Differentiable Surrogates: Theory & Applications

• Differentiable surrogates are smooth, convex proxy functions that enable gradient-based optimization of complex, non-differentiable objectives.
• They facilitate nonlinear dimension reduction by approximating high-dimensional functions through low-dimensional, geometrically faithful mappings.
• They offer robust optimization guarantees by employing convex majorants and concentration inequalities, ensuring stable performance even with limited data.

A differentiable surrogate is a mathematically smooth (i.e., differentiable) function—often convex—that acts as a tractable proxy for a more complex, non-differentiable, or non-convex objective in optimization, regression, or representation learning. Differentiable surrogates enable the deployment of gradient-based algorithms on problems originally defined by losses, constraints, or geometric regularities that are otherwise difficult to handle directly. In the context of “Surrogate to Poincaré inequalities on manifolds for dimension reduction in nonlinear feature spaces” (Nouy et al., 3 May 2025), differentiable convex surrogates are constructed to facilitate dimension reduction via nonlinear feature learning, leveraging geometric inequalities such as the Poincaré inequality and incorporating concentration-of-measure phenomena for robust optimization.

1. Differentiable Surrogate Construction and Properties

Poincaré inequalities provide upper bounds on the variance of a function $u$ in terms of the "average" size of its gradient on a manifold: $$\operatorname{Var}_{\mu}(u) \leq C \int_{\mathbb{R}^d} |\nabla u(x)|^2\, d\mu(x)$$ where $\mu$ is a probability measure and $C$ is domain-dependent. These inequalities are a natural tool for feature learning and dimension reduction, but the associated optimization problems are often nonconvex and involve complicated dependencies on both $u$ and its gradient.

To address the intrinsic difficulty of directly minimizing a Poincaré-inspired loss $\mathcal{J}(g)$—where $\mathcal{J}(g)$ encapsulates the relationships among a candidate mapping $g$, the function $u$, and their respective gradients—the paper introduces convex, differentiable surrogate losses $\widetilde{\mathcal{J}}(g)$. These surrogates are constructed such that:

• $\widetilde{\mathcal{J}}(g) \geq \mathcal{J}(g)$ for feasible $g$
• $\widetilde{\mathcal{J}}(g)$ is convex and smooth in $g$ (facilitating reliable gradient-based optimization)
• The surrogate respects the geometric information of the original inequality, embedding essential variance–gradient relationships

Formally, the original and surrogate losses may take generic forms

$$\mathcal{J}(g) = \mathbb{E}_\mu\left[\phi(u, g, \nabla u, \nabla g)\right],\qquad \widetilde{\mathcal{J}}(g) = \mathbb{E}_\mu\left[\psi(u, g, \nabla u, \nabla g)\right]$$

with $\psi$ chosen as a convex majorant of $\phi$.

2. Nonlinear Dimension Reduction with Compositional Models

The goal is to approximate a high-dimensional function $u: \mathbb{R}^d \to \mathbb{R}$ by a composition $f \circ g$ where $g: \mathbb{R}^d \rightarrow \mathbb{R}^m$ (with $m \ll d$), extracting a lower-dimensional, nonlinear representation. For a fixed $g$, $f$ is learned via classical regression methods using labeled evaluations of $u$.

The differentiable surrogate is leveraged to search over mappings $g$ such that the resulting representation:

• Encodes the variability of $u$ as reflected in its gradient (i.e., respects the manifold geometry)
• Admits efficient and stable optimization
• Enables downstream recovery of $u$ via $f \circ g$ with controlled approximation error

This surrogate-based strategy provides a theoretically grounded workflow for nonlinear dimension reduction that remains practical for large $d$ due to smoothness and convexity.

3. Surrogate-Based Loss Minimization: Theoretical and Algorithmic Guarantees

Direct minimization of the Poincaré-inspired loss $\mathcal{J}(g)$ is typically intractable due to nonconvex geometry and challenging interactions among $u$, $g$, and their gradients. By constructing convex surrogates $\widetilde{\mathcal{J}}(g)$ and replacing nonconvex terms with tractable approximations, the following properties are achieved:

• Optimization becomes robust to local minima and numerically stable, suitable for gradient-based solvers
• Concentration inequalities are employed to provide probabilistic guarantees on the deviation of surrogate loss minimizers from those of the original loss, for broad classes of functions $g$ (including polynomials) and measures $\mu$
• Sub-optimality bounds for the surrogate minimize risk of overfitting or poor generalization, important for small sample sizes and high-dimensional regimes

A generic form of the loss objective is

$$\widetilde{\mathcal{J}}(g) = \frac{1}{N} \sum_{i=1}^N \left[ L(u(x_i), f(g(x_i))) + \lambda\, R(g(x_i), \nabla g(x_i)) \right]$$

where $L$ is a regression loss and $R$ is a convex regularizer inspired by the Poincaré inequality.

4. Empirical Performance and Regimes of Benefit

Extensive benchmark experiments demonstrate that the surrogate-based approach:

• Provides lower approximation errors than standard (non-surrogate) iterative minimization of $\mathcal{J}(g)$, especially in small-sample regimes
• Shows particularly strong performance for low-dimensional embeddings ($m=1$), where accurately capturing the manifold’s geometry is critical yet nonconvex optimization is typically harder
• Exhibits greater computational efficiency and numerical stability, attributed to the convexity and smoothness of $\widetilde{\mathcal{J}}(g)$
• Outperforms standard iterative methods that directly target the training Poincaré-based loss, notably in terms of both convergence and final approximation quality

This suggests that convex surrogates are especially valuable under limited data or when robust optimization is paramount.

5. Broader Implications and Applications

The convex, differentiable surrogate methodology developed for Poincaré-inequality-based manifold learning has implications across several domains:

• High-dimensional function approximation: Enables tractable and geometrically faithful regression and representation learning in spaces where direct optimization is prohibitive.
• Nonlinear feature extraction: Provides a principled approach for learning nonlinear embeddings that preserve essential geometric characteristics, useful in manifold learning and scientific computing.
• Optimization with geometric constraints: The surrogate framework can potentially be adapted to other cases where optimization is impeded by hard-to-optimize constraints or variational regularities, such as control, inverse problems, and reinforcement learning.
• Theoretical guarantees: The use of concentration inequalities and sub-optimality results gives practitioners tools to assess the risks and reliability of surrogate-based optimization, strengthening trust in empirical successes.

In summary, by constructing and minimizing convex, differentiable surrogates to Poincaré-inequality-derived geometric losses, the approach supports efficient and robust nonlinear dimension reduction with theoretical performance guarantees and strong practical results—even in challenging, data-constrained, or high-dimensional regimes (Nouy et al., 3 May 2025).