Propensity Straight-Through Gradients for Discrete Stochastic Systems

Continuous-time Markov chains (CTMCs) provide the backbone for modeling discrete stochastic dynamics across applied, physical, and biological sciences. Their integration with modern gradient-based machine learning, however, is limited by the hard categorical event selection intrinsic to Gillespie-type simulation algorithms. We exploit the affine state update to obtain the exact one-step conditional-mean sensitivity by differentiating normalized reaction propensities. We pair this backward rule with exact forward trajectories to define the propensity straight-through (PST) estimator. At the trajectory level, we show that one-step sensitivities composed across events can depart from the exact multistep sensitivity. We derive the resulting per-step discrepancy in closed form and prove that it vanishes identically for affine downstream dependence. PST matches the accuracy of Gumbel-Softmax straight-through across all benchmarks: reversible dimerization (0.06% error), a genetic oscillator (1.7% error), a 50-task repressilator suite (0.17% median error), and patch-clamp ion-channel recordings ($R^2$ = 0.988). Under matched settings, PST converges 3.0-fold faster on the oscillator and 2.1-fold faster on the ion channel. At deep-learning scale, PST trains a 203,796-parameter stochastic reaction network with hard sampling, reaching 98.22% MNIST digit classification accuracy. By differentiating an exact conditional mean rather than a relaxed sample, PST offers a temperature- and Gumbel-free path to scalable gradient-based learning through exact stochastic trajectories.

Accelerating Evolutionary Strategy via Rao-Blackwellizing Realization of Uncertain Input

We investigate Optimization under Input Uncertainty (OIU), in which the input to the objective function, rather than the objective function itself, is subject to uncertainty. OIU appears in manufacturing processes with production tolerance, control of physical systems with actuation noise, Mixture of Experts, and Reinforcement Learning (RL). Most of the existing approaches solve OIU by using the value of the objective function but discard the information of the realized input, even though the realized input is observable in various applications. The question here is whether the discarded information of the realized input is useful to accelerate the optimization process. We affirmatively answer this question for Evolutionary Strategy (ES) by theoretically showing that the information of the realized input can reduce the variance of the gradient estimator via Rao-Blackwellization. Using the Rao-Blackwellized gradient estimator, we propose Phenotype-Accelerated Evolutionary Strategy (PAES), which is a refinement of ES for OIU. Numerical experiments show that PAES converges faster than the usual ES from simple continuous optimization problems to RL benchmarks.

Exposure-Based Reinforcement Learning to Rank

Reinforcement learning (RL) methods for learning-to-rank (LTR) can optimize (almost) any ranking goal, e.g., from precision or discounted cumulative gain to fairness-of-exposure or ranking distillation. However, standard RL is ineffective and computationally costly due to the enormous action space in LTR settings. Existing methods reach computational efficiency through custom gradient computation algorithms, but they are very complex to implement and often clash with auto-differentiation. Consequently, existing RL for LTR is not attractive to many practitioners. We reconsider RL for LTR while actively avoiding reliance on custom gradients. Contrary to the existing approaches, we focus on variance reduction and GPU computation. In doing so, we discover that high sample-efficiency can be reached through baseline corrections and partial marginalization. Furthermore, we propose an abstraction that places gradient estimation behind a document-exposure distribution, this enables seamless plug-and-play integration with auto-differentiation. Thereby, one only has to implement a loss as a differentiable function of exposure and RL for LTR can optimize it using auto-differentiation. Our experimental results reveal that our new exposure-based RL for LTR approach converges considerably faster and at significantly higher ranking performance than existing custom gradients, with no additional costs in computation time when using GPUs. In contrast, existing custom gradients result in severe stability issues when converging over many epochs, which never occur for our methods. Thus, we considerably improve RL for LTR methodology by increasing its effectiveness, efficiency, and ease of application.

GradInf: Gradient Estimation as Probabilistic Inference

Gradient estimation -- the task of computing the gradient of the expected value of a probabilistic program -- has diverse applications in scientific computing, but is notoriously difficult because of issues such as high-dimensional integration, discrete random choices, and complex stochastic dependencies. This article introduces gradient inference, a new approach to developing sound and efficient gradient estimators for probabilistic programs. Gradient inference rests on a formal reduction from a gradient estimation problem to a closely related probabilistic inference problem, whose solution can be differentiated to obtain a gradient estimator. This inference problem is obtained by applying two powerful statistical operations -- coupling and factorization -- to the input probabilistic program. Our reduction lets us leverage the rich toolkit of probabilistic inference algorithms to design novel gradient estimators that extend and improve upon existing methods. We introduce GradInf, a probabilistic programming system that facilitates the sound and automated implementation of gradient inference. GradInf is centered around programmable source-to-source transformations for coupling and factorizing higher-order probabilistic programs, whose soundness is proven in terms of a denotational semantics. Key to our development is the use of information-flow typing to allow random choices in a probabilistic program to be factored out and partially evaluated, which improves our ability to deploy sophisticated probabilistic inference algorithms. The resulting system offers practitioners a principled framework for designing gradient estimators. We apply GradInf to several challenging case studies, showing that it can express prominent gradient estimators from the literature and enables the construction of new state-of-the-art estimators that outperform the best existing baselines.

