# Perspective Proximal Fine-Tuning Solver

- ID: 1274
- Canonical URL: https://synthcore.org/idea/1274/perspective-proximal-fine-tuning-solver
- API JSON: https://synthcore.org/api/idea/1274.json
- API Markdown: https://synthcore.org/api/idea/1274.md
- Verification status: unverified
- Source: [arXiv:2607.22278](https://arxiv.org/abs/2607.22278)
- Category: optimization
- Solves: stability, sample-efficiency, accuracy
- ML areas: optimizer, fine-tuning, training
- Math tags: convex-analysis, optimization, dynamical-systems
- Ratings: usefulness 5/10; difficulty 6/10; novelty 7/10

## Idea description

Replace ordinary projected-gradient updates for a convex neural subproblem with a homogeneous perspective formulation and Douglas-Rachford splitting. The additional scale variable makes the update less sensitive to large variations in loss or parameter scale and can expose infeasible combinations of constraints instead of producing unstable iterates.

## Mathematical statement

The paper uses the perspective of a convex function f, defined for p>0 by p f(x/p), and represents a convex constraint f_i(x)\le 0 through the homogeneous inequality p f_i(x/p)\le q together with p\ge 0; fixing p=1 and q=0 recovers the original constraint. Here x is a parameter vector, p is a nonnegative homogenizing scalar, q is a slack or certificate coordinate, and f_i is a proper lower-semicontinuous convex function. For a convex split objective F(x)=f(x)+g(x), Douglas-Rachford uses A=\partial f and B=\partial g with iteration z^{k+1}=z^k+\operatorname{prox}_{\gamma A}(2\operatorname{prox}_{\gamma B}(z^k)-z^k)-\operatorname{prox}_{\gamma B}(z^k), where \gamma>0 is the step size, z is the splitting state, and \operatorname{prox}_{\gamma h}(v)=\arg\min_u[h(u)+(2\gamma)^{-1}\|u-v\|_2^2]. The paper's structural result is that projection onto the perspective embedding reduces to solving a scalar equation and evaluating proximity operators of functions from the original problem. The adaptation applies this to a convex neural head or adapter, with f as the data-fit loss and g as a regularizer or constraint indicator; the final parameter is x/p when p>0.

## Key formulas

- $$p f_i(x/p)\le q,\qquad p\ge 0,$$
- $$p=1,\qquad q=0,$$
- $$z^{k+1}=z^k+\operatorname{prox}_{\gamma A}\!\left(2\operatorname{prox}_{\gamma B}(z^k)-z^k\right)-\operatorname{prox}_{\gamma B}(z^k),\qquad A=\partial f,\;B=\partial g,$$
- $$x_{\mathrm{new}}=\tilde{x}/\tilde{p}\quad\text{when }\tilde{p}>0.$$

## Implementation notes

Integrate this as an inner optimizer for a convex parameter block, rather than replacing backpropagation through all nonlinear network weights. A minimal target is a frozen transformer with a trainable linear classification head, linear probe, or low-dimensional convex adapter. Let w be the trainable vector, let f(w)=(1/N)\sum_i \ell(y_i,\phi_i^    op w) be logistic or squared data-fit loss, and let g(w) be an L2, group-L1, or box-constraint indicator. At each outer epoch, form the fixed feature matrix phi and run a perspective Douglas-Rachford inner solve. Pseudocode: initialize the splitting state z and p=1; repeat u=prox_{gamma g}(z); set v=2u-z; compute (x_tilde,p_tilde) by projecting v onto the perspective embedding; solve the projection's scalar p equation with safeguarded Newton or bisection; evaluate prox_{gamma f} for the parameter component; update z <- z + (x_tilde,p_tilde)-u; stop on primal and dual residuals; return w=x_tilde/p_tilde if p_tilde>epsilon. For squared loss, implement the proximal step with a linear solve; for logistic loss, use 3-5 Newton steps or a one-dimensional separable prox approximation. Estimate the scalar root numerically, while computing residuals and certificates directly from the embedding. Track p, primal residual, dual residual, objective gap, and whether p approaches zero with a nonzero normalized certificate. First experiment: train a linear head on frozen MNIST or CIFAR-10 features with box constraints, comparing projected gradient, projected Adam, and ordinary Douglas-Rachford at equal matrix-vector FLOPs. Success means fewer iterations to a fixed validation loss, lower sensitivity when multiplying the loss by 10 or 100, and correct detection of deliberately contradictory constraints. A negative result is no improvement over ordinary proximal splitting or frequent collapse of p toward zero on feasible problems.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.