On Cost-Aware Sequential Hypothesis Testing with Random Costs and Action Cancellation

TL;DR

The paper extends cost-aware sequential hypothesis testing to scenarios where actions incur random costs and can be canceled via per-action deadlines. It shows that, under ex-post costs, deadlines do not change the cost, while under ex-ante costs, deadlines induce an effective per-action cost and can reduce total expense depending on cost tails. The authors derive conditions for when deadlines help, analyze several cost-tail families (Erlang, Hyperexponential, Pareto, Log-Logistic), and validate findings with numerical experiments demonstrating preserved $\Theta(\log(1/\delta))$ scaling. The work informs design of cost-aware SHT policies by incorporating timing strategies for action cancellation in realistic, heterogeneous-cost environments.

Abstract

We study a variant of cost-aware sequential hypothesis testing in which a single active Decision Maker (DM) selects actions with positive, random costs to identify the true hypothesis under an average error constraint, while minimizing the expected total cost. The DM may abort an in-progress action, yielding no sample, by truncating its realized cost at a smaller, tunable deterministic limit, which we term a per-action deadline. We analyze how this cancellation option can be exploited under two cost-revelation models: ex-post, where the cost is revealed only after the sample is obtained, and ex-ante, where the cost accrues before sample acquisition. In the ex-post model, per-action deadlines do not affect the expected total cost, and the cost-error tradeoffs coincide with the baseline obtained by replacing deterministic costs with cost means. In the ex-ante model, we show how per-action deadlines inflate the expected number of times actions are applied, and that the resulting expected total cost can be reduced to the constant-cost setting by introducing an effective per-action cost. We characterize when deadlines are beneficial and study several families in detail.

Figures (7)

• Figure 1: System model. The DM seeks to identify the correct hypothesis indexed by $\theta\in\mathcal{H}$. By taking action $A_n$ at time step $n$, the DM obtains a realization of $X_n\sim f_\theta^{A_n}$ after a random cost $C_{A_n}\sim f_{C_{A_n}}$ is incurred.
• Figure 2: Illustrating Proposition \ref{['proposition: Pareto Example']} when $C_a\sim \mathrm{Pareto}(1, 3/2)$. $\kappa_a > \mathbb{E}\left[C_a\right] = 3$ for any $T_a < 3/2$ and $\kappa_a \leq \mathbb{E}\left[C\right]$ when $T_a\geq 3/2$. The optimal per-action deadline $T_a^* \approx 3.41825$ from Lemma \ref{['lemma: Pareto cost optimal deadline']}, is also depicted.
• Figure 3: Illustrating Proposition \ref{['proposition: LogLogistic Example']} when $C_a\sim \mathrm{LogLogistic}(4, 3/2)$. The updated fixed cost, $\kappa_a$, increases with $T_a$, implying that it cannot be optimized. Taking the distribution median $\alpha_a = 4$ as the per-action deadline ensures that $\kappa_a(\alpha_a)\leq\mathbb{E}\left[C_a\right]$.
• Figure 4: Simulation results for a scenario where $C_a\sim\mathrm{LogLogistic}(\alpha_a, \alpha_a)$ for any $a$. The use of the per-action deadline and $T_a = F_{C_a}^{-1}(0.5) = \alpha_a$ reduces expected total cost of the vanilla CASHT algorithms ($T_a = \infty$).
• Figure 5: Simulation results for a scenario where $C_a\sim\mathrm{Pareto}(x_{\min,a}, \alpha_a)$ for any $a$. The use of optimal per-action deadline (computed as in Lemma \ref{['lemma: Pareto cost optimal deadline']}) improves the performance of CA algorithms.

Theorems & Definitions (8)

• Theorem 1
• Corollary 1
• Lemma 1
• Proposition 1
• Proposition 2
• Proposition 3
• Lemma 2
• Proposition 4