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Vittorio Colao

Publications and source records attributed to Vittorio Colao.

10 recordsLinked to original sources
arXiv2026

Nash-Bargaining HalpernSGD via Limited-Memory MultiLRSGA: A Two-Phase Optimizer for Multi-Objective Learning

We propose NB-HalpernSGD via LM-MultiLRSGA, a two-phase optimizer for multi-objective optimization. The process starts with a competitive optimization phase by applying a limited-memory variant of MultiLRSGA, named LM-MultiLRSGA, designed to reduce the memory footprint of the original optimizer while preserving its rotational correction mechanism. It addresses multi-objective tasks by considering the competitive game associated with the original multi-objective problem, approximating a Nash equilibrium. This point defines a Nash-equilibrium-induced disagreement point: we formulate the Nash bargaining problem associated with the original losses and rewrite its Nash product as a logarithmic minimization surrogate. This minimization problem is solved using HalpernSGD, anchored at the computed competitive reference point. Therefore, the method uses the Nash equilibrium as a principled reference point for the bargaining stage and then moves toward a Pareto-oriented solution of the original multi-objective problem. We validate the proposed optimizer on a PINN-inspired neural model, where it outperforms established multi-objective optimizers, including PCGrad, MultiAdam, and DualConeGD. Finally, while the convergence properties of HalpernSGD have been extensively studied, we discuss the convergence and stability properties of the proposed LM-MultiLRSGA phase.

math.OC↗
arXiv2026

MultiLRSGA: A method for multi-player differentiable games

We propose MultiLRSGA, an $h$-player extension of LRSGA for the computation of stable Nash equilibria in differentiable games. The method originates from the decomposition of the game Jacobian into symmetric and antisymmetric components, which motivates symplectic corrections designed to attenuate the rotational part of the dynamics. In the two-player setting, LRSGA replaces mixed second-order blocks with low-rank secant approximations. The passage to the multi-player case, however, is not a mere blockwise reformulation: the antisymmetric correction is no longer determined by a single pair of cross-interactions, but by a block antisymmetric operator collecting all pairwise couplings among the players. On this basis, we formulate MultiLRSGA by constructing, for each player, a low-rank approximation of the Jacobian of the partial gradient and extracting from it the blocks required to define an approximate antisymmetric correction. Under standard local assumptions around a stable Nash equilibrium, we prove local linear convergence of the method. The key technical ingredient is a lemma controlling the distance between the exact antisymmetric correction and its secant approximation in the $h$-player setting, thereby extending to the multi-player framework the convergence mechanism previously available for LRSGA. The proposed formulation preserves the computational advantages of low-rank symplectic corrections and is naturally suited to numerical validation on differentiable games with explicit payoffs and more than two agents.

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arXiv2026

On the Convergence of HalpernSGD

We study HalpernSGD for minimizing a convex Fr\'echet differentiable function with Lipschitz gradient on a real Hilbert space. Its deterministic orbit converges strongly to the anchor-selected minimizer \(P_S(u)\) under vanishing steps that need not be square summable. For stochastic oracles, weighted summability of the scaled martingale fluctuation and the \(\F_n\)-measurable bias yields almost sure strong convergence, while expected last-iterate bounds require suffix regularity of the error budget. In the unbiased case, for the critical choice \(\varepsilon_n=n^{-1/2}\), \(\alpha_n=(\log n)/n\) eventually, a uniform conditional moment bound of order \(p>4\) gives both conclusions. We also discuss finite-difference and zeroth-order applications for which scaled, rather than raw, noise is the natural quantity.

math.OC↗
arXiv2025

A Multi-Phase Dual-PINN Framework: Soft Boundary-Interior Specialization via Distance-Weighted Priors

