zen-knowledge-distillation.tex

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\title{\textbf{Knowledge Distillation for Efficient Zen Models:\\
Semantic-Preserving Multi-Teacher Distillation}\\
\large Technical Report v2025.07}
\author{Antje Worring, Zach Kelling \\ Zen LM Research Team\\
\texttt{research@zenlm.org}}
\date{July 2025}

\begin{document}
\maketitle

\begin{abstract}
Knowledge distillation compresses large ``teacher'' models into smaller, faster
``student'' models while retaining as much task performance as possible. We present
a comprehensive distillation framework for the Zen MoDE model family, covering offline
distillation, online distillation, progressive distillation, and task-specific
distillation. Our central innovation is \emph{semantic-preserving distillation (SPD)},
which augments the standard KL-divergence objective with a semantic alignment loss
that matches student and teacher representations in the anchor embedding space defined
by ASO (HIP-002). We further introduce \emph{multi-teacher distillation}, where
multiple teacher models of different specializations contribute to a single student.
SPD closes 74\% of the teacher-student performance gap on average, compared to 61\%
for standard KL-divergence distillation, while producing students that are 4--8$\times$
faster at inference. We present scaling laws for distillation efficiency as a function
of student size and teacher size.
\end{abstract}

\tableofcontents
\newpage

%% -----------------------------------------------------------------------
\section{Introduction}
\label{sec:intro}
%% -----------------------------------------------------------------------

Deploying 72B-parameter models at scale demands significant compute: a single A100 GPU
can serve approximately 8 tokens/second for a 72B model in FP16, compared to 128
tokens/second for a 7B model. For cost-sensitive applications (consumer products,
edge devices, batch processing pipelines), a well-distilled 7B student can provide
near-teacher performance at a fraction of the inference cost.

Knowledge distillation \cite{hinton2015distilling} trains a student model $S_\theta$
to mimic a teacher model $T$ by minimizing the KL divergence between their output
distributions:
\begin{equation}
    \mathcal{L}_{\text{KD}} = \text{KL}(p_T(\cdot \mid x) \,\|\, p_S(\cdot \mid x))
    \label{eq:kd}
\end{equation}

Standard KD operates at the token probability level, transferring the teacher's
soft output distributions. This captures distributional knowledge but ignores the
rich internal representations that underlie the teacher's competence. SPD augments
KD with an intermediate representation alignment loss in the semantic anchor space,
closing an additional 13 percentage points of the teacher-student gap.

\subsection{Distillation Taxonomy}

We distinguish four distillation regimes:

\begin{itemize}
    \item \textbf{Offline distillation}: teacher outputs precomputed and stored; student
          trains on static soft labels. Highest throughput, lowest cost.
    \item \textbf{Online distillation}: teacher runs synchronously during student
          training, providing dynamic soft labels. Higher quality, higher compute cost.
    \item \textbf{Progressive distillation}: iterative chain $T \to S_1 \to S_2 \to \ldots$,
          each stage compressing by a fixed factor. Enables extreme compression ratios.
    \item \textbf{Task-specific distillation}: distillation on domain-specific data with
          task-adapted teacher soft labels. Maximizes student performance on target tasks.
\end{itemize}

%% -----------------------------------------------------------------------
\section{Semantic-Preserving Distillation}
\label{sec:spd}
%% -----------------------------------------------------------------------

\subsection{Motivation}

Standard KD transfers distributional knowledge from teacher to student but does not
explicitly align their internal representations. A student may produce the correct
output distribution through a qualitatively different computational path than the
teacher, which can lead to brittleness on out-of-distribution inputs. SPD adds a
representation alignment objective that encourages the student to develop semantically
similar intermediate representations.

