Two Training Clocks in Machine Learning

• Two Training Clocks are a framework that decomposes learning into two distinct phases—fast behavioral adaptation (e.g., classifier or rule clocks) and slower structural or representational refinement.
• The approach is exemplified in grokking, generative modeling, and reinforcement learning, where metrics like cross-entropy loss, stable rank, and state-specific step sizes reveal asymmetrical training dynamics.
• Beyond ML, similar dual-clock concepts appear in high-precision metrology and Mars timekeeping, highlighting the broad applicability of using separate temporal measures to capture complex system behaviors.

“Two Training Clocks” denotes a family of research formulations in which learning or inference is decomposed into two distinct temporal indices or stopping times. In contemporary machine learning, the phrase has been formalized most explicitly for grokking as a fast classifier clock and a slow representation clock, separating label fitting from structural simplification of the learned map (Tan et al., 4 Jun 2026). Closely related formulations appear in generative modeling as a rule-learning clock $\tau_{\mathrm{rule}}$ and a memorization clock $\tau_{\mathrm{mem}}$, whose separation defines an innovation window (Wang et al., 11 May 2026), and in reinforcement learning as a distinction between global and local clocks for step-size indexing in differential temporal-difference learning (Antrobius et al., 7 May 2026). Outside machine learning, the same phrase also appears in unrelated domains, including two Earth-orbiting atomic clocks for tests of the Einstein Equivalence Principle (Litvinov et al., 2021) and Earth-day training clocks for public Mars timekeeping (Flesch et al., 11 Jul 2025). The unifying motif is temporal duality: two clocks are introduced because a single time parameter does not adequately capture the relevant dynamics, whether those dynamics concern optimization, stochastic approximation, relativistic metrology, or pedagogy.

1. Conceptual scope and formal meanings

In the grokking literature, two training clocks are stopping times attached to two distinct objectives: fitting and simplification. The classifier clock is defined by

$$T_{\mathrm{cls}}(\epsilon):=\inf\{t\ge 0:L_{\mathrm{CE}}(A_t)\le \epsilon\},$$

while the representation clock is

$$T_{\mathrm{rep}}(\eta):=\inf\{t\ge 0:S(A_t)\le \eta\},$$

where $L_{\mathrm{CE}}(A_t)$ is the empirical cross-entropy and $S(A_t)$ is a nonnegative structural gap such as a spectral penalty gap, a trailing singular-value tail, or a stable-rank proxy (Tan et al., 4 Jun 2026). In this usage, grokking is interpreted as a temporal mismatch: the classifier clock completes early, whereas the representation clock completes much later.

In generative modeling, the same basic separation is expressed with different observables. The rule-learning clock $\tau_{\mathrm{rule}}$ is “the first training step at which model generations (after discretization) become reliably rule-valid,” and the memorization clock $\tau_{\mathrm{mem}}$ is “the first training step at which models begin to reproduce training samples exactly” (Wang et al., 11 May 2026). Their gap defines the innovation window $[\tau_{\mathrm{rule}},\tau_{\mathrm{mem}}]$, during which generations are rule-valid yet novel.

In average-reward reinforcement learning, the phrase refers not to stopping times but to indexing schemes for step sizes. A global clock uses a step-size sequence $\alpha_t$ that depends only on the global time index $\tau_{\mathrm{mem}}$0, whereas a local clock uses $\tau_{\mathrm{mem}}$1, where

$\tau_{\mathrm{mem}}$2

so each state receives the base sequence indexed by its own visit count (Antrobius et al., 7 May 2026). Here the distinction is not between two phases of training but between two update clocks that induce different limiting ODEs.

These meanings are not interchangeable. A plausible implication is that “two training clocks” is best understood as an umbrella phrase for settings in which two temporally distinct processes are materially relevant, rather than as a single fixed technical definition across fields.

