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Econ-288 · economics · Economics

Adaptive experimental inference

LLM status Open unverified

Statement

Original subject group: Econometrics, causal inference and measurement

Audit classification: Background; proposed extension. Correct authors verified; published2025 DOI added. Adaptive inference already has solutions, and a bounded IPS martingale gives simultaneous coverage here even under unrestricted conditional drift. Only the explicitly specified optimization frontier is a proposed research task.

Definitions and precise research target

Consider a sequential experiment over $T$ participants with two treatments $a\in\{0,1\}$ and bounded potential outcomes $Y_t(a)\in[0,1]$. Conditional mean rewards $\mu_{a,t}=E[Y_t(a)\mid\mathcal H_{t-1}]$ may change with time, where $\mathcal H_{t-1}$ is recorded history. Assume conditional randomization given history and an almost-sure drift budget $\sum_{t=2}^{T}\max_a|\mu_{a,t}-\mu_{a,t-1}|\leq V$. Assignment probability $p_t$, measurable from history, satisfies $\epsilon\leq p_t\leq1-\epsilon$ for fixed $0<\epsilon<1/2$. Define the running average effect $\tau_t=t^{-1}\sum_{s=1}^{t}(\mu_{1,s}-\mu_{0,s})$ and welfare regret $R_T=\sum_{t=1}^{T}\{\max_a\mu_{a,t}-\mu_{A_t,t}\}$, where $A_t$ is the assigned treatment. Develop assignment and inference rules whose confidence intervals $C_t$ satisfy $\Pr(\tau_t\in C_t\text{ for every }t\leq T)\geq1-\alpha$, for stated $\alpha\in(0,1)$, uniformly over the bounded-outcome drift class. Characterize attainable trade-offs between expected regret and interval width as functions of $T,V,\epsilon$, and compare them with fixed randomization. Regret compares one-step choices along realized history, rather than alternative state trajectories. Fixed positive exploration can imply linear regret even without drift. Arbitrary unbounded drift is excluded explicitly. Adaptive inference already has established solutions; the nonstationary joint welfare-and-inference optimization is a proposed extension, and the supporting reference does not verify its exact open status. Outcomes are assessed in a prespecified participant population with consistent measurement across the observation horizon.

Definitions, assumptions and mathematical qualifications. Let $\mathcal H_{t-1}$ contain all prior assignments/outcomes. Require $A_t\perp(Y_t(0),Y_t(1))\mid\mathcal H_{t-1}$, with $p_t=\Pr(A_t=1\mid\mathcal H_{t-1})$. Observed $Y_t=Y_t(A_t)$ yields score $X_t=A_tY_t/p_t-(1-A_t)Y_t/(1-p_t)$, whose conditional mean is $\mu_{1,t}-\mu_{0,t}$. For fixed horizon $T$, Hoeffding's martingale bound and a union bound give valid intervals centered at $t^{-1}\sum_{s\leq t}X_s$, intersected with $[-1,1]$, with radius $\epsilon^{-1}\sqrt{2\log(2T/\alpha)/t}$. Thus simultaneous coverage exists without the drift restriction; existence is not open. Study the attainable frontier of $(\sup_M E_MR_T,\sup_M E_M\sum_t|C_t|)$ over a specified outcome/drift class and admissible designs. Fixed exploration yields $E R_T\geq\epsilon\Delta T$ when stationary arm means differ by $\Delta>0$.

Variants and assumption changes

  1. Editorial, fixed exploration benchmark. Keep $\epsilon>0$; compare the displayed valid intervals and unavoidable linear regret with stronger variance-adaptive inference or tighter Pareto bounds.
  2. Editorial, declining exploration. Permit $\epsilon_t\downarrow0$ according to a stated rule and impose a drift budget $V$; characterize how increasing inverse-probability variance trades against dynamic regret and anytime interval length.

