We study split-conformal prediction for regression when the reported prediction set must be a single interval, at target marginal coverage $1-α$, where $α$ is the nominal miscoverage level. Under this reporting constraint, the natural conditional target is the shortest interval with conditional mass at least $1-α$, rather than an equal-tailed interval or a possibly disconnected high-probability set. We parameterize this single-interval oracle by a lower-tail allocation, which determines how the nominal miscoverage $α$ is split between the two endpoints, and propose tail-allocation conformalized quantile regression (TA-CQR). TA-CQR estimates this allocation by searching over quantile-defined cores and then applies nonnegative additive split-conformal calibration, retaining exact finite-sample marginal coverage under exchangeability. The main contribution is theoretical. We characterize the oracle geometry, including its highest-density interpretation under unimodality and the positive connectedness cost induced by disconnected highest-density sets. We prove local recovery of the selected allocation and core, establish that calibration radii are asymptotically negligible under endpoint-density conditions, and give a finite-sample calibrated length oracle inequality with explicit grid, endpoint-quantile estimation, and calibration-sampling terms. Simulations and real-data examples report coverage and length jointly.

Greedy optimization methods such as Matching Pursuit (MP) and Frank-Wolfe (FW) algorithms regained popularity in recent years due to their simplicity, effectiveness and theoretical guarantees. MP and FW address optimization over the linear span and the convex hull of a set of atoms, respectively. In this paper, we consider the intermediate case of optimization over the convex cone, parametrized as the conic hull of a generic atom set, leading to the first principled definitions of non-negative MP algorithms for which we give explicit convergence rates and demonstrate excellent empirical performance. In particular, we derive sublinear (O(1/t)) convergence on general smooth and convex objectives, and linear convergence (O(e^{-t})) on strongly convex objectives, in both cases for general sets of atoms. Furthermore, we establish a clear correspondence of our algorithms to known algorithms from the MP and FW literature. Our novel algorithms and analyses target general atom sets and general objective functions, and hence are directly applicable to a large variety of learning settings.

Fast, stealthy, and cheap--autonomous, semisubmersible drone boats carrying tons of cocaine could be international law enforcement's nightmare scenario. A big one just came ashore. Colombian military officials intercepted this 40-foot-long uncrewed fiberglass "narco sub" in the ocean just off Tayrona National Park. On a bright morning last April, a surveillance plane operated by the Colombian military spotted a 40-foot-long shark-like silhouette idling in the ocean just off Tayrona National Park. It was, unmistakably, a "narco sub," a stealthy fiberglass vessel that sails with its hull almost entirely underwater, used by drug cartels to move cocaine north. The plane's crew radioed it in, and eventually nearby coast guard boats got the order, routine but urgent: Intercept. In Cartagena, about 150 miles from the action, Captain Jaime González Zamudio, commander of the regional coast guard group, sat down at his desk to watch what happened next.

From acomputational point of view,this problem is equivalent to the quadratic assignment problem, and as such is an NP-hard problem (Burkard et al., 1998).

The flexibility of these methods is often limited bytherequirement thatthecutsbeaxisaligned.

We categorize existing implementations2 into 2 kinds: (1) for verification only (typically implemented on CPUs, including DeepZ[35], and DeepPoly[37])3 (2) for training certified defense (typically using more efficient, yet weaker or approximated bounds: convex outer4 adversarial polytope[45], DiffAI[28], IBP[9] andCROWN-IBP[50]). Ourcontributionisnot8 to improve tightness of LiRPA bounds, but the first framework that generalizes to general computational graphs in9 anautomatic manner. In CROWN[50], the quadratic bound is only applied to 2-layer networks and is hard to extend to14 multiplelayers,aswhenpropagatingaquadratic boundtothe3rdlayeritbecomes quadratic (x4)duetocorrelations15 between twoquadratic terms ("order explosion").

Greedy optimization methods such as Matching Pursuit (MP) and Frank-Wolfe (FW) algorithms regained popularity in recent years due to their simplicity, effectiveness and theoretical guarantees. MP and FW address optimization over the linear span and the convex hull of a set of atoms, respectively. In this paper, we consider the intermediate case of optimization over the convex cone, parametrized as the conic hull of a generic atom set, leading to the first principled definitions of non-negative MP algorithms for which we give explicit convergence rates and demonstrate excellent empirical performance. In particular, we derive sublinear (O(1/t)) convergence on general smooth and convex objectives, and linear convergence (O(e^{-t})) on strongly convex objectives, in both cases for general sets of atoms. Furthermore, we establish a clear correspondence of our algorithms to known algorithms from the MP and FW literature. Our novel algorithms and analyses target general atom sets and general objective functions, and hence are directly applicable to a large variety of learning settings.

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In this paper, we address the graph matching problem.