Study improves H H H -consistency bounds for regression analysis.
problem Improving H H H -consistency bounds for regression analysis. method Generalized theorems and novel H H H -consistency bounds for various surrogate loss functions. result Derives principled surrogate losses for adversarial regression.
Study of loss functions for learning to defer, proving consistency.
problem Learning to defer in machine learning.
method Introduced a family of surrogate losses parameterized by Ψ Ψ Ψ and proved their consistency. result Proved realizable H H H -consistency and Bayes-consistency of specific surrogate losses. Enhanced H H H -consistency bounds derived under relaxed conditions.
problem Quantifying the relationship between zero-one estimation error and surrogate loss estimation error.
method Relaxing the condition on the surrogate loss conditional regret and presenting a general framework for establishing enhanced H H H -consistency bounds. result Derivation of more favorable H H H -consistency bounds in various scenarios. Study on H H H -consistency bounds for machine learning surrogates.
problem Estimating target loss error relative to surrogate loss error in machine learning.
method Developed H H H -consistency bounds for various surrogates and loss functions. result Stronger guarantees than existing methods, offering distribution-dependent and -independent bounds.
Enhanced consistency bounds derived for classification under a new noise condition.
problem Enhanced consistency bounds for classification under a new noise condition.
method Model Margin Noise (MM noise) assumption, derived enhanced H-consistency bounds.
result Enhanced H-consistency bounds under MM noise condition, interpolates between linear and square-root regimes.
This paper tackles deferral learning with multiple experts, providing strong theoretical guarantees.
problem Optimizing input assignment to experts balancing accuracy and computational cost.
method Introducing new surrogate loss functions and efficient algorithms with strong theoretical learning guarantees.
result Realizable H H H -consistency, H H H -consistency bounds, and Bayes-consistency for deferral learning. Unified surrogate loss framework for multi-label learning with strong consistency guarantees.
problem Improving consistency and accounting for label correlations in multi-label learning.
method Introducing multi-label logistic loss and extending it to comprehensive multi-label comp-sum losses, proving strong consistency guarantees for any multi-label loss.
result Unified surrogate loss framework benefiting from strong consistency guarantees for any multi-label loss.
Study on top- k k k classification with new loss functions and algorithms.
problem Improving multi-class classification accuracy and cardinality trade-off.
method Introducing cardinality-aware loss functions and deriving their consistency bounds.
result New cardinality-aware algorithms for top- k k k classification. Paper establishes a universal growth rate for smooth surrogate losses in classification.
problem Analyzing growth rates of consistency bounds for various surrogate losses.
method Proves square-root growth rate for smooth margin-based losses; extends to multi-class classification.
result Demonstrates a universal square-root growth rate for smooth comp-sum and constrained losses.
New method improves consistency in preference learning for neural networks.
problem Inconsistent surrogate losses in preference learning for neural networks.
method Formulated a margin-shifted ranking framework and introduced Structure-Aware H H H -consistency. result Proved superior consistency guarantees for capacity-bounded models using heavy-tailed surrogates.
Study of estimation errors in surrogate loss minimizers, providing stronger guarantees than existing methods.
problem Estimation errors in surrogate loss minimizers for various hypothesis sets.
method Detailed study of H \mathscr{H} H -consistency estimation error bounds, proving general theorems for distribution-dependent and independent settings. result Explicit bounds for zero-one and adversarial losses, showing enhancements under distributional assumptions.
This paper introduces new loss functions for balanced multi-class classification.
problem Balancing class imbalance in multi-class classification.
method Introduces two new surrogate loss families: GLA and GCA.
result GCA losses offer stronger theoretical guarantees in imbalanced settings.
Develops algorithms for optimizing multi-label metrics with provable guarantees.
problem Optimizing complex multi-label metrics like F-measure and Jaccard index.
method Principled learning algorithms based on H-consistency for generalized metrics.
result Provable H H H -consistency bounds for multi-label metric optimization. Study on learning to defer with multiple experts using new surrogate losses.
problem Learning to defer with multiple experts in a machine learning context.
method Introducing a new family of surrogate losses for the multiple-expert setting, proving H H H -consistency bounds, and designing learning algorithms. result Explicit guarantees for new learning to defer algorithms based on minimization of these surrogate losses.
