On the stability of learning

John J. Vastola

q-bio.NCOct 26, 2026

On the stability of learning

Authors: John J. Vastola

Organizations: Department of Neurobiology, Harvard Medical School · Department of Psychology and Center for Brain Science, Harvard University · Kempner Institute for the Study of Natural and Artificial Intelligence, Harvard University

Abstract

Learning algorithms are usually analyzed under the assumption that informative gradients are continuously available. But this is not always true---learning signals in biological neural networks, for example, may be sparse, local, and only intermittently available at any given synapse. In this work, we show that if gradient information is noisy and only intermittently available, learning may be unstable. In particular, even if average learning dynamics converge to the correct optimum, variance in learning may be asymptotically infinite. We explore this point in a number of analytically tractable settings, including single-layer and multi-layer linear networks, and derive explicit conditions under which learning is stable. These conditions complement, but are conceptually distinct from, existing conditions related to the stability of stochastic gradient descent. Our work demonstrates that unbiased credit assignment is not enough for reliable learning; sufficiently frequent and informative feedback is also required, which yields an additional constraint on models of biologically plausible learning.

Figures & tables

Figure 1: Sparse gradient arrivals: motivation and intuition. ( a ) A simple example of sparse gradient arrivals: a weight receives intermittent updates, in between which the regularization term produces relaxation toward zero. Three example stochastic simulations are shown. ( b ) Error as a function of number of arrivals in a toy model. If

s/\rho<1

, error always decreases (left); if

1<s/\rho<2

, error alternates in sign but decreases in magnitude (center); and if

s/\rho>2

, learning becomes unstable, and error both alternates in sign and increases in magnitude (right).

Figure 2: Instability in a linear model. ( a ) Average dynamics are unconditionally stable and do not depend on the arrival rate

\rho

. ( b ) Output variance converges more slowly as

\rho

decreases and diverges to infinity when

\rho<\rho_{c}

. ( c ) The steady state output variance diverges like

1/(\rho-\rho_{c})

. ( d )

\rho_{c}

decreases as

\gamma

increases, so one way to improve stability is to increase the regularization strength.

Figure 3: Instability in a two-layer network. ( a ) We trained a two-layer network to reproduce the output of an identical teacher network. To ensure our linear model provides a (locally) reasonable description of its dynamics, we initialized it near the teacher’s weights. ( b ) Normalized objective and output variance throughout learning. Note that for

\rho<\rho_{c}

(dark red),

J

blows up. For

\rho

somewhat above the threshold (red, orange), learning still looks unstable on this finite time scale. Learning only appears stable well beyond the threshold (green). ( c ) Instantaneous growth rate of the covariance dynamics near the fixed point. The black line indicates the linear model’s prediction and the red dots indicate numerical measurements. Below

\rho=\rho_{c}

, this local growth rate can be positive, which indicates instability.

Figure 4: Depth benignly modulates stability in a deep linear network. ( a )

\rho_{c}

can increase, decrease, or remain constant with

L

depending on the value of the target

W_{*}

. In all cases, the difference between

\rho_{c}(L)

and its asymptotic value

\rho_{c}(\infty)

is not substantial. ( b )

\rho_{c}

is non-monotonic as a function of

\gamma

(

L=10

shown), but this is mainly because sufficiently high regularization destabilizes the nontrivial fixed point.

Figure 5: Pooling parameters can improve stability. ( a ) If nearby synapses store similar information, neurons may benefit from sharing information with their spatial neighbors. Pictured is a line of synaptic weights; arrows indicate bidirectional information sharing. ( b ) The eigenvalues

d_{q}

\bm{D}

are indexed by spatial frequencies

|q|

, and

d_{q}

decreases as

|q|

increases. ( c ) The

q

-dependence of

d_{q}

causes higher spatial frequencies of the fixed point to be damped, and the effect size increases as the amount of diffusion

k

increases (more purple

\to

higher

k

). ( d ) One effect of this frequency-dependence is to smooth the fixed point of the average dynamics. In the pictured example,

w_{i}^{*}

is highly oscillatory (black), but dynamics converge to a smoothed version of it (shades of purple). ( e )

\rho_{c}

decreases as

k

increases, so although parameter-pooling produces bias, it also improves stability.

Figure 6: Averaging over gradient arrivals can improve stability. ( a ) Intuitively, one can reduce variance by performing a temporal average over recent gradient arrivals. ( b ) Mean dynamics are linear but not monotonic. As the gradient-averaging time scale

\tau_{z}

increases, they converge more slowly. (black:

\tau_{z}=0

; lightest green:

\tau_{z}=100

) ( c , d ) Increasing

\tau_{z}

improves stability since

\rho_{c}

decreases as long as

\gamma>0

. But if

\gamma=0

, there is no stability benefit.

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Open paper AskPDF arXiv

Stable training is often treated as evidence that learning is succeeding, but stability characterizes optimization behavior rather than correctness relative to an external objective. We study what happens when the signal being optimized remains persistently biased. We define Stable but Wrong (SBW) as a learning state in which optimization remains active or stabilizes while the learned outcome stays systematically displaced from an independently defined objective. A minimal theoretical model shows that persistent systematic bias can separate a stable convergence point from the true objective even under standard conditions for stable convergence. We then introduce static and dynamic bias in reinforcement learning, supervised learning, and continual fine-tuning of large language models. Across these settings, optimization of the current learning signal continues or stabilizes while correctness relative to the external objective declines or remains displaced. Finally, two controlled challenge-correction experiments with external ground truth test whether a prespecified error can be exposed and changed. In controlled scientific discovery and CIFAR-10, targeted evidence corrects or reduces established errors more effectively than equal-budget general evidence, with no evident harm to paired correct states under the tested protocols. These experiments do not provide a general method for automatically discovering unknown errors. Overall, SBW identifies a basic limit of convergence and other internal optimization metrics as evidence of correctness in systems that learn through persistent bias and feedback.

Abstract

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