References and source audit

Checked source. arXiv:2405.01281v1 (2 May 2024), Sections 2--5; Annual Review of Statistics and Its Application 12 (2025), pp. 407--423. The survey describes existing valid-inference methods; it does not declare this nonstationary frontier open. Primary preprint text and published metadata checked. Preprint cover misspells Demistifying; published title is Demystifying. Published DOI 10.1146/annurev-statistics-040522-015431. https://arxiv.org/pdf/2405.01281.
  1. Aurélien Bibaut; Nathan Kallus. 2024. Demystifying Inference After Adaptive Experiments. arXiv:2405.01281v1 (2 May 2024), Sections 2--5; Annual Review of Statistics and Its Application 12 (2025), pp. 407--423. https://arxiv.org/pdf/2405.01281. DOI: https://doi.org/10.48550/arXiv.2405.01281.

Common conventions: Source mathematical conventions

These conventions apply to each entry unless explicitly replaced.

Models and identification. A primitive model $m$ specifies all probability laws, feasible choices, information, equilibrium and measurement rules used by its target. The admissible class $\mathcal M$ is fixed independently of outcomes. If $P_O(m)$ is its observable probability law and $\tau(m)$ the target, its identified set at observable law $P$ is \[ \mathcal I_\tau(P)=\{\tau(m):m\in\mathcal M,\ P_O(m)=P\}. \] Point identification means that this set is a singleton. An empty set signals incompatibility of data and assumptions. Coordinate bounds need not describe the joint set. Bounds are sharp only if every retained target is attainable or arbitrarily approachable by compatible models. Finite bounds require explicit restrictions on unobserved quantities; unrestricted confounding, hidden wealth, extrapolation or arbitrary equilibrium selection can yield uninformative sets.

Causal interventions. A potential outcome $Y_i(p)$ is the outcome under a complete policy regime $p$, including timing, financing, information and market spillovers. The target population and units are fixed before treatment. With interference, use the complete assignment vector or a justified exposure mapping; individual no-interference notation alone cannot identify an economy-wide intervention. A difference of mean potential outcomes does not require the cross-world joint distribution. Conditioning on post-treatment migration, survival, enrollment or employment changes the estimand and needs separate justification.

Statistical statements. An estimator, confidence set, forecast or test specifies a sampling law, model class and loss/coverage criterion. Uniform $(1-\alpha)$-coverage means $\inf_{m\in\mathcal M}\Pr_m\{\tau(m)\in C_n\}\geq1-\alpha$, or an explicitly asymptotic version. The whole parameter space trivially has coverage, so informativeness requires a stated width, risk or power objective. A forecasting improvement is relative to a named benchmark, information set and evaluation population.

Equilibrium and optimization. An equilibrium comprises feasible plans, optimality, beliefs where needed, budget constraints and all market/resource clearing. The primitives or a stated benchmark define each correspondence $\mathcal E(p;m)$. Nonemptiness is assumed or proved, never inferred just from compact policy sets. A selected-equilibrium target uses a declared selection rule; a robust target quantifies over all permitted equilibria. Use suprema or infima unless attainment is established. An $\varepsilon$-optimal policy is feasible and within $\varepsilon$ of the finite optimum value. Report an empty feasible set separately.

Welfare and accounting. Normative utility, interpersonal normalization, social weights, numeraire, discounting and the use of public receipts are primitives. Behavioral utility need not coincide with the welfare criterion. Household welfare includes ownership income and rents once; adding profits again to compensating variation generally double-counts them. A monetary equivalent is defined by an explicit transfer and price convention, with monotonicity and range conditions. Average effects, mechanism contributions, transfers and welfare losses are different targets. Infinite sums require convergence and dynamic feasible plans require terminal or no-Ponzi conditions.

