#01Oct 30, 2026
math.OC
Exact Theory and Optimal Stepsizes for Stochastic Gradient Descent in Smooth Convex Minimization
Benjamin Grimmer, Qining Shen
This work considers the design of stepsize schedules for stochastic gradient descent in smooth convex optimization. Throughout, we assume that gradient noise is conditionally zero-mean with bounded variance. Among problems with dimension $d\geq N+1$, we provide an exact worst-case analysis of the expected suboptimality of stochastic gradient descent's final iterate. Our analysis applies to any (potentially nonconstant) schedule of $N$ stepsizes in $[0,1.5]^N$. Here, exact means that we provide a matching objective function and noise distributions where our guarantee is attained. Whenever the variance bound is at least $Ω(1/\sqrt{N})$, our theory provides the minimax optimal stepsize schedule for stochastic gradient descent and proves its uniqueness. These optimal stepsizes monotonically decrease, ruling out any accelerations in expected final suboptimality from fixed fractal schedules observed in the deterministic case.
#02Oct 30, 2026
cs.AI
ACCT: Asymmetric Cross-LLM Context Transfer
Yuzhe Liu, Haolin Tian, Tonghan Wang
In real-world deployments, large language model (LLM) systems are increasingly evolving from single-model invocation toward heterogeneous deployments in which different models are dynamically invoked for inference. When the same context is reused across models, however, each newly selected target model typically has to process it again, repeatedly incurring substantial computational costs. Therefore, we propose ACCT (Asymmetric Cross-LLM Context Transfer), which targets KV cache transfer between models of different scales within the same model family. ACCT transforms the KV cache produced by a source model into sparse KV states that can be directly consumed by a target model, thereby enabling cross-model reuse of context computation and decoupling context computation from downstream inference. ACCT consists of three lightweight components: the Cartographer identifies query-relevant context tokens for each target layer and KV head; the Allocator determines how many context tokens each target layer and KV head retains under a shared total KV budget; and the Compiler transforms selected source KV into sparse target KV. The target model can therefore perform inference using the transferred KV without reprocessing the original context. We evaluate ACCT in two complementary settings comprising five general-capability benchmarks and six highly context-dependent tasks. Experiments show that ACCT matches or exceeds the target model's original inference performance even when reducing the KV cache size by 75%. At a context length of approximately 19K tokens, ACCT's KV conversion achieves a 6.5-21.2 times speedup over target context prefill. These results demonstrate the feasibility of transferring context computation across models under a limited KV budget.
#03Oct 30, 2026
cs.LG
Hyperbolic Inverse Reinforcement Learning
Iulia Rusu, Anoop Praturu, Sergei Gepshtein and 1 more
Inverse reinforcement learning (IRL) seeks to infer the reward function underlying observed behavior, but its predictive accuracy and interpretability depend on how that reward is computed from the underlying features. While prior IRL reward functions have been implemented through a variety of techniques including linear weights, neural networks, or other function approximators, none make assumptions about the geometry of the underlying feature space. Inspired by geometric organization observed in neural circuits, we introduce Hyperbolic inverse reinforcement learning (Hyperbolic IRL), which parameterizes reward as a function of Hyperbolic distance between the feature vector and a learned parameter vector, as well as its Euclidean distance variant. We tested our methods against a range of linear and nonlinear IRL baselines under both deterministic and stochastic policies. Our Hyperbolic approach achieved comparable performance to state-of-the-art models on established benchmarks and excelled in the low data limit and across dense environments. Applied to human navigation among multiple exhibits in a museum room, Hyperbolic IRL reveals reward structure that depends not only on individual exhibit-distance features, but also on non-separable conjunctions between them. Together, these results demonstrate that a Hyperbolic reward parameterization can improve both the predictive accuracy and interpretability of IRL, providing a framework for studying how non-separable, conjunctive structure can begin to explain behavior.
#04Oct 30, 2026
cs.LG
There Is No Free Loop: When Does Looping a Trained Model Pay for Its Inference Compute?
Rhea Sanjay Sukthanker, Pashmina Cameron
Looping re-applies a block of layers so that each token passes through the block more than once. Adding a loop to a trained model is attractive because it leaves the parameter count, and hence the disk size, the memory footprint and the training cost unchanged. Unfortunately, every generated token pays for the loop. Re-applying m of N layers makes each token about (N+m)/N times as expensive. Loops that a model learns during large-scale pretraining justify this price, but whether loops added after pretraining justify this cost remains unclear. We compare a looped model with two simple baselines. We first compare it with the same model trained identically without the loop (equal parameters). Secondly, we compare it with the best unlooped use of the same inference floating-point operations (equal FLOPs). Across Qwen3 and Qwen3.5 models from 0.8B to 9B parameters, only the looped 0.8B model beats both baselines. At 4B and 9B, the looped model does not beat the first baseline, i.e. the same model trained without the loop. This is not because the benchmarks are too easy, since on the same benchmarks the unlooped 9B model still scores up to 13 points higher than the 4B model, showing that there is ample room for improvement. Introducing loops in an already pretrained model therefore only pays for small models with a fixed parameter budget.
#05Oct 30, 2026
cs.LG
Fast Privacy Loss Accounting for Random Allocation via Multiscale FFT
Vitaly Feldman, Moshe Shenfeld
In the random allocation sampling scheme each user's data is used in $k$ steps chosen randomly and uniformly from a sequence of $t$ steps. This scheme is known to improve on Poisson sampling in private optimization and in communication-efficient aggregation. The existing privacy loss distribution (PLD) accountant for this scheme runs in time quadratic in the inverse of the accuracy parameter $α$ (Feldman and Shenfeld, 2026). We give an algorithm, \emph{multiscale FFT}, that represents likelihood ratios on a grid that is uniform within each dyadic interval and computes the exact rounding of every pairwise sum using FFTs within each interval. It keeps the guarantees of the accountant and improves the dependence of its running time on $α$ from $O(α^{-2})$ to $O(α^{-1}\log(1/α))$. In addition, we demonstrate that rounding can be performed via the mean-preserving connect-the-dots projection of (Doroshenko et al., 2026). This reduces the errors that rounding introduces in the mean of the privacy loss and in the $χ^2$ divergence of the computed pair from the order of $α$ to $O(α^2/\log(t))$. In our experiments, our algorithm achieves speedups of several orders of magnitude at high accuracy and gives improved privacy bounds for DP-SGD with PREAMBLE.