New algorithms optimize metrics for binary classification with class imbalance.
problem Optimizing metrics like Fβ, AM, Jaccard for imbalanced classes.
method Reformulates metric optimization as cost-sensitive learning, using surrogate loss functions.
result METRO algorithms provide strong theoretical guarantees and outperform baselines.
Study on calibration and consistency of adversarial surrogate losses.
problem Designing robust classifiers with theoretical guarantees.
method Extensive analysis of H-calibration and H-consistency of adversarial surrogate losses.
result Some convex loss functions and supremum-based convex losses are not H-calibrated for important hypothesis sets.
This thesis tackles learning with multi-class abstention and multi-expert deferral, improving model reliability and efficiency.
problem Improving model reliability and efficiency in large language models (LLMs) by leveraging multiple experts.
method Developed new surrogate losses and consistency guarantees for multi-class classification and regression with deferral.
result Strong consistency guarantees for surrogate losses in multi-class classification and regression with deferral.
The paper revisits discriminative vs. generative classifiers, showing naive Bayes requires fewer samples.
problem Comparing discriminative and generative classifiers in multiclass settings.
method Theoretical analysis and simulations of naive Bayes vs. logistic regression.
result Multiclass naive Bayes requires fewer samples to approach asymptotic error compared to logistic regression.
Theoretical analysis of cross-entropy loss functions and their robustness.
problem Guarantees for using cross-entropy as a surrogate loss function.
method Theoretical analysis of a broad family of loss functions, including cross-entropy.
result First H H H -consistency bounds for comp-sum losses and smooth adversarial comp-sum losses. A framework for ranking with abstention, offering theoretical guarantees and practical effectiveness.
problem Making predictions with limited cost when uncertain.
method Introduces a novel ranking framework with abstention, analyzing theoretical consistency bounds.
result Extensive theoretical analysis including H H H -consistency bounds for linear and neural network models. Classifies special homogeneous curves with polynomial equations.
problem Identifying and classifying special homogeneous curves.
method Analyzing homogeneous polynomials and their level sets with group actions.
result All special homogeneous curves are classified.
Given a hyperbolic subgroup H H H of a hyperbolic group G G G for which a Cannon-Thurston map $\hat i:\partial H \ra \partial G$ exists, we study the limit set Λ H Λ_H Λ H of H H H with respect to its action on ∂ G \partial G ∂ G . We prove that the set of conical limit points is exactly the subset of Λ H Λ_H Λ H consisting of the points to wh…
Study improves top-k set prediction with low cardinality.
problem Improving top-k set prediction accuracy with low cardinality.
method Introduces new target loss function and surrogate losses.
result Demonstrates effectiveness of cardinality-aware algorithms.
A new method for learning to defer decisions with expert advice improves over standard methods.
problem Learning to defer decisions with expert advice in systems where expert information can be modified after selection.
method An augmented surrogate that operates on the composite expert-advice action space, providing consistency guarantees and excess-risk bounds.
result The method improves over standard Learning-to-Defer and adapts its advice acquisition behavior to the cost regime.
Unified RMOT framework for non-modelable risk factors reduces audit bounds.
problem Infinite audit bounds for exotic derivatives pricing with sparse market data.
method Rough Martingale Optimal Transport (RMOT) with rough volatility regularization.
result Finite, explicit, and asymptotically tight extrapolation bounds for non-modelable risk factors.