Source versions. A working-paper date, revision date and journal publication year are distinguished. A DOI for a journal version is not automatically the DOI of an earlier manuscript. Locators refer to the stated version and printed pages where specified. A metadata-only check does not verify a claim about a full-text conclusion. Every entry's source subsection narrows the provenance claim accordingly.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-288-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-288

Adaptive experimental inference Dynamic-regret upper bounds with simultaneous inference and a width lower bound

Question and scope

In a two-arm adaptive experiment, the goal is joint control of participant regret and uncertainty about the running average treatment effect under bounded drift. Existence of valid intervals is already established. This attempt proves an explicit pair of bounds using restarted exponential weights and a variance-sensitive martingale interval, then derives a stationary regret--width obstruction. It does not characterize the exact nonstationary Pareto frontier.

Model

At dates $t=1,\ldots,T$, potential outcomes $Y_t(a)\in[0,1]$ have conditional means $\mu_{a,t}={\mathbb{E}}[Y_t(a)\mid\mathcal H_{t-1}]$. Conditional randomization means $A_t$ is independent of the current potential outcomes given recorded history, with predictable probability $p_t$ of arm 1. Assume \[ \epsilon\le p_t\le1-\epsilon,\quad 0<\epsilon<1/2, \qquad \sum_{t=2}^T\max_a|\mu_{a,t}-\mu_{a,t-1}|\le V \] almost surely. Means may depend on past assignments and outcomes. Regret is evaluated along the realized history: $R_T=\sum_t[\max_a\mu_{a,t}-\mu_{A_t,t}]$. The inferential target is the predictable running average $\tau_t=t^{-1}\sum_{s\le t}(\mu_{1,s}-\mu_{0,s})$.

A self-contained Bernstein bound

Lemma (Bounded martingale Bernstein inequality): If $D_t$ are martingale differences, $|D_t|\le b$, and $\sum_{s\le t}{\mathbb{E}}[D_s^2\mid\mathcal H_{s-1}]\le v_t$ for deterministic $v_t$, then for $L>0$, \[ {\mathbb{P}}\left(\left|\sum_{s\le t}D_s\right|> \sqrt{2v_tL}+\frac{2bL}{3}\right)\le2e^{-L}. \]
Proof: For $|x|<3$, the exponential series and $k!\ge2\,3^{k-2}$ for integers $k\ge2$ give $e^x\le1+x+x^2/[2(1-|x|/3)]$. Thus for $0<\lambda<3/b$, \[ {\mathbb{E}}[e^{\lambda D_s}\mid\mathcal H_{s-1}] \le\exp\left\{\frac{\lambda^2{\mathbb{E}}[D_s^2\mid\mathcal H_{s-1}]}{2(1-\lambda b/3)}\right\}. \] Iteration and Markov's inequality yield the upper tail $\exp\{-x^2/[2(v_t+bx/3)]\}$ by choosing $\lambda=x/(v_t+bx/3)$ (the zero-variance case is trivial). The positive root of $x^2=2L(v_t+bx/3)$ is $bL/3+\sqrt{2v_tL+b^2L^2/9}$, which is at most the cutoff in the lemma. Apply the same calculation to $-D_s$ and add tails. ∎

Define the inverse-probability score $X_t=A_tY_t/p_t-(1-A_t)Y_t/(1-p_t)$ and let $L=\log(2T/\alpha)$, $0<\alpha<1$. Use the confidence interval \[ C_t=\left[\frac1t\sum_{s\le t}X_s-r_t, \frac1t\sum_{s\le t}X_s+r_t\right]\cap[-1,1], \quad r_t=\sqrt{\frac{2L}{t\epsilon(1-\epsilon)}}+ \frac{4L}{3\epsilon t}. \] An empty intersection is allowed on the failure event and has length zero.

