In machine learning, the loss functions optimized during training often differ from the target loss that defines task performance due to computational intractability or lack of differentiability. We present an in-depth study of the target loss estimation error relative to the surrogate loss estimation error. Our analysis leads to $H$-consistency bounds, which are guarantees accounting for the hypothesis set $H$. These bounds offer stronger guarantees than Bayes-consistency or $H$-calibration and are more informative than excess error bounds. We begin with binary classification, establishing tight distribution-dependent and -independent bounds. We provide explicit bounds for convex surrogates (including linear models and neural networks) and analyze the adversarial setting for surrogates like $ρ$-margin and sigmoid loss. Extending to multi-class classification, we present the first $H$-consistency bounds for max, sum, and constrained losses, covering both non-adversarial and adversarial scenarios. We demonstrate that in some cases, non-trivial $H$-consistency bounds are unattainable. We also investigate comp-sum losses (e.g., cross-entropy, MAE), deriving their first $H$-consistency bounds and introducing smooth adversarial variants that yield robust learning algorithms. We develop a comprehensive framework for deriving these bounds across various surrogates, introducing new characterizations for constrained and comp-sum losses. Finally, we examine the growth rates of $H$-consistency bounds, establishing a universal square-root growth rate for smooth surrogates in binary and multi-class tasks, and analyze minimizability gaps to guide surrogate selection.

Human beings are able to transfer the policies swiftly to a structurally similar task.

We formalize and study the natural approach of designing convex surrogate loss functions via embeddings for problems such as classification or ranking.

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To optimize this loss function, we propose two families of surrogate losses: cost-sensitive comp-sum losses and cost-sensitive constrained losses.

Mechanical metamaterials enable unconventional and programmable mechanical responses through structural design rather than material composition. In this work, we introduce a multistable mechanical metamaterial that exhibits a toggleable stiffness effect, where the effective shear stiffness switches discretely between stable configurations. The mechanical analysis of surrogate beam models of the unit cell reveal that this behavior originates from the rotation transmitted by the support beams to the curved beam, which governs the balance between bending and axial deformation. The stiffness ratio between the two states of the unit cell can be tuned by varying the slenderness of the support beams or by incorporating localized hinges that modulate rotational transfer. Experiments on 3D-printed prototypes validate the numerical predictions, confirming consistent stiffness toggling across different geometries. Finally, we demonstrate a monolithic soft clutch that leverages this effect to achieve programmable, stepwise stiffness modulation. This work establishes a design strategy for toggleable stiffness using multistable metamaterials, paving the way for adaptive, lightweight, and autonomous systems in soft robotics and smart structures.

Human beings are able to transfer the policies swiftly to a structurally similar task.