Let G / H G/H G / H be a compact homogeneous space, and let g ^ 0 \hat{g}_0 g ^ 0 and g ^ 1 \hat{g}_1 g ^ 1 be G G G -invariant Riemannian metrics on G / H G/H G / H . We consider the problem of finding a G G G -invariant Einstein metric g g g on the manifold G / H × [ 0 , 1 ] G/H\times [0,1] G / H × [ 0 , 1 ] subject to the constraint that g g g restricted to G / H × { 0 } G/H\times \{0\} G / H × { 0 } and G / H × { 1 } G/H\times \{1\} G / H × { 1 } co…
Linear-Core Surrogates combine fast optimization and statistical efficiency in classification and structured prediction.
problem The trade-off between smoothness and margin-based losses in classification and structured prediction.
method Linear-Core (LC) Surrogates, a family of convex loss functions that stitch a linear core to a smooth tail.
result LC Surrogates achieve fast linear consistency rates while maintaining differentiability and strict H H H -consistency bounds. Online L2D algorithm for multiclass classification with varying experts.
problem Handling streaming data, changing expert availability, and shifting expert distribution.
method First online L2D algorithm with O ( ( n + n e ) T 2 / 3 ) O((n+n_e)T^{2/3}) O (( n + n e ) T 2/3 ) and O ( ( n + n e ) T ) O((n+n_e)\sqrt{T}) O (( n + n e ) T ) regret guarantees. result Effective extension of standard L2D to settings with varying expert availability and reliability.
We improve adversarial robustness calibration analysis for broader hypothesis sets.
problem Improving calibration for adversarial robustness in machine learning.
method A finer definition of calibration for adversarial robustness.
result Our results cover most common hypothesis sets in machine learning.
A novel framework for regression with multiple experts, addressing challenges in infinite and continuous label spaces.
problem Challenges in regression with multiple experts due to the infinite and continuous nature of the label space.
method Introduces a novel framework for regression with deferral, analyzing both single-stage and two-stage scenarios with new surrogate loss functions.
result Proves H H H -consistency bounds for both single-stage and two-stage methods, providing stronger guarantees than Bayes consistency. We study Tian's α α α -invariant in comparison with the α 1 α_1 α 1 -invariant for pairs ( S d , H ) (S_d,H) ( S d , H ) consisting of a smooth surface S d S_d S d of degree d d d in the projective three-dimensional space and a hyperplane section H H H . A conjecture of Tian asserts that α ( S d , H ) = α 1 ( S d , H ) α(S_d,H)=α_1(S_d,H) α ( S d , H ) = α 1 ( S d , H ) . We show that this is indeed true for d = 4 d=4 d = 4 (the res…
We introduce spherical T-duality, which relates pairs of the form ( P , H ) (P,H) ( P , H ) consisting of a principal S U ( 2 ) SU(2) S U ( 2 ) -bundle P → M P\rightarrow M P → M and a 7-cocycle H H H on P P P . Intuitively spherical T-duality exchanges H H H with the second Chern class c 2 ( P ) c_2(P) c 2 ( P ) . Unless d i m ( M ) ≤ 4 dim(M)\leq 4 d im ( M ) ≤ 4 , not all pairs admit spherical T-duals and the spheric…
The holonomy algebra $\g$ of an n + 2 n+2 n + 2 -dimensional Lorentzian manifold ( M , g ) (M,g) ( M , g ) admitting a parallel distribution of isotropic lines is contained in the subalgebra $\simil(n)=(\Real\oplus\so(n))\zr\Real^n\subset\so(1,n+1)$ . An important invariant of $\g$ is its $\so(n)$ -projection $\h\subset\so(n)$ , which is a Riemannian…
Let T T T be a circle and L T LT L T be its loop group. Let M \mathcal{M} M be an infinite dimensional manifold equipped with a nice L T LT L T -action. We construct an analytic L T LT L T -equivariant index for M \mathcal{M} M , and justify it in terms of noncommutative geometry. More precisely, we construct a Hilbert space H \mathcal{H} H consis…
New framework for learning from imbalanced data with theoretical guarantees.
problem Class imbalance in machine learning, especially in multi-class problems.
method Theoretical framework and new margin loss function for imbalanced classification.
result Proves strong H H H -consistency of the proposed margin loss function. Study of groups and their quasi-isometrically embedded subgroups.
problem Understanding the structure and properties of groups and their subgroups.
method Abstracting the notion of A/QI triples and using methods from geometric group theory.
result Stability of quasi-isometrically embedded subgroups in finitely generated groups.
