research traditions. From conservative contextual bandits it inherits a safety-oriented refusal to act on fragile estimates; from markdown pricing it inherits the idea that downward moves must be made cautiously because an aggressive move can create irreversible loss. What is specific to CMRUP is that these ideas are re-engineered for censored contextual pricing with discontinuous residual demand, yielding a one-dimensional residual-grid UCB method whose conservatism is localized, geometric, and discontinuity-aware rather than baseline-budgeted or globally monotone (Deb et al., 2024, Jia et al., 2023, Xu et al., 7 May 2026).