2. Grokking: fast fitting and slow representation simplification

The most formalized use of the term occurs in “Deciphering Two Training Clocks in Grokking via Deep Linear Network Theory with Conditional ReLU Reduction” (Tan et al., 4 Jun 2026). The paper studies a $\tau_{\mathrm{mem}}$3-class training set $\tau_{\mathrm{mem}}$4 with effective linear classifier $\tau_{\mathrm{mem}}$5, row vectors $\tau_{\mathrm{mem}}$6, and incorrect-class logit gap

$\tau_{\mathrm{mem}}$7

The sample loss and empirical cross-entropy admit the pure-gap form

$\tau_{\mathrm{mem}}$8

Because the softmax tail is controlled by the margins, the paper associates the fast clock with gap growth. If all incorrect gaps are at least $\tau_{\mathrm{mem}}$9, then

$$T_{\mathrm{cls}}(\epsilon):=\inf\{t\ge 0:L_{\mathrm{CE}}(A_t)\le \epsilon\},$$0

hence $$T_{\mathrm{cls}}(\epsilon):=\inf\{t\ge 0:L_{\mathrm{CE}}(A_t)\le \epsilon\},$$1.

The representation clock is tied to structural energy. For the deep linear surrogate, the regularized end-to-end energy is

$$T_{\mathrm{cls}}(\epsilon):=\inf\{t\ge 0:L_{\mathrm{CE}}(A_t)\le \epsilon\},$$2

where the Schatten quasi-norm is

$$T_{\mathrm{cls}}(\epsilon):=\inf\{t\ge 0:L_{\mathrm{CE}}(A_t)\le \epsilon\},$$3

The paper also tracks effective dimension by stable rank,

$$T_{\mathrm{cls}}(\epsilon):=\inf\{t\ge 0:L_{\mathrm{CE}}(A_t)\le \epsilon\},$$4

which is presented as a robust proxy for low-rank structure. A natural structural gap is $$T_{\mathrm{cls}}(\epsilon):=\inf\{t\ge 0:L_{\mathrm{CE}}(A_t)\le \epsilon\},$$5 or a normalized singular-value tail.

The central theoretical separation is rate-theoretic. Under post-margin gap growth,

$$T_{\mathrm{cls}}(\epsilon):=\inf\{t\ge 0:L_{\mathrm{CE}}(A_t)\le \epsilon\},$$6

or under one-step tail contraction,

$$T_{\mathrm{cls}}(\epsilon):=\inf\{t\ge 0:L_{\mathrm{CE}}(A_t)\le \epsilon\},$$7

the classifier clock satisfies a logarithmic bound:

$$T_{\mathrm{cls}}(\epsilon):=\inf\{t\ge 0:L_{\mathrm{CE}}(A_t)\le \epsilon\},$$8

By contrast, under a sharp late-time Kurdyka–Łojasiewicz tail near $$T_{\mathrm{cls}}(\epsilon):=\inf\{t\ge 0:L_{\mathrm{CE}}(A_t)\le \epsilon\},$$9,

$$T_{\mathrm{rep}}(\eta):=\inf\{t\ge 0:S(A_t)\le \eta\},$$0

with $$T_{\mathrm{rep}}(\eta):=\inf\{t\ge 0:S(A_t)\le \eta\},$$1, the structural energy decays polynomially:

$$T_{\mathrm{rep}}(\eta):=\inf\{t\ge 0:S(A_t)\le \eta\},$$2

and if $$T_{\mathrm{rep}}(\eta):=\inf\{t\ge 0:S(A_t)\le \eta\},$$3 is comparable to $$T_{\mathrm{rep}}(\eta):=\inf\{t\ge 0:S(A_t)\le \eta\},$$4, then

$$T_{\mathrm{rep}}(\eta):=\inf\{t\ge 0:S(A_t)\le \eta\},$$5

The paper’s comparison theorem makes the mismatch explicit: if $$T_{\mathrm{rep}}(\eta):=\inf\{t\ge 0:S(A_t)\le \eta\},$$6 for some $$T_{\mathrm{rep}}(\eta):=\inf\{t\ge 0:S(A_t)\le \eta\},$$7, then

$$T_{\mathrm{rep}}(\eta):=\inf\{t\ge 0:S(A_t)\le \eta\},$$8

while $$T_{\mathrm{rep}}(\eta):=\inf\{t\ge 0:S(A_t)\le \eta\},$$9 (Tan et al., 4 Jun 2026).