In earlier papers, we introduced spherical T-duality, which relates pairs of the form ( P , H ) (P,H) ( P , H ) consisting of an oriented S 3 S^3 S 3 -bundle P → M P\rightarrow M P → M and a 7-cocycle H H H on P P P called the 7-flux. Intuitively, the spherical T-dual is another such pair ( P ^ , H ^ ) (\hat P, \hat H) ( P ^ , H ^ ) and spherical T-duality exchanges the 7-flux with …
The paper studies a special Grassmannian space and shows it's an orbit of a unitary group.
problem Investigating a specific Grassmannian space of infinite-dimensional subspaces.
method Analyzing the restricted p p p -Schatten class Grassmannian and showing it's an affine coadjoint orbit of a unitary group. result The restricted p p p -Schatten class Grassmannian is shown to be an affine coadjoint orbit of an infinite-dimensional restricted unitary group. ORAT improves model robustness against outliers and adversarial attacks.
problem Challenges of training data like outliers and adversarial samples.
method Bi-level optimization with robust rank-based loss function.
result ORAT achieves theoretical consistency and uniform convergence rates.
Unified model for prediction and deferral selects top-k entities efficiently.
problem Efficiently selecting top-k entities for deferral in machine learning.
method One-stage Top- k k k Learning-to-Defer framework with a convex surrogate. result Unified model achieves superior accuracy-cost trade-offs.
The paper classifies differentiable structures on a line with two origins.
problem Classifying differentiable structures on a non-Hausdorff line with two origins.
method Using homeomorphisms and diffeomorphisms, the paper establishes a bijection between structures and coset classes.
result The line with two origins admits uncountably many non-diffeomorphic structures for each differentiability class.
Unified framework for deferring queries to top-k experts, improving accuracy-cost trade-offs.
problem Limitation of existing L2D frameworks to single-expert deferral.
method Top- k k k Learning-to-Defer framework, including adaptive Top- k ( x ) k(x) k ( x ) variant. result Superior accuracy-cost trade-offs with multi-expert deferral.
Let T T T be a circle group, and L T LT L T be its loop group. We hope to establish an index theory for infinite-dimensional manifolds which L T LT L T acts on, including Hamiltonian L T LT L T -spaces, from the viewpoint of K K KK K K -theory. We have already constructed several objects in the previous paper \cite{T}, including a Hilbert space $…
The study optimizes bounds for comparing training and population loss.
problem Optimizing bounds for comparing training and population loss.
method Derives generic information-theoretic and PAC-Bayesian generalization bounds using convex comparator functions.
result The tightest possible bound is obtained with the comparator being the convex conjugate of the CGF of the bounding distribution.
Introduces bounded scale measure and generalizes property A.
problem Defining property A for large scale spaces with bounded geometry.
method Introduces bounded scale measure, shows its coarse invariance, and generalizes property A.
result Definition of property A for large scale spaces with bounded scale measure is a coarse invariant.
Paper improves PAC-Bayes bounds for various loss types.
problem Improving PAC-Bayes bounds for different types of losses.
method Introducing new high-probability PAC-Bayes bounds for bounded and general tail behaviors losses, and extending to anytime-valid bounds.
result New fast-rate and mixed-rate bounds for losses with bounded ranges, and parameter-free bounds for losses with general tail behaviors.
Improved bounds for Monte Carlo Rademacher Averages using self-bounding functions.
problem Proving sharper concentration bounds for MCERA.
method Deriving new bounds through self-bounding functions and concentration of measure.
result Novel bounds depend on data-dependent quantities, improving over standard methods.
Study bounds on self-shrinkers with bounded HA for applications.
problem Understanding bounds on self-shrinkers with bounded HA.
method Integral and pointwise bounds on the second fundamental form of self-shrinkers.
result Gap and compactness results for self-shrinkers.