Theorem (Simultaneous coverage and cumulative width): For every admissible assignment policy, whether or not it is chosen for regret, \[ {\mathbb{P}}(\tau_t\in C_t\text{ for all }t\le T)\ge1-\alpha. \] Moreover, pathwise, \[ \sum_{t=1}^T|C_t|\le \min\left\{2T, 4\sqrt{\frac{2TL}{\epsilon(1-\epsilon)}}+ \frac{8L}{3\epsilon}(1+\log T)\right\}. \]
Proof: Conditional randomization gives ${\mathbb{E}}[X_t\mid\mathcal H_{t-1}]=\mu_{1,t}-\mu_{0,t}$. Its centered difference is bounded by $1/\epsilon+1\le2/\epsilon$. Also \[ {\mathbb{E}}[X_t^2\mid\mathcal H_{t-1}] \le\frac1{p_t}+\frac1{1-p_t} =\frac1{p_t(1-p_t)}\le\frac1{\epsilon(1-\epsilon)}. \] The lemma with $b=2/\epsilon$, $v_t=t/[\epsilon(1-\epsilon)]$ gives failure probability at most $\alpha/T$ at each time. A union bound proves coverage, since the true target belongs to $[-1,1]$. Width is at most $\min(2,2r_t)$. Sum using $\sum_{t\le T}t^{-1/2}\le2\sqrt T$ and $\sum_{t\le T}t^{-1}\le1+\log T$. ∎

A compatible assignment rule

Partition time into consecutive blocks of length at most $h$, where $1\le h\le T$ is an integer. At a block start set arm weights to one. Let $q_{a,t}$ be normalized weights and assign arm $a$ with probability $p_{a,t}=\epsilon+(1-2\epsilon)q_{a,t}$. Observe loss $L_t=1-Y_t$ and set $\widehat\ell_{a,t}={\mathbf 1}\{A_t=a\}L_t/p_{a,t}$. Update the weight of arm $a$ by multiplication by $e^{-\eta\widehat\ell_{a,t}}$, with $\eta=\sqrt{(1-2\epsilon)\log2/h}$. The same observations supply the preceding confidence intervals.
Theorem (Explicit dynamic-regret bound): The stated rule satisfies \[ {\mathbb{E}} R_T\le\epsilon T+ 2\left\lceil\frac Th\right\rceil\sqrt{(1-2\epsilon)h\log2} +2hV. \] Consequently the excess above the unavoidable exploration term can be bounded by optimizing the displayed finite set of $h$. For $V=0$, $h=T$ gives $O(\sqrt T)$. Away from endpoint truncations, $h$ of order $(T/V)^{2/3}$ gives $O(T^{2/3}V^{1/3})$, with an additional $O(\sqrt T)$ term covering small $V$.
Proof: Within one block, comparison of the total exponential weight with the weight of a fixed arm, using $e^{-x}\le1-x+x^2/2$ for $x\ge0$ and $\log(1+x)\le x$, gives \[ \sum_t\sum_a q_{a,t}\widehat\ell_{a,t} -\sum_t\widehat\ell_{j,t} \le\frac{\log2}{\eta} +\frac\eta2\sum_t\sum_aq_{a,t}\widehat\ell_{a,t}^2. \] This is an algebraic inequality for every realized sequence. Conditional expectations of the estimates are the arm's current expected loss. Furthermore ${\mathbb{E}}[\sum_aq_{a,t}\widehat\ell_{a,t}^2\mid\mathcal H_{t-1}] \le\sum_aq_{a,t}/p_{a,t}\le2/(1-2\epsilon)$. A block comparator may be selected from history at the start of that block; it is then fixed during the block, so the same expectation argument applies.

The expected loss of the actual assignment, less comparator loss, is at most $1-2\epsilon$ times the corresponding $q$-weighted difference plus $\epsilon$ at each date, because each of the two arm losses lies in $[0,1]$. For a block of length $b\le h$, this gives regret against a fixed comparator at most $\epsilon b+(1-2\epsilon)\log2/\eta+\eta b \le\epsilon b+2\sqrt{(1-2\epsilon)h\log2}$. Choose the comparator as the arm with highest conditional mean at the block's first date. If $V_b$ is the realized total variation within that block, each later mean differs from its initial value by at most $V_b$. The per-date loss of using this initial best arm instead of the current best is at most $2V_b$. Hence the additional block regret is at most $2hV_b$. Summing and using $\sum_bV_b\le V$ proves the bound. Balancing $T/\sqrt h$ with $hV$, and clipping $h$ to $[1,T]$, gives the stated orders. ∎