This establishes an asymptotic distinction between “fitting” and “simplification.” In this framework, late generalization is not attributed merely to delayed optimization of the empirical objective, but to the slower decay of a structural energy associated with rule-aligned geometry.

3. Spectral regularization, deep linear networks, and conditional ReLU transfer

The deep linear core of the grokking theory derives its slow clock from the interaction between layerwise weight decay and end-to-end spectral bias. For a depth-$L_{\mathrm{CE}}(A_t)$0 deep linear network with

$L_{\mathrm{CE}}(A_t)$1

trained with cross-entropy loss and layerwise $L_{\mathrm{CE}}(A_t)$2 weight decay, the paper uses the equivalence

$L_{\mathrm{CE}}(A_t)$3

Thus layerwise decay becomes an end-to-end Schatten-$L_{\mathrm{CE}}(A_t)$4 quasi-norm penalty (Tan et al., 4 Jun 2026). For deeper $L_{\mathrm{CE}}(A_t)$5, the induced exponent satisfies $L_{\mathrm{CE}}(A_t)$6, which intensifies shrinkage of small singular values.

On a smooth spectral stratum where $L_{\mathrm{CE}}(A_t)$7 is simple, the local singular-value dynamics obey

$L_{\mathrm{CE}}(A_t)$8

For $L_{\mathrm{CE}}(A_t)$9, the factor $S(A_t)$0 diverges as $S(A_t)$1, so weak singular directions experience stronger shrinkage. The paper presents this as the mechanism by which the spectrum concentrates and effective dimension falls, with the limiting stable rank satisfying

$S(A_t)$2

where $S(A_t)$3 is the number of nonzero singular values of the limit (Tan et al., 4 Jun 2026).

The same paper then provides a conditional transfer of this mechanism to ReLU multilayer perceptrons. If the activation masks on the training set remain fixed over an interval $S(A_t)$4, with

$S(A_t)$5

then for each training sample $S(A_t)$6,

$S(A_t)$7

In a fixed-activation region, the nonlinear model therefore reduces to an active linear subsystem on the training set.

For a two-layer ReLU embedding model

$S(A_t)$8

the sample-wise gradients are

$S(A_t)$9

with bounds

$\tau_{\mathrm{rule}}$0

while

$\tau_{\mathrm{rule}}$1

Under controlled downstream norms and nonvanishing hidden features, this supports a head-first training regime in which the classifier head moves faster than the embedding block (Tan et al., 4 Jun 2026). The paper states this only conditionally and does not claim a global proof for nonlinear training dynamics. That caveat is important: the two-stage ReLU story is framed as an explanatory reduction compatible with observed behavior, not as a universal theorem.

4. Generative models: rule clock, memorization clock, and the innovation window

A second machine-learning use of two training clocks appears in “The two clocks and the innovation window: When and how generative models learn rules” (Wang et al., 11 May 2026). The paper studies synthetic rule-governed distributions and distinguishes a rule-learning clock $\tau_{\mathrm{rule}}$2 from a memorization clock $\tau_{\mathrm{rule}}$3. The first is operationally defined as the first training step at which sample-level rule accuracy exceeds $\tau_{\mathrm{rule}}$4, sustained beyond transient spikes; the second is the first training step at which the sample-level memorization ratio exceeds $\tau_{\mathrm{rule}}$5, with an adaptive threshold $\tau_{\mathrm{rule}}$6 used in scaling fits.

The main testbed is group parity in a $\tau_{\mathrm{rule}}$7 dimensional Boolean space reshaped to $\tau_{\mathrm{rule}}$8 images. The image is partitioned into $\tau_{\mathrm{rule}}$9 groups of size $\tau_{\mathrm{mem}}$0, each of which must have even parity. Samples are generated in $\tau_{\mathrm{mem}}$1, then assessed for closeness to the Boolean cube via

$\tau_{\mathrm{mem}}$2

with invalid samples declared by thresholding at $\tau_{\mathrm{mem}}$3. After binarization to $\tau_{\mathrm{mem}}$4, parity accuracy is measured at group and sample levels. Sample-level chance is $\tau_{\mathrm{mem}}$5.

