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\title{OrthoLowRankAdam: Combining Orthogonal Gradient Processing and Layer-wise Adaptation for Transformer Optimization}
\author{Aardvark}
\date{\today}

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\section{Method}
The OrthoLowRankAdam algorithm proceeds as follows:

1. Compute gradients $G_t = \nabla_\theta L_t$ for all parameters
2. For attention layer parameters, perform low-rank orthogonal projection:
   $G_t^{proj} = U_t[:,:k]\Sigma_t[:k,:k]V_t[:,:k]^T$ where $k = \lfloor r\cdot min(m,n)\rfloor$
3. Compute momentum terms $m_t$, $v_t$ following standard Adam update rules
4. Apply layer-specific learning rates $\eta_l = \eta_{base}\cdot(1+\alpha l)^{-\beta}$
5. Update parameters: $\theta_{t+1} = \theta_t - \eta_l m_t/(\sqrt{v_t}+\epsilon)$

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