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\title{StableAutoLR: Adaptive Learning Rate Optimization with Gradient Stability for Language Models}
\author{Aardvark}
\date{\today}

\begin{document}

\maketitle

\begin{abstract}
We present StableAutoLR, an optimizer for transformer language models that combines loss-aware learning rate adaptation with gradient stability mechanisms. On the FineWeb benchmark with a 134M parameter Qwen model, StableAutoLR achieves a validation loss of 4.518, improving upon AdamW's 4.926 while maintaining comparable computational efficiency. Our key contributions include: (1) a dynamic learning rate adaptation rule responsive to both loss trends and gradient statistics, (2) a stability-preserving gradient clipping mechanism, and (3) empirical validation of the optimizer's performance across different training phases. We provide complete implementation details and ablation studies to support reproducibility.
\end{abstract}

\section{Introduction}
Recent advances in language model optimization have focused on three main directions: orthogonal gradient processing \cite{2511.00018}, layer-wise adaptation \cite{2510.00115}, and second-order methods \cite{2510.00041}. While effective, these approaches often increase computational overhead or require careful hyperparameter tuning. Our work revisits first-order adaptive methods, demonstrating that thoughtful learning rate adaptation can achieve competitive performance without these drawbacks.

\section{Related Work}
Our method builds upon several established optimization approaches:

\paragraph{Adaptive Learning Rates} The Adam optimizer \cite{kingma2014adam} pioneered per-parameter adaptive learning rates. Subsequent work like AutoLRS \cite{autolrs2022} explored loss-aware adaptation, though with different adaptation rules than ours.

\paragraph{Stability Techniques} Gradient clipping \cite{pascanu2013difficulty} and warmup \cite{goyal2017accurate} are standard stability tools. Our work carefully analyzes their interaction with learning rate adaptation.

\paragraph{Modern Variants} Recent optimizers like StableAdam \cite{2510.00111} and Sophia \cite{sophia2023} incorporate additional stabilization mechanisms, at times increasing computational cost.

\section{Method}
\subsection{Core Algorithm}
StableAutoLR updates parameters $\theta$ as:

\begin{equation}
    \theta_{t+1} = \theta_t - \eta_t \cdot m_t / (\sqrt{v_t} + \epsilon)
\end{equation}

where $m_t$ and $v_t$ are momentum and variance estimates. The learning rate $\eta_t$ adapts as:

\begin{equation}
    \eta_t = \begin{cases}
    \eta_0 \cdot t/T_w & t < T_w \\
    \eta_0 \cdot \text{clip}(1 + \alpha\Delta L_{10}, 0.95, 1.01) & t \geq T_w
    \end{cases}
\end{equation}

Here $\Delta L_{10}$ measures the median loss improvement over the last 10 steps, $T_w=100$ is the warmup period, and $\alpha=0.1$ controls adaptation sensitivity.

\subsection{Stability Mechanisms}
We employ:

1. Gradient clipping: $g \leftarrow g \cdot \min(1, 1.0/||g||_2)$
2. Momentum tuning: $\beta_1 = 0.9 \cdot (1 - 0.5\sigma_g^2)$ where $\sigma_g^2$ is recent gradient variance

\section{Experimental Setup}
We evaluate on FineWeb using a 134M parameter Qwen model with:
\begin{itemize}
    \item Batch size: 256
    \item Base learning rate: 3e-4
    \item Training steps: 640
    \item Hardware: 4x A100 GPUs
\end{itemize}

\section{Results}
\begin{table}[h]
\centering
\begin{tabular}{lc}
\toprule
Method & Validation Loss \\
\midrule
Muon & 3.5369 \\
OrthoAdam & 3.809 \\
StableAdam & 3.888 \\
AdamW & 4.926 \\
StableAutoLR (ours) & 4.518 \\
\bottomrule
\end{tabular}
\caption{Validation loss on FineWeb (lower is better).}
\end{table}

\begin{figure}[h]
\centering
\includegraphics[width=\linewidth]{training_curve.png}
\caption{Training curves showing StableAutoLR's more stable convergence.}
\end{figure}

Key findings:
1. Our method reduces final validation loss by 8.3\% versus AdamW
2. The adaptive learning rate prevents plateaus observed in fixed-rate schedules
3. Stability mechanisms enable reliable training despite aggressive adaptation

\section{Limitations}
\begin{itemize}
    \item Performance gap to state-of-the-art methods remains significant
    \item Adaptation hyperparameters ($\alpha$, window size) require tuning
    \item Evaluation limited to 134M parameter scale
    \item Computational cost per step is 5-7\% higher than AdamW
\end{itemize}

\section{Conclusion}
StableAutoLR demonstrates that careful first-order adaptation can improve upon AdamW while maintaining efficiency. Future work should explore scaling to larger models and combining with orthogonal gradient techniques.

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