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Imagine a model choosing its next word, with several
continuations that fit the same context.

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Our first bar is the observed label; the others are alternatives
in this original teaching example.

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A training example records one continuation, without proving
that every other plausible continuation is wrong.

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Imagine an update pushing nearly all probability onto the label,
making the alternatives difficult to sample.

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These probabilities are illustrative, but increasing target
confidence can also erase other useful possibilities.

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Can supervised adaptation learn the observed answer while
preserving useful relationships among the other possibilities?

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The paper studies pretrained distributions, rather than treating
every non-label token as uninformative tail noise.

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Entropy analysis finds comparable candidate probabilities,
corresponding to different effective support sizes.

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Shannon and Renyi entropies describe effective support size and
how close the normalized profile is to uniform.

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Small groups can reflect semantic branching; larger groups can
reflect structural ambiguity among plausible digits.

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Reported patterns span instruction, code, and mathematics
datasets across different model families and architectures.

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The goal preserves plausible alternative structure, while
allowing their absolute probabilities to change.

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LP-SFT selects local candidates from the frozen base model,
using its highest ranked next-token possibilities.

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It removes the supervised label, leaving only non-label
candidates as the explicit objects of preservation.

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Cross-entropy pushes the label upward; preserving that same
label could directly oppose its supervision.

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The target learns from the example, while the preservation term
protects relationships among the alternatives.

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The default uses ten base candidates, covering several plausible
regimes without a separate per-token schedule.

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This is a training objective, rather than an inference-time
requirement to filter generated words.

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Return to the toy distribution, with an update raising the label
while retaining alternative proportions.

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The alternatives start at thirty, twenty, and ten percent:
relative proportions of three to two to one.

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Their updated probabilities are fifteen, ten, and five percent,
so their combined mass falls by half.

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Inside the non-label set, the relative profile remains;
meanwhile, the supervised label gains probability.

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Local normalization compares the alternative set shape,
independently of the total probability assigned to it.

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These ratios illustrate the goal; real training encourages
preservation through a loss, rather than invented values.

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The objective combines target cross-entropy with KL over locally
normalized non-label candidate distributions.

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Cross-entropy learns observed response tokens from supervised
instruction and response examples.

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The second term uses fixed base information to protect relative
preferences among selected alternatives.

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Gradient analysis explains why unnormalized matching instead
pushes more probability mass toward the alternatives.

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Local normalization removes that pressure; target exclusion
prevents direct conflict with the label supervision.

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The frozen base is a local structural reference, rather than a
stronger teacher producing new answers.

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A naive implementation would keep both models in memory and run
the frozen base at each step.

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LP-SFT uses the fact that its base model and training examples
are fixed before optimization begins.

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An offline pass computes base candidate logits and caches the
information needed for local preservation sets.

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Fine-tuning compares current model logits with that cache,
restricting the comparison to relevant non-label candidates.

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This avoids an extra frozen-model forward pass at each step
while retaining the local base preferences.

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Cache construction remains a separate, one-time preparation cost
alongside the ongoing fine-tuning computation.

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Table one uses Qwen version three, with fourteen billion
parameters.

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It is fine-tuned on mixed mathematics, code, and general
instruction data.

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The aggregate averages displayed single-sample, finite-budget,
and general knowledge metrics.

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The base scores sixty-six point seven nine; cross-entropy
fine-tuning reaches fifty-eight point five five.

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LP-SFT reaches sixty-five point three two, reducing the gap
while remaining below this configuration base average.

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MMLU accuracy likewise stays closer to the base under LP-SFT
than cross-entropy in this matched comparison.

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These scoped retention results use a mean of displayed metrics,
rather than pooled benchmark task accuracy.

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Separate the supervised lesson from the alternative structure
that the pretrained model already considers plausible.

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Target exclusion protects supervision; local normalization
preserves relationships without fixing alternative mass.

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These choices strengthen the label while encouraging non-label
preferences to survive domain adaptation.

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Evaluation tests one sampled answer and access to correct
solutions within a finite sampling budget.

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Matched results support preservation, while the objective
specifies which relationships receive protection.

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As the observed answer improves, LP-SFT asks which other
plausible continuations should remain possible after training.