Physics-informed neural networks (PINNs) often struggle with multi-scale PDEs featuring sharp gradients and nontrivial boundary conditions, as the physics residual and boundary enforcement compete during optimization. We present a dual-network framework that decomposes the solution as $u = u_{\text{D}} + u_{\text{B}}$, where $u_{\text{D}}$ (domain network) captures interior dynamics and $u_{\text{B}}$ (boundary network) handles near-boundary corrections. Both networks share a unified physics residual while being softly specialized via distance-weighted priors ($w_{\text{bd}} = \exp(-d/\tau)$) that are cosine-annealed during training. Boundary conditions are enforced through an augmented Lagrangian method, eliminating manual penalty tuning. Training proceeds in two phases: Phase~1 uses uniform collocation to establish network roles and stabilize boundary satisfaction; Phase~2 employs focused sampling (e.g. ring sampling near $\partial\Omega$) with annealed role weights to efficiently resolve localized features. We evaluate our model on four benchmarks, including the 1D Fokker-Planck equation, the Laplace equation, the Poisson equation, and the 1D wave equation. Across Laplace and Poisson benchmarks, our method reduces error by $36-90\%$, improves boundary satisfaction by $21-88\%$, and decreases MAE by $2.2-9.3\times$ relative to a single-network PINN. Ablations isolate contributions of (i)~soft boundary-interior specialization, (ii)~annealed role regularization, and (iii)~the two-phase curriculum. The method is simple to implement, adds minimal computational overhead, and broadly applies to PDEs with sharp solutions and complex boundary data.

math.NA↗
arXiv2025

Limited-Memory LRSGA: An Iterative Method for Computing Nash Equilibria in Competitive Optimization Problems

We introduce LMLRSGA, a limited memory variant of Low Rank Symplectic Gradient Adjustment (LRSGA) for differentiable games. It is an iterative scheme for approximating Nash equilibria with first order like cost while retaining the stabilizing effect of symplectic second order corrections via low rank information. By storing only a limited history of curvature pairs, LMLRSGA is well suited to high parameter competitive models such as GANs. In particular, we provide a per iteration spectral stability condition for LRSGA near Nash equilibria, a limited memory implementation (LMLRSGA) based on adapted two loop recursions together with a local convergence analysis for fixed history length, and an empirical evaluation on GAN training on MNIST and FashionMNIST, including spectral diagnostics of the training dynamics.

math.OC↗
arXiv2025

On the Rate of Asymptotic Regularity of Iterative Methods for Nonexpansive Mappings in CAT(0) Spaces and Hyperbolic Optimization

The Krasnosel'ski\u{\i}--Mann and Halpern iterations are classical schemes for approximating fixed points of nonexpansive mappings in Banach spaces, and have been widely studied in more general frameworks such as $CAT(\kappa)$ and, more generally, geodesic spaces. Convergence results and convergence rate estimates in these nonlinear settings are already well established. The contribution of this paper is twofold: first, we extend to complete $CAT(0)$ spaces proof techniques originally developed in the linear setting of Banach and Hilbert spaces, thereby recovering the same asymptotic regularity bounds; second, we introduce a Halpern--type optimizer for hyperbolic optimization as a nonlinear counterpart of the Euclidean HalpernSGD scheme.

math.OC↗
arXiv2025

Solutions to Second-Order Nonlocal Evolution Equations Governed by Non-Autonomous Forms

Our main contributions include proving sufficient conditions for the existence of solution to a second order problem with nonzero nonlocal initial conditions, and providing a comprehensive analysis using fundamental solutions and fixed-point techniques. The theoretical results are illustrated through applications to partial differential equations, including vibrating viscoelastic membranes with time-dependent material properties and nonlocal memory effects.

math.AP↗
arXiv2025

A Low-Rank Symplectic Gradient Adjustment Method for Computing Nash Equilibria

This work presents a theoretical and numerical investigation of the symplectic gradient adjustment (SGA) method and of a low-rank SGA (LRSGA) method for efficiently solving revviolet optimization problems arising from two-player Nash games. The SGA method outperforms the gradient method by including second-order mixed derivatives computed at each iterate, which requires considerably larger computational effort. For this reason, an LRSGA method is proposed where the approximation to second-order mixed derivatives is obtained by rank-one updates. The theoretical analysis presented in this work focuses on novel convergence estimates for the SGA and LRSGA methods, including parameter bounds. The numerical experiments complement the theory by studying the behavior of LRSGA on explicit deterministic games with known equilibria and by evaluating its computational advantage over exact SGA on a CLIP-inspired neural-network training task, where LRSGA achieves comparable loss values lower CPU time than SGA with explicitly assembled mixed-derivative blocks.

math.OC↗
arXiv2021

Solutions to nonlocal evolution equations governed by non-autonomous forms and demicontinuous nonlinearities

We deal with the existence of solutions having L2 regularity for a class of non autonomous evolution equations. Associated with the equation, a general non local condition is studied. The technique we used combines a finite dimensional reduction together with the Leray-Schauder continuation principle. This approach permits to consider a wide class of nonlinear terms by allowing demicontinuity assumptions on the nonlinearity.

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