K-ABENA: K-Adaptive Backpropagation with Error-based N-exclusion Algorithm : (Compensated Loss-Based Sample Exclusion with Unbiased Gradient Estimation)

We present K-ABENA (K-Adaptive Backpropagation with Error-based N-exclusion Algorithm), a selective gradient computation framework that reduces per-iteration training cost by excluding a fraction of low-loss ("minor") observations from the backward pass. Its canonical form (v3) combines a defensive-mixture sampling design over the minor set with Horvitz-Thompson inverse-probability reweighting, yielding a design-unbiased Horvitz-Thompson gradient estimator (Lemma 2) and whose self-normalized practical variant carries a bias of order O(1/m) with an explicit constant (Lemma 3). We prove an O(1/sqrt(T)) non-convex convergence guarantee for SGD under the estimator, with an additive term that quantifies the residual bias (Theorem 1). We further prove that uncompensated loss-based selection - a family that includes OHEM, SBP, and the two earlier K-ABENA variants - admits no stationary point at any minimizer where its selection bias is bounded away from zero (Proposition 2), and we quantify this failure empirically: at 0.17% class imbalance, uncompensated variants reach test AUC 0.53-0.62 versus 0.9998 for full-batch SGD, while the compensated estimator attains 0.9991 at identical 28.4% compute savings. On real datasets (Breast Cancer, Digits, Wine, Diabetes) the compensated estimator is statistically indistinguishable from full-batch SGD (paired permutation tests, p >= 0.5; Section 7) while saving 28-54% of per-epoch gradient computation. A biased "regularized mode" (the earlier half-domain variant) is retained as an option with a proven exact bias decomposition (Lemma 5) and quantified contraindications: it collapses to 0.386 accuracy under 40% label noise (baseline: 0.832) and to 0.53 AUC under extreme imbalance. Every advantage and every limitation reported in this paper is either proved or measured; all experiments are CPU-scale (NumPy/scikit-learn) and their scope is stated explicitly.

Unbiased Derivative Estimation for Stationary Mean of Parameterized Markov chains

We propose a new approach to unbiased estimation of the gradients of the stationary means associated with parametrized families of Markov chains. Our estimators are particularly efficient when the Markov chains have slow mixing rate. Our approach does not require a specific parametrization except for an oracle to evaluate the transition density and its gradient at a given data point without any additional knowledge about the density function itself. It makes our estimator suitable for parametrizations associated with neural networks. The estimator can potentially achieve large improvement in terms of efficiency. Numerical experiments confirm the good performance predicted by the theory.

Adaptive directional gradients for parameterised quantum circuits

Training parameterised quantum circuits (PQCs) on quantum hardware is bottlenecked by the measurement cost of gradient estimation, which under the parameter-shift rule scales linearly in the number of trainable parameters and dominates the total shot budget of training at scale. In this work, we propose a framework of forward gradient estimators for PQCs, based on the forward mode of automatic differentiation, that yields an unbiased estimator of the gradient by averaging a freely tunable number of random directional derivatives and recovers SPSA, random coordinate descent, and the parameter-shift rule as limiting cases, with no ancilla qubits or controlled-gate overhead. We prove that stochastic quantum forward gradient descent converges under standard assumptions, with an explicit second-moment expansion that interpolates between the single-direction extreme of SPSA and the full-gradient extreme of parameter-shift. Within this framework we derive QUIVER (Quantum Iterative V-adaptive Estimator Rule), an adaptive optimiser for parameterised circuits whose update rule follows from a closed-form minimum measurement-cost allocation. We show numerically that forward gradients train Hamming-weight-preserving orthogonal quantum neural networks with up to 60 qubits and 1770 parameters on the ECG5000 and MNIST datasets orders of magnitude more efficiently than the parameter-shift rule. We also demonstrate that our proposed QUIVER optimiser can outperform iCANS and gCANS measurement-frugal optimisers on optimisation problems using the quantum approximate optimisation algorithm and quantum simulation with the variational quantum eigensolver.