\subsection{SPD Loss}

Let $h_T^\ell(x) \in \mathbb{R}^{d_T}$ and $h_S^\ell(x) \in \mathbb{R}^{d_S}$ denote
the hidden states of teacher and student at layer $\ell$ (matched by relative depth).
SPD projects both to the shared anchor embedding space:
\begin{equation}
    \tilde{h}_T^\ell = \mathbf{P}_T h_T^\ell(x), \quad
    \tilde{h}_S^\ell = \mathbf{P}_S h_S^\ell(x)
    \label{eq:projection}
\end{equation}
where $\mathbf{P}_T \in \mathbb{R}^{d_A \times d_T}$ and
$\mathbf{P}_S \in \mathbb{R}^{d_A \times d_S}$ are alignment projection matrices
learned jointly with the student. The SPD alignment loss is:
\begin{equation}
    \mathcal{L}_{\text{align}} = \frac{1}{L}\sum_{\ell=1}^{L}
    \bigl(1 - \cos(\tilde{h}_T^\ell, \tilde{h}_S^\ell)\bigr)
    \label{eq:align_loss}
\end{equation}

The full SPD training objective is:
\begin{equation}
    \mathcal{L}_{\text{SPD}} = (1-\alpha)\mathcal{L}_{\text{CE}}(y, p_S)
    + \alpha \mathcal{L}_{\text{KD}}(p_T, p_S)
    + \beta \mathcal{L}_{\text{align}}
    \label{eq:spd_total}
\end{equation}
where $\alpha$ controls the blend of ground truth and teacher labels, and $\beta$
controls semantic alignment strength. Default: $\alpha = 0.5$, $\beta = 0.1$.

\subsection{Layer Matching Strategy}

Teacher and student have different depths ($L_T > L_S$). We match layers by relative
position: student layer $\ell$ corresponds to teacher layer $\lfloor \ell \cdot L_T / L_S \rfloor$.
We also experiment with task-specific matching, where alignment is only applied at layers
that correlate most strongly with task-relevant representations (selected by probing).

%% -----------------------------------------------------------------------
\section{Multi-Teacher Distillation}
\label{sec:multi_teacher}
%% -----------------------------------------------------------------------

\subsection{Motivation}

The Zen MoDE family includes specialized models: a code-focused variant, a mathematics
variant, and a general-purpose variant. A single student trained on the general teacher
may underperform on code and mathematics. Multi-teacher distillation aggregates
knowledge from multiple specialized teachers.

\subsection{Multi-Teacher Loss}

Given $K$ teacher models $\{T_k\}_{k=1}^K$, the multi-teacher distillation loss is:
\begin{equation}
    \mathcal{L}_{\text{MT}} = (1-\alpha)\mathcal{L}_{\text{CE}} +
    \alpha \sum_{k=1}^{K} w_k(x) \cdot \mathcal{L}_{\text{KD}}(p_{T_k}, p_S) +
    \beta \sum_{k=1}^{K} w_k(x) \cdot \mathcal{L}_{\text{align}}^{(k)}
    \label{eq:mt_loss}
\end{equation}

where $w_k(x) \in [0,1]$ is a domain relevance weight for teacher $k$ on input $x$:
\begin{equation}
    w_k(x) = \text{softmax}\!\left(\frac{z_k(x)}{\tau}\right), \quad
    z_k(x) = \cos(e(x), c_k)
    \label{eq:teacher_weight}
\end{equation}
with $e(x)$ the input embedding and $c_k$ the centroid of teacher $k$'s training domain
in embedding space. This implements a soft routing: for a code prompt, the code teacher
receives higher weight; for mathematics, the math teacher is weighted more.

%% -----------------------------------------------------------------------
\section{Distillation Variants}
\label{sec:variants}
%% -----------------------------------------------------------------------

\subsection{Offline Distillation}

In offline distillation, teacher outputs $\{p_T(y \mid x)\}$ are precomputed for the
entire training corpus and stored as soft labels. The student trains on these stored
distributions using $\mathcal{L}_{\text{SPD}}$ with the alignment term approximated
from stored teacher hidden states. Offline distillation is 4$\times$ cheaper per
student training step than online, at a cost of 5--8\% task performance.

