We develop a geometric framework that links objective accuracy to structural recovery in prototype-based clustering. The analysis is algorithm-agnostic and applies to a broad class of admissible loss functions. We define a clustering condition number that compares within-cluster scale to the minimum loss increase required to move a point across a cluster boundary. When this quantity is small, any solution with a small suboptimality gap must also have a small misclassification error relative to a benchmark partition. The framework also clarifies a fundamental trade-off between robustness and sensitivity to cluster imbalance, leading to sharp phase transitions for exact recovery under different objectives. The guarantees are deterministic and non-asymptotic, and they separate the role of algorithmic accuracy from the intrinsic geometric difficulty of the instance. We further show that errors concentrate near cluster boundaries and that sufficiently deep cluster cores are recovered exactly under strengthened local margins. Together, these results provide a geometric principle for interpreting low objective values as reliable evidence of meaningful clustering structure.

A probabilistic classifier is considered "well calibrated" when its predicted probabilities closely align with the empirical frequencies of the corresponding labels [

From the perspective of statistical learning theory, (1) is rather intriguing. Moreover, they do not provide instance-wise matching lower bounds to verify the tightness of the upper bounds.

We introduce a novel cyclic Markov decision process (MDP) framework for multi-step decision problems with heterogeneous stage-specific dynamics, transitions, and discount factors across the cycle. In this setting, offline learning is challenging: optimizing a policy at any stage shifts the state distributions of subsequent stages, propagating mismatch across the cycle. To address this, we propose a modular structural framework that decomposes the cyclic process into stage-wise sub-problems. While generally applicable, we instantiate this principle as CycleFQI, an extension of fitted Q-iteration enabling theoretical analysis and interpretation. It uses a vector of stage-specific Q-functions, tailored to each stage, to capture within-stage sequences and transitions between stages. This modular design enables partial control, allowing some stages to be optimized while others follow predefined policies. We establish finite-sample suboptimality error bounds and derive global convergence rates under Besov regularity, demonstrating that CycleFQI mitigates the curse of dimensionality compared to monolithic baselines. Additionally, we propose a sieve-based method for asymptotic inference of optimal policy values under a margin condition. Experiments on simulated and real-world Type 1 Diabetes data sets demonstrate CycleFQI's effectiveness.

LetAbean nHermitian matrixandletBbea(n 1) (n 1)matrixwhich is constructed by deleting thei-th row andi-th column ofA. Denote thatΦ = [ϕ(x1),...,ϕ(xn)] Rn D, where D is the dimension of feature spaceH. Performing rank-n singular value decomposition (SVD) onΦ, we have Φ = HΣV, where H Rn n, Σ Rn n is a diagonal matrix whose diagonal elements are the singular values of Φ,andV RD n. F(α) in Eq.(21) is proven differentiable and thep-th component of the gradient is F(α) αp = Then, a reduced gradient descent algorithm [26] is adopted to optimize Eq.(21). The three deep neural networks are pre-trained on the ImageNet[5].

Let us defineK Rn n a random diagonal sampling matrix whereKj,j Bernoulli(pj) for 1 j n. Therefore, Bernoulli-CRS will perform on average the same amount of computations as in the fixed-rankCRS. This formulation immediately hints atthe possibility tosample over the input channeldimension, similarly to sampling column-row pairs in matrices. Let ` be a β-Lipschitz loss function, and let the network be trained with SGD using properly decreasing learning rate. Let us denote the weight, bias and activation gradients with respect to a loss function` by Wl, bl, al respectively.