Alpha-Investing — Sequential FDR Control

What goes wrong with a naive approach

The runtime maintains dozens of simultaneous monitors: conformal frame-guard alerts, allocation-budget CUSUMs, flake detectors, BOCPD regime flips, fairness interventions. Each one tests at some significance level $\alpha$. If they all test at $\alpha = 0.05$ independently and you have 20 monitors, the expected number of false positives per frame is $1$. Multiply by 60 fps and you are drowning.

The usual answers are all bad:

• Bonferroni correction — divide $\alpha$ by the number of tests. Works for fixed-$n$ testing but is catastrophically conservative when tests run continuously and new ones come online at runtime.
• Manual prioritisation — pick three “important” alerts, mute the rest. Fragile: the muted alerts are exactly where the next outage will come from.
• Unthrottled — accept the noise. Users stop trusting the alert system, which is worse than not having one.

The right abstraction is sequential false-discovery-rate control: a global budget that is spent when you reject (fire an alert) and recovered when rejections are themselves rejected in their own local tests.

Mental model

Think of it as a gambling game:

• You start with wealth $W_0$ (an $\alpha$ budget).
• To run a test at level $\alpha_i$, you spend $\alpha_i$ from your wealth.
• If the test rejects (alert fires), you earn back a reward $\psi \alpha_i$ — you were “right” to spend on it.
• If the test accepts, the $\alpha_i$ is lost.

As long as the wealth stays non-negative and you stay within the rules, the marginal false discovery rate (mFDR) is bounded:

where $V$ is the number of false rejections and $R$ the total rejections.

The math

Wealth update

For test $i$ with level $\alpha_i$:

where $R_i \in \{0, 1\}$ is the rejection indicator and $\psi \in (0, 1)$ is the reward fraction.

Admissibility

A test at level $\alpha_i$ is admissible iff:

where $\phi \in (0, 1)$ is the investment fraction. This is the “don’t bet the farm” rule — you can’t spend your entire wealth on one test.

mFDR guarantee

If $\psi + \phi \le 1$ and $W_0 \le \alpha$, then:

for any stopping rule, any test order, any number of tests — the guarantee is sequential.

Defaults

With these, $\psi + \phi = 0.6 \le 1$ and $W_0 = 0.5$ so the guarantee holds with $\alpha = 0.5$.

Why you need it when running dozens of monitors

E[false alerts per frame] = 20 × 0.05 = 1.0
E[false alerts per second at 60 fps] = 60

→ alert log is 99% noise.
→ users ignore alerts, outages slip through.

W_0 = 0.5, ψ = 0.5, φ = 0.1
per-test α_i ≤ 0.05 admissible only while W_t ≥ 0.5

mFDR ≤ 0.5 globally.
Alerts land in a trickle and clear when things recover.

Rust interface

use ftui_runtime::alpha_investing::{AlphaInvestor, InvestmentConfig};
 
let mut investor = AlphaInvestor::new(InvestmentConfig {
    initial_wealth:     0.5,
    reward_fraction:    0.5,
    investment_fraction: 0.1,
});
 
// Subsystem proposes a test at level 0.03
if let Some(ticket) = investor.request(0.03) {
    let rejected = run_local_test();
    investor.resolve(ticket, rejected);
} else {
    // Not enough wealth — defer or pick a cheaper test.
}

Every subsystem that wants to raise an alert first asks the investor for a ticket. If the wealth can’t support the $\alpha_i$, the test is suppressed for this frame — the budget speaks.

How to debug

Each admissible test emits an alpha_invest line:

{"schema":"alpha_invest","test_id":"budget_alert",
 "alpha_i":0.03,"wealth_before":0.48,
 "rejected":true,"wealth_after":0.465,
 "reward":0.015}

FTUI_EVIDENCE_SINK=/tmp/ftui.jsonl cargo run -p ftui-demo-showcase
 
# Which monitors are bleeding wealth?
jq -c 'select(.schema=="alpha_invest" and .rejected==false)
       | .test_id' /tmp/ftui.jsonl | sort | uniq -c | sort -rn

If wealth trends to zero and stays there, the system is chronically “unhealthy” — the monitors are firing real alerts. Investigate the subsystem with the highest rejection rate first.

Pitfalls

Cross-references

• E-processes — per-test anytime-valid machinery that feeds into the investor.
• Conformal frame gating — a major consumer of alpha-investing wealth.
• /runtime/evidence-sink — how alpha_invest lines are emitted.

Where next