X6 Criteria

From the documentation of the field

This page sets out what evidence of possessing X₆, Feedback Adaptation, looks like. It states criteria only. It answers the situation called P₆, nonstationarity. The standard asks for all six primitives of Ismail's Primitives in Class C.

What the primitive requires

The plain meaning is to discard stale evidence after the world changes. A structure possesses X₆ when it has a mechanism that notices a change and stops the structure leaning on evidence from before it. Only a mechanism that performs this function counts; a document that resembles one does not. Error theory explains the difference.

The situation it answers

P₆ is present when the relationship the structure relies on has actually changed, in the rewards it pays or in how the situation moves. The formal definition is HasP₆ in SixPrimitives/Phase0.lean, at line 210.

A single surprising outcome that is consistent with an unchanged process is not a change of regime. Ordinary variance around a strategy the evidence still supports is not one either. That holds provided the same explanation is not being reached for every time. A pattern of surprises, each dismissed as noise on its own, is no longer a single surprising outcome.

What lacking it means

A structure that lacks X₆ incurs linear regret wherever the situation arises: its loss grows in proportion to time. The proof demonstrates this in the smallest environment that exhibits the situation. The failure condition below says precisely what lacking it means.

The failure condition is LacksX₆, at line 99 of the same file. It is stated about a changepoint at time τ. It says that, for a window after the change whose length is proportional to τ, the probability that the structure still plays the option that was best before the change stays at or above one half plus a positive margin. The structure keeps leaning on pre-change evidence for as long as a fixed share of the time the old regime lasted. In the witness environment there are two options, and halfway through the horizon the two swap, once. A structure that lacks X₆ keeps playing the old favourite after the swap, and its loss grows in proportion to time. Audit it yourself walks through the definition.

What possession looks like

Phase0 defines two levels of possession.

Ordinary possession, PossessesX₆, means the structure is not lacking. The stale option does not dominate the window after the change.

Robust possession, RobustX₆, adds a condition at the level of cycles. The structure is not lacking, and at every cycle the summary it actually holds equals a clean summary. This is the paper's evidence-reset form of possession.

Evidence that counts

• Change detection. A recorded means by which a change in the world is noticed, with the signals it responds to.
• Triggered revision. When a change is noticed, the assumptions that rested on the old regime are revisited, and each revision is dated and tied to the change.
• Speed. The time from a change to a changed action is recorded, and it is short relative to how long the old regime lasted.
• Retirement of old evidence. Evidence from before the change is retired or down-weighted in later decisions, and the record shows it.
• Different decisions. Decisions after a change differ from decisions before it in the way the change requires.

What does not count

• A review calendar with no trigger tied to change.
• A strategy refresh carried out as a routine, whatever has or has not changed.
• Reacting to a single outcome consistent with an unchanged process.
• Assumptions that were revised but whose decisions were not.
• An old summary left in use because it is convenient.

Where the criteria come from

LacksX₆ (SixPrimitives/Phase0.lean, line 99), PossessesX₆ and RobustX₆ (lines 148 and 151), HasP₆ (line 210), the necessity theorem necessity_x6 (SixPrimitives/Phase3.lean, line 2575), the witness environment env₆_nonstat (SixPrimitives/Phase3.lean, line 2078), and the paper's section on the swapping-options environment. The plain meaning and the boundary cases are on the field definition page. The criteria shared by all six primitives are on the criteria page.