A stationary regret--width obstruction

This lower bound applies to any adaptive policy and any terminal interval, not only the above construction. Let arm 1 have Bernoulli mean $3/4$, and arm 0 have Bernoulli mean $\theta$, independently over dates and arms. Let $P_0$ denote $\theta=1/4$. Write $n_0={\mathbb{E}}_0\sum_t{\mathbf 1}\{A_t=0\}$, so ${\mathbb{E}}_0 R_T=n_0/2$. Suppose a terminal interval $C$ covers $3/4-\theta$ with probability at least $1-\alpha$ for all $\theta\in[1/4,3/8]$, where $0<\alpha<1/2$.
Theorem (Learning the underassigned arm requires regret): For every $h_0\in(0,1/8]$, \[ {\mathbb{E}}_0|C|\ge h_0\{1-2\alpha-2h_0\sqrt{n_0}\}_+. \] In particular choose $h_0=\min\{1/8,(1-2\alpha)/(4\sqrt{n_0})\}$, interpreting the second entry as infinity when $n_0=0$. Then ${\mathbb{E}}_0|C|\ge(1-2\alpha)h_0/2$.
Proof: Compare $P_0$ with $P_1$ having $\theta=1/4+h_0$. The adaptive assignment kernels are identical functions of observed history under both laws, so the chain rule for log likelihood gives ${\operatorname{KL}}(P_0,P_1)=n_0\,\mathrm{kl}(1/4,1/4+h_0)$. The elementary inequality $\log x\le x-1$ gives $\mathrm{kl}(p,q)\le(p-q)^2/[q(1-q)]\le8h_0^2$ in this interval. Thus ${\operatorname{TV}}(P_0,P_1)\le\sqrt{{\operatorname{KL}}/2}\le2h_0\sqrt{n_0}$. To recall the last inequality's justification, collapse each law onto a set achieving total variation; the log-sum inequality decreases relative entropy, and the binary relative entropy is at least twice the squared probability difference, by its second derivative $1/[p(1-p)]\ge4$ and integration from its minimum.

Under $P_0$, the interval contains the $P_0$ target with probability at least $1-\alpha$, and the $P_1$ target with probability at least $1-\alpha-{\operatorname{TV}}(P_0,P_1)$. The intersection of these events has probability at least $1-2\alpha-{\operatorname{TV}}$. On it the interval length is at least the target separation $h_0$. This proves the first bound. The chosen $h_0$ makes the bracket at least $(1-2\alpha)/2$. ∎

Under fixed exploration $n_0\ge\epsilon T$, so regret is already at least $\epsilon T/2$ in this example. The lower theorem does not match every upper-bound dependence on $V,\epsilon$ or accumulated width. It identifies a concrete obstruction to obtaining arbitrarily narrow effect intervals while almost never assigning the inferior arm.

Remaining gap and references

The joint bounds are compatible and hold with conditional drift, but no optimality claim is made for the full frontier. Sharper variance adaptation, unknown drift budgets, declining exploration and confidence-width objectives at every time remain to be optimized. This work concerns regret along realized histories, not policy values under alternative state trajectories.

The primary preprint and published metadata of the survey below were checked. It describes established adaptive-inference methods; the exponential-weights and martingale calculations above are proved here as a restricted construction, without a novelty claim.

Completion Estimate

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This is a post hoc, heuristic estimate for the full catalogue target. The attempt combines a restarted adaptive assignment rule with simultaneous treatment-effect intervals and explicit dynamic-regret and cumulative-width bounds, and proves a stationary regret-width obstruction. The nonstationary Pareto frontier still lacks matching lower bounds and optimized dependence on drift, exploration, and width at every date.

References

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