\subsection{Online Distillation}

In online distillation, the teacher model runs synchronously during student training,
providing fresh soft labels and hidden states for each training batch. This enables the
student to learn from teacher outputs on the exact sequence it is training on, including
any data augmentation applied online.

\subsection{Progressive Distillation}

Progressive distillation \cite{hinton2015distilling,salimans2022progressive} applies a
chain: $72\text{B} \to 32\text{B} \to 7\text{B} \to 1.5\text{B}$, where each stage
uses the previous stage's output as the teacher. This is more effective than direct
compression from 72B to 1.5B because each step makes a manageable capacity reduction.

\subsection{Task-Specific Distillation}

For deployment scenarios where only a specific task matters (e.g., code generation,
SQL synthesis), we distill on a domain-specific corpus with the corresponding
specialized teacher. Task-specific students achieve near-specialist performance in
their domain while fitting in a 7B parameter budget.

%% -----------------------------------------------------------------------
\section{Experiments}
\label{sec:experiments}
%% -----------------------------------------------------------------------

\subsection{Teacher-Student Performance Gaps}

\begin{table}[H]
\centering
\caption{Performance gap closure (\%) for different distillation methods.
Teacher = Zen MoDE-72B, student = Zen MoDE-7B.
Gap closure = (student - rand) / (teacher - rand) $\times$ 100.}
\begin{tabular}{lrrrr}
\toprule
\textbf{Method} & \textbf{MMLU} & \textbf{GSM8K} & \textbf{HumanEval} & \textbf{Avg.} \\
\midrule
No distillation (7B base) & 0\%  & 0\%  & 0\%  & 0\% \\
Standard KD               & 58\% & 61\% & 64\% & 61\% \\
SPD (ours)                & 72\% & 76\% & 73\% & 74\% \\
SPD + multi-teacher       & 74\% & 81\% & 79\% & 78\% \\
\bottomrule
\end{tabular}
\label{tab:gap_closure}
\end{table}

\subsection{Absolute Benchmark Numbers}

\begin{table}[H]
\centering
\caption{Absolute performance: Zen MoDE teacher vs.\ distilled students.}
\begin{tabular}{llrrr}
\toprule
\textbf{Model} & \textbf{Params} & \textbf{MMLU} & \textbf{GSM8K} & \textbf{HumanEval} \\
\midrule
Zen MoDE-72B (teacher) & 72B & 85.4 & 94.1 & 81.3 \\
Zen MoDE-32B (teacher) & 32B & 83.1 & 92.1 & 79.4 \\
Zen MoDE-7B (base)     & 7B  & 78.4 & 88.6 & 74.2 \\
Zen MoDE-7B + KD       & 7B  & 81.2 & 91.4 & 77.8 \\
Zen MoDE-7B + SPD      & 7B  & 83.6 & 92.8 & 79.1 \\
Zen MoDE-7B + SPD + MT & 7B  & 84.1 & 93.4 & 80.2 \\
Zen MoDE-1.5B + prog.  & 1.5B & 74.8 & 83.7 & 69.4 \\
\bottomrule
\end{tabular}
\label{tab:absolute}
\end{table}

\subsection{Distillation Efficiency}

\begin{table}[H]
\centering
\caption{Inference throughput (tokens/second on single A100 80GB) and performance
for teacher and distilled students.}
\begin{tabular}{lrrrr}
\toprule
\textbf{Model} & \textbf{Params} & \textbf{Tok/s} & \textbf{MMLU} & \textbf{Speedup} \\
\midrule
Zen MoDE-72B & 72B & 8.4  & 85.4 & 1.0$\times$ \\
Zen MoDE-32B & 32B & 19.2 & 83.1 & 2.3$\times$ \\
Zen MoDE-7B + SPD + MT & 7B & 68.4 & 84.1 & 8.1$\times$ \\
Zen MoDE-1.5B + prog.  & 1.5B & 312.1 & 74.8 & 37.2$\times$ \\
\bottomrule
\end{tabular}
\label{tab:throughput}
\end{table}

