Tikhonov-Extragradient Bilevel Optimizer

Integrate this at the inner optimizer for a bilevel task. Use a small pretrained encoder with a trainable linear classifier or LoRA adapter as $x$; define $F(x)=\nabla_xL_{\mathrm{train}}(x)$ and $H(x)=L_{\mathrm{val}}(x)$, preferably with the trainable block isolated so the inner objective is close to convex or locally monotone. At iteration $k$, compute a training gradient $g=\nabla_xL_{\mathrm{train}}(x)$ and validation gradient $v=\nabla_xL_{\mathrm{val}}(x)$, form $r=g+\lambda_kv$, and create look-ahead parameters $y=x-\eta r$. Recompute both gradients at $y$: $g_y=\nabla_yL_{\mathrm{train}}(y)$ and $v_y=\nabla_yL_{\mathrm{val}}(y)$, then update $x\leftarrow x-\eta(g_y+\lambda_kv_y)$. Use $\lambda_k=\lambda_0(k+1)^{-p}$, gradient clipping, and stop-gradient through $y$ for the first implementation. The mathematical quantities supplied by the method are the regularized operator and extragradient recursion; estimate approximate monotonicity empirically using random pairs, via $\langle F(x)-F(z),x-z\rangle/\|x-z\|^2$. Compare against SGD, AdamW, and ordinary extragradient at equal gradient evaluations on a frozen-feature classifier or LoRA model on AG News. Track validation loss, iterate displacement, gradient-angle oscillation, and final last-iterate accuracy. Success means reduced oscillation and better final validation accuracy at equal compute, with a target of 20% lower validation-loss variance or faster last-iterate convergence than SGD.