Memorization is defined by exact equality to a training sample after binarization. To separate genuine novelty from trivial overlap, the paper compares against the expected sample-level memorization ratio under uniform sampling from the parity-constrained support and under uniform sampling from the full Boolean cube, and constructs held-out valid-novel sets disjoint from the training set by Hamming distance at least $\tau_{\mathrm{mem}}$6 (Wang et al., 11 May 2026).

The empirical scaling laws define the two-clock structure sharply. For DiT-mini, the memorization clock satisfies

$\tau_{\mathrm{mem}}$7

and for GPT-mini,

$\tau_{\mathrm{mem}}$8

while GPT-B yields

$\tau_{\mathrm{mem}}$9

These fits are described as near-linear in dataset size $[\tau_{\mathrm{rule}},\tau_{\mathrm{mem}}]$0 with architecture-dependent prefactors. Memorization trajectories collapse when plotted against steps-per-sample, with onset around $[\tau_{\mathrm{rule}},\tau_{\mathrm{mem}}]$1 steps-per-example.

The rule clock depends instead on rule complexity and capacity. For DiT-mini, parity with $[\tau_{\mathrm{rule}},\tau_{\mathrm{mem}}]$2 is learned at approximately $[\tau_{\mathrm{rule}},\tau_{\mathrm{mem}}]$3 steps, $[\tau_{\mathrm{rule}},\tau_{\mathrm{mem}}]$4 at approximately $[\tau_{\mathrm{rule}},\tau_{\mathrm{mem}}]$5, and $[\tau_{\mathrm{rule}},\tau_{\mathrm{mem}}]$6 at approximately $[\tau_{\mathrm{rule}},\tau_{\mathrm{mem}}]$7; for $[\tau_{\mathrm{rule}},\tau_{\mathrm{mem}}]$8, the transition becomes unreliable or collides with memorization within $[\tau_{\mathrm{rule}},\tau_{\mathrm{mem}}]$9 steps. Increasing model capacity moves $\alpha_t$0 earlier, but the paper notes that even the largest $\alpha_t$1-layer DiT fails to learn $\alpha_t$2 within $\alpha_t$3 steps (Wang et al., 11 May 2026).

The interval

$\alpha_t$4

is called the innovation window. It widens with increasing $\alpha_t$5, narrows with rule complexity $\alpha_t$6, and may vanish when $\alpha_t$7. This formulation parallels the grokking account: a fast phase associated with acquiring a coarse rule-compatible behavior is followed by a slower phase associated with training-set attraction and exact memorization. The paper also reports the same two-clock structure beyond parity, including exact-$\alpha_t$8, row-$\alpha_t$9, Latin squares, and $\tau_{\mathrm{mem}}$00 Sudoku.

5. Clocks as update indices in differential temporal-difference learning

In reinforcement learning, “two training clocks” designates a different but structurally analogous distinction. “On the Divergence of Differential Temporal Difference Learning without Local Clocks” separates a global clock from a local clock in the step-size schedule of temporal-difference updates (Antrobius et al., 7 May 2026). The global clock uses $\tau_{\mathrm{mem}}$01; the local clock uses $\tau_{\mathrm{mem}}$02, where the per-state visit count $\tau_{\mathrm{mem}}$03 indexes the base sequence separately for each state.

The paper emphasizes that the correspondence between local and global clocks depends on the problem class. In discounted policy evaluation, for fixed target policy $\tau_{\mathrm{mem}}$04 and discount $\tau_{\mathrm{mem}}$05, the tabular off-policy TD(0) error is

$\tau_{\mathrm{mem}}$06

with importance ratio

$\tau_{\mathrm{mem}}$07

The associated ODEs are

$\tau_{\mathrm{mem}}$08

and

$\tau_{\mathrm{mem}}$09

Because $\tau_{\mathrm{mem}}$10 is a nonsingular $\tau_{\mathrm{mem}}$11-matrix and left multiplication by a positive diagonal matrix preserves positive stability, both systems are globally asymptotically stable.

