Projective Pareto Continuation for Multi-Task Training
Implementation & benchmark of arXiv:2607.02803 — Singularities in Multi-Objective Optimization and their Crossing during Continuation
Source paper: Singularities in Multi-Objective Optimization and their Crossing during Continuation arXiv:2607.02803 ⓘ · analyzed Aug 30, 2026
AI-generated research hypothesis, automatically tested. Not peer-reviewed.
Idea description
Replace repeated multi-task training runs at different loss weights with pseudo-arclength continuation over stationary solutions of the weighted objective. Use homogeneous objective weights so that the algorithm can cross points where the conventional ratio of task weights diverges, then store the resulting network checkpoints as an approximate Pareto set.
Formulas
Mathematical statement
The paper defines Pareto-critical points by the KKT condition $$\sum_{i=1}^{m}\lambda_i\nabla f_i(\bm{x}^{\star})=\bm{0},\qquad \lambda_i\geq0,\qquad \sum_{i=1}^{m}\lambda_i=1.$$ For two losses L_1(theta) and L_2(theta), use the equivalent homogeneous stationarity system $$F(\theta,a,b)=a\nabla_\theta L_1(\theta)+b\nabla_\theta L_2(\theta)=0,\qquad a^2+b^2=1,\quad a,b\geq0.$$ Here theta is the trainable parameter vector, a and b are homogeneous nonnegative objective weights, and F is a vector residual. In the chart a greater than zero, the ordinary scalarization ratio is r=b/a. In the chart b greater than zero, use q=a/b and solve $$q\nabla_\theta L_1(\theta)+\nabla_\theta L_2(\theta)=0.$$ Thus r can diverge without the projective point becoming singular. Pseudo-arclength continuation adds $$((z-z_k)^Tt_k)-\Delta s=0,$$ where z=(theta,a,b), z_k is the previous solution, t_k is the normalized tangent to the solution branch, and Delta s is the continuation step. The tangent is estimated from the null space of the Jacobian of the augmented residual. The transferable property is that projective coordinates compactify the weight ratio while pseudo-arclength parameterizes the branch by arc length rather than by the scalarization weight.
Implementation notes
Integrate this as an offline optimizer for a two-task model, initially using the shared trunk and task heads of a small MLP. Let L_1(theta) and L_2(theta) be full-batch task losses, and let z=(theta,a,b). First obtain a converged solution for a chosen weight pair, such as (a,b)=(1,0) or (1/sqrt(2),1/sqrt(2)). At each continuation step, compute task gradients g_1=grad_theta L_1 and g_2=grad_theta L_2, then form the residual H(z)=[a*g_1+b*g_2, a^2+b^2-1]. Estimate the tangent t_k using the smallest right singular vector of the Jacobian D_z H(z_k); obtain Jacobian-vector products with automatic differentiation or finite differences. Predict z_pred=z_k+Delta_s*t_k. Correct this prediction with 3 to 8 damped Newton or Gauss-Newton iterations on H(z)=0 together with the pseudo-arclength equation (z-z_k)^T*t_k-Delta_s=0. Use Levenberg-Marquardt damping or a trust region because neural-network Hessians can be ill-conditioned. Do not divide by a or b inside the solver. For logging, report r=b/a when a is above 10^-4 and q=a/b otherwise. Keep only a,b greater than or equal to zero for the standard Pareto-weight quadrant. The mathematical quantities computed exactly are the homogeneous stationarity residual, circle normalization, projective chart representation, and arclength constraint; Jacobians, damping, and step sizes are numerical approximations. Begin with deterministic full-batch losses, then test minibatch noise. The first experiment should use a two-task regression or MNIST/Fashion-MNIST shared MLP with 1,000 to 10,000 trainable parameters. Compare one continuation run against 20 independent AdamW runs at fixed ratios r in {10^-3,...,10^3}, using equal total gradient evaluations. Measure the number of nondominated checkpoints, hypervolume of the two-loss front, and whether continuation passes through regions where fixed-ratio continuation fails. Success means a better loss-front hypervolume or more nondominated solutions at equal compute, with stable behavior when r becomes numerically very large.
Verification
Stage 1 · Toy mechanism gate: Failed ✗
Stage 2 · Mechanism transferred to benchmark: Not tested
Stage 2 · Practical benchmark result: Not run
Methodology: Toy-system gate first; the benchmark stage runs only after a pass. How verification works
Stage 1 — Mechanism check agent confidence 9/10
Built a deterministic pseudo-arclength continuation solver for the homogeneous two-task stationarity system, including SVD tangent estimation, damped Gauss-Newton correction, projective weights, Pareto filtering, and hypervolume comparison. The core math check passed: H norm 2.29e-16, tangent residual ||Jt|| 1.34e-16, circle error 2.22e-16, and maximum correction residual 2.72e-16. On this toy quadratic problem continuation produced 31 nondominated checkpoints with hypervolume 0.2042, while 20 fixed-ratio Adam runs produced hypervolume 0.2141, so the promised Pareto-front improvement was not observed.
- Agent confidence
- 9/10
- Baseline
- 20 fixed-ratio Adam runs, ratios 1e-3 to 1e3: 20 nondominated points, hypervolume 0.2141392207, 620 optimizer steps/gradient evaluations
- Idea
- 31 pseudo-arclength checkpoints: 31 nondominated points, hypervolume 0.2041732971, 31 continuation corrections; max observed b/a ratio 6.13, min a 0.1611, min b 0.7071
Limitations: Only a two-parameter deterministic convex quadratic regression problem was tested, not a shared-trunk MLP or minibatch/noisy training. The continuation path was initialized near equal weights and did not approach a truly divergent ratio chart; baseline optimization schedules and compute accounting are only approximate, and no repeated-seed or larger-scale test was performed.
How to run: python3 experiment.py
Verdict computed by deterministic test code from paired-seed statistics — not by the language model.
Artifacts
- experiment.py 5.3 KB View Raw
- report.md 1.6 KB View
- results.json 6.5 KB View Raw