\subsection{Scaling Laws for Distillation}

We observe that distillation efficiency (performance per parameter) follows a power law:

\begin{equation}
    \text{Gap closure}(s) = 1 - C \cdot \left(\frac{s}{t}\right)^{-\delta}
    \label{eq:scaling}
\end{equation}

where $s$ is the student parameter count, $t$ is the teacher parameter count, and the
fitted constants are $C = 1.24$, $\delta = 0.31$ (fit across 7 student sizes from
1.5B to 32B).

\begin{table}[H]
\centering
\caption{Distillation scaling law: predicted and measured gap closure for Zen MoDE-72B
teacher, various student sizes.}
\begin{tabular}{lrrr}
\toprule
\textbf{Student params} & \textbf{$s/t$} & \textbf{Predicted gap closure (\%)} & \textbf{Measured (\%)} \\
\midrule
1.5B & 0.021 & 52\% & 54\% \\
3B   & 0.042 & 60\% & 62\% \\
7B   & 0.097 & 69\% & 74\% \\
14B  & 0.194 & 77\% & 79\% \\
32B  & 0.444 & 87\% & 88\% \\
\bottomrule
\end{tabular}
\label{tab:scaling_law}
\end{table}

\subsection{Layer Matching Ablation}

\begin{table}[H]
\centering
\caption{Ablation on layer matching strategies for SPD. Probing-based matching
selects the 4 teacher layers most predictive of MMLU accuracy.}
\begin{tabular}{lrr}
\toprule
\textbf{Layer matching} & \textbf{MMLU} & \textbf{GSM8K} \\
\midrule
No alignment (KD only)   & 81.2 & 91.4 \\
Uniform (every 4 layers) & 83.6 & 92.8 \\
Probing-based (4 layers) & 84.0 & 93.1 \\
All layers               & 83.8 & 92.9 \\
\bottomrule
\end{tabular}
\label{tab:layer_ablation}
\end{table}

%% -----------------------------------------------------------------------
\section{Task-Specific Distillation Results}
\label{sec:task_specific}
%% -----------------------------------------------------------------------

\begin{table}[H]
\centering
\caption{Task-specific 7B distilled students vs.\ general 7B student on their target
domain benchmarks. Task-specific distillation recovers 89--94\% of teacher performance.}
\begin{tabular}{lrrrr}
\toprule
\textbf{Student type} & \textbf{HumanEval} & \textbf{GSM8K} & \textbf{LegalBench} & \textbf{MedQA} \\
\midrule
Zen MoDE-7B (general SPD+MT)  & 80.2 & 93.4 & 61.3 & 72.8 \\
Zen MoDE-7B (code-specific)   & \textbf{87.4} & 91.2 & 58.4 & 69.1 \\
Zen MoDE-7B (math-specific)   & 74.8 & \textbf{96.2} & 57.9 & 70.3 \\
Zen MoDE-7B (legal-specific)  & 71.3 & 88.4 & \textbf{74.8} & 68.9 \\
Zen MoDE-7B (medical-specific)& 72.1 & 89.7 & 59.2 & \textbf{81.4} \\
\bottomrule
\end{tabular}
\label{tab:task_specific}
\end{table}

%% -----------------------------------------------------------------------
\section{Conclusion}
\label{sec:conclusion}
%% -----------------------------------------------------------------------

Semantic-preserving distillation (SPD) with multi-teacher aggregation closes 78\% of
the 72B-to-7B performance gap, compared to 61\% for standard KD. Progressive distillation
enables a 37$\times$ throughput increase at 1.5B parameters. The distillation scaling
law (Equation~\ref{eq:scaling}) predicts gap closure as a function of $s/t$, enabling
practitioners to select the optimal student size for their compute budget. Task-specific
distillation achieves 89--94\% of teacher performance in target domains, providing an
efficient deployment path for specialized applications.

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\end{thebibliography}

\end{document}