The average-reward setting is different. Let $\tau_{\mathrm{mem}}$12 be the average reward and $\tau_{\mathrm{mem}}$13 the bias function satisfying

$\tau_{\mathrm{mem}}$14

Differential TD evaluates $\tau_{\mathrm{mem}}$15 and $\tau_{\mathrm{mem}}$16 simultaneously using

$\tau_{\mathrm{mem}}$17

With global clock,

$\tau_{\mathrm{mem}}$18

and with local clock,

$\tau_{\mathrm{mem}}$19

After eliminating the fast component of $\tau_{\mathrm{mem}}$20, the limiting ODEs are

$\tau_{\mathrm{mem}}$21

and

$\tau_{\mathrm{mem}}$22

The matrix $\tau_{\mathrm{mem}}$23 is positive stable for all $\tau_{\mathrm{mem}}$24, but $\tau_{\mathrm{mem}}$25 need not be (Antrobius et al., 7 May 2026).

The paper’s counterexample demonstrates that global-clock DTD may diverge in average-reward RL even though the local-clock version converges for all $\tau_{\mathrm{mem}}$26. For a family of MDPs indexed by integer $\tau_{\mathrm{mem}}$27, it constructs a positive stationary distribution with

$\tau_{\mathrm{mem}}$28

and derives the exact stability region

$\tau_{\mathrm{mem}}$29

where

$\tau_{\mathrm{mem}}$30

For $\tau_{\mathrm{mem}}$31, $\tau_{\mathrm{mem}}$32, so any $\tau_{\mathrm{mem}}$33 makes the global-clock ODE unstable. The underlying $\tau_{\mathrm{mem}}$34-dimensional block is governed by the Hurwitz polynomial

$\tau_{\mathrm{mem}}$35

which is Hurwitz iff $\tau_{\mathrm{mem}}$36 (Antrobius et al., 7 May 2026).

This usage of two clocks is conceptually distinct from the grokking and generative-model formulations, yet the shared theme remains temporal asymmetry. A plausible implication is that local clocks act as a normalization of update speed across states, whereas global clocks encode visitation frequencies directly into the mean dynamics.

6. Experimental observables, diagnostics, and recurring motifs

Across the machine-learning usages, two-clock formulations are empirically tied to measurable observables rather than to latent phases inferred post hoc. In grokking experiments on modular addition modulo prime $\tau_{\mathrm{mem}}$37, with main figures using $\tau_{\mathrm{mem}}$38, the classifier clock is tracked by the training loss $\tau_{\mathrm{mem}}$39 and the representation clock by stable rank $\tau_{\mathrm{mem}}$40 or spectral tail statistics of the learned map (Tan et al., 4 Jun 2026). The observed pattern is that training loss drops early across weight-decay settings, test loss improves much later, and stable rank decreases late in step with test-loss improvement. The architecture used is a symmetric ReLU MLP with trainable token embeddings of dimension $\tau_{\mathrm{mem}}$41, trained with Adam at learning rate $\tau_{\mathrm{mem}}$42, betas $\tau_{\mathrm{mem}}$43, full-batch training for $\tau_{\mathrm{mem}}$44 epochs, and weight decay swept over $\tau_{\mathrm{mem}}$45.

In the generative-model setting, the observables are sample-level rule accuracy, sample-level memorization ratio, invalid fraction measured via $\tau_{\mathrm{mem}}$46, and auxiliary diagnostics such as nearest-neighbor Hamming distance to the training set and DSM loss by noise scale (Wang et al., 11 May 2026). The paper reports that DSM loss splits precede the sample-level transitions and are concentrated at intermediate scales. For DiT, the critical ranges first appear near $\tau_{\mathrm{mem}}$47 and propagate to adjacent scales; for GPT, per-position cross-entropy collapses first at positions $\tau_{\mathrm{mem}}$48, reflecting the deterministic last bit of each parity group once the rule is learned.

A notable shared motif is that the fast clock is associated with coarse compatibility with the task objective, while the slow clock is associated with geometry. In the grokking theory, geometry is spectral and low-rank, expressed via singular-value shrinkage and stable rank. In the generative-model study, geometry is basin structure in the denoising vector field: around $\tau_{\mathrm{mem}}$49, basins of rule-valid vertices expand, whereas around $\tau_{\mathrm{mem}}$50, basins of training samples begin to dominate (Wang et al., 11 May 2026). In both cases, the later clock marks a refinement of the learned representation or energy landscape after an earlier behavioral transition has already occurred.

Another recurring motif is that optimization hyperparameters modulate the gap between clocks. In grokking, layerwise $\tau_{\mathrm{mem}}$51 decay induces the Schatten-$\tau_{\mathrm{mem}}$52 penalty and thus drives late simplification (Tan et al., 4 Jun 2026). In generative models, increasing dataset size pushes memorization later approximately linearly in $\tau_{\mathrm{mem}}$53, while GPT weight decay in the explored sweep “substantially delays $\tau_{\mathrm{mem}}$54” and moderate learning rates are most favorable for DiT (Wang et al., 11 May 2026). These observations suggest that two-clock formulations provide not merely descriptive diagnostics but also levers for intervention.

7. Other uses of the phrase and limits of unification

Outside machine learning, “Two Training Clocks” has appeared in two unrelated senses. In gravitational redshift metrology, it denotes a test of Local Position Invariance using two Earth-orbiting stable atomic clocks on synchronized eccentric orbits, $\tau_{\mathrm{mem}}$55 out of phase, linked by a coherent space-to-space link with Doppler compensation (Litvinov et al., 2021). The signal model is

$\tau_{\mathrm{mem}}$56

with residual fit

$\tau_{\mathrm{mem}}$57

The optimal configuration has perigee altitude approximately $\tau_{\mathrm{mem}}$58 and orbital period $\tau_{\mathrm{mem}}$59--$\tau_{\mathrm{mem}}$60. Reported accuracies after $\tau_{\mathrm{mem}}$61 years are $\tau_{\mathrm{mem}}$62 for the VCH-1010 hydrogen maser, $\tau_{\mathrm{mem}}$63 for the PHARAO cesium fountain, and $\tau_{\mathrm{mem}}$64 for a future optical clock (Litvinov et al., 2021). Here the clocks are physical instruments, not phases of training.

In Mars horology, the phrase refers to Earth-day “training clocks” that teach the use of public Mars clocks while preserving the SI second (Flesch et al., 11 Jul 2025). The paper identifies two such clocks: a $\tau_{\mathrm{mem}}$65-hour clock with standard analog motion and a $\tau_{\mathrm{mem}}$66-hour “Martian” clock with convergent hand motion. The operational Mars day is taken as

$\tau_{\mathrm{mem}}$67

while the true sol is

$\tau_{\mathrm{mem}}$68

For the $\tau_{\mathrm{mem}}$69-hour design, the Earth training version uses $\tau_{\mathrm{mem}}$70 and the Mars version uses $\tau_{\mathrm{mem}}$71. For the $\tau_{\mathrm{mem}}$72-hour Martian design, the hand ratios are $\tau_{\mathrm{mem}}$73 in the $\tau_{\mathrm{mem}}$74-minute variant or $\tau_{\mathrm{mem}}$75 in the relaxed $\tau_{\mathrm{mem}}$76-minute variant, with the defining property that at each exact hour all three hands converge on the hour mark (Flesch et al., 11 Jul 2025). This usage is pedagogical and mechanical rather than algorithmic.

These non-ML usages place a limit on any attempt to unify the phrase too strongly. The common wording does not imply a common theory. What is shared is only the structural decision to introduce two clocks because a single temporal description is insufficient for the problem at hand. In machine learning, this insufficiency concerns distinct rates of optimization, generalization, or memorization. In relativistic metrology and Mars timekeeping, it concerns measurement geometry and human training, respectively.