Variance Reduction

Machine learning models involving discrete latent variables require gradient estimators to facilitate backpropagation in a computationally efficient manner. The most recent addition to the Straight-Through family of estimators, ReinMax, can be viewed from a numerical ODE perspective as incorporating an approximation via Heun's method to reduce bias, but at the cost of high variance. In this work, we introduce the ReinMax-Rao and ReinMax-CV estimators which incorporate Rao-Blackwellisation and control variate techniques into ReinMax to reduce its variance. Our estimators demonstrate superior performance on training variational autoencoders with discrete latent spaces. Furthermore, we investigate the possibility of leveraging alternative numerical methods for constructing more accurate gradient approximations and present an alternative view of ReinMax from a simpler numerical integration perspective.

Normalizing flows can be used to construct unbiased, reduced-variance estimators for lattice field theory observables that are defined by a derivative with respect to action parameters. This work implements the approach for observables involving gluonic operator insertions in the SU(3) Yang-Mills theory and two-flavor Quantum Chromodynamics (QCD) in four space-time dimensions. Variance reduction by factors of $10$-$60$ is achieved in glueball correlation functions and in gluonic matrix elements related to hadron structure, with demonstrated computational advantages. The observed variance reduction is found to be approximately independent of the lattice volume, so that volume transfer can be utilized to minimize training costs.

Bayesian inference for inverse problems involves computing expectations under posterior distributions--e.g., posterior means, variances, or predictive quantities--typically via Monte Carlo (MC) estimation. When the quantity of interest varies significantly under the posterior, accurate estimates demand many samples--a cost often prohibitive for partial differential equation-constrained problems. To address this challenge, we introduce conditional neural control variates, a modular method that learns amortized control variates from joint model-data samples to reduce the variance of MC estimators. To scale to high-dimensional problems, we leverage Stein's identity to design an architecture based on an ensemble of hierarchical coupling layers with tractable Jacobian trace computation. Training requires: (i) samples from the joint distribution of unknown parameters and observed data; and (ii) the posterior score function, which can be computed from physics-based likelihood evaluations, neural operator surrogates, or learned generative models such as conditional normalizing flows. Once trained, the control variates generalize across observations without retraining. We validate our approach on stylized and partial differential equation-constrained Darcy flow inverse problems, outperforming classical Stein control variates and achieving substantial variance reduction, even when the analytical score is replaced by a learned surrogate.

Gradient-based hyperparameter optimization (HPO) have emerged recently, leveraging bilevel programming techniques to optimize hyperparameter by estimating hypergradient w.r.t. validation loss. Nevertheless, previous theoretical works mainly focus on reducing the gap between the estimation and ground-truth (i.e., the bias), while ignoring the error due to data distribution (i.e., the variance), which degrades performance. To address this issue, we conduct a bias-variance decomposition for hypergradient estimation error and provide a supplemental detailed analysis of the variance term ignored by previous works. We also present a comprehensive analysis of the error bounds for hypergradient estimation. This facilitates an easy explanation of some phenomena commonly observed in practice, like overfitting to the validation set. Inspired by the derived theories, we propose an ensemble hypergradient strategy to reduce the variance in HPO algorithms effectively. Experimental results on tasks including regularization hyperparameter learning, data hyper-cleaning, and few-shot learning demonstrate that our variance reduction strategy improves hypergradient estimation. To explain the improved performance, we establish a connection between excess error and hypergradient estimation, offering some understanding of empirical observations.

Asynchronous reinforcement learning has become increasingly central to scaling LLM post-training, delivering major throughput gains by decoupling rollout generation from policy updates. However, widely used policy-gradient objectives such as REINFORCE and GRPO suffer under high asynchrony: stale rollouts produce heavy-tailed importance weights, so a small number of trajectories dominate updates and the policy-gradient estimator becomes markedly higher variance. Through systematic analysis on math, reasoning, and tool-use benchmarks, we find that this increasing variance is reliably predicted by collapsing effective sample size (ESS), which prior stabilization methods largely fail to address. Motivated by this diagnosis, we introduce $\textbf{V}$ariance $\textbf{C}$ontrolled $\textbf{P}$olicy $\textbf{O}$ptimization ($\textbf{VCPO}$), a method that (i) dynamically scales the learning rate with ESS to dampen unreliable updates and (ii) applies a closed-form minimum-variance baseline for off-policy settings, without a critic model and adding minimal overhead. Empirically, across math and general reasoning benchmarks, this enables robustly stable asynchronous training compared to previous stabilization and algorithmic methods, even in highly off-policy regimes (128 steps off-policy). In a long-horizon, tool-use task, VCPO matches synchronous performance while delivering a 2.5$\times$ speedup in training time. Code is available at: https://github.com/mit-han-lab/vcpo

Off-policy evaluation (OPE) is essential for assessing ranking and recommendation systems without costly online interventions. Self-Normalised Inverse Propensity Scoring (SNIPS) is a standard tool for variance reduction in OPE, leveraging a multiplicative control variate. Recent advances in off-policy learning suggest that additive control variates (baseline corrections) may offer superior performance, yet theoretical guarantees for evaluation are lacking. This paper provides a definitive answer: we prove that $β^\star$-IPS, an estimator with an optimal additive baseline, asymptotically dominates SNIPS in Mean Squared Error. By analytically decomposing the variance gap, we show that SNIPS is asymptotically equivalent to using a specific -- but generally sub-optimal -- additive baseline. Our results theoretically justify shifting from self-normalisation to optimal baseline corrections for both ranking and recommendation.

While flow matching is elegant, its reliance on single-sample conditional velocities leads to high-variance training targets that destabilize optimization and slow convergence. By explicitly characterizing this variance, we identify 1) a high-variance regime near the prior, where optimization is challenging, and 2) a low-variance regime near the data distribution, where conditional and marginal velocities nearly coincide. Leveraging this insight, we propose Stable Velocity, a unified framework that improves both training and sampling. For training, we introduce Stable Velocity Matching (StableVM), an unbiased variance-reduction objective, along with Variance-Aware Representation Alignment (VA-REPA), which adaptively strengthen auxiliary supervision in the low-variance regime. For inference, we show that dynamics in the low-variance regime admit closed-form simplifications, enabling Stable Velocity Sampling (StableVS), a finetuning-free acceleration. Extensive experiments on ImageNet $256\times256$ and large pretrained text-to-image and text-to-video models, including SD3.5, Flux, Qwen-Image, and Wan2.2, demonstrate consistent improvements in training efficiency and more than $2\times$ faster sampling within the low-variance regime without degrading sample quality. Our code is available at https://github.com/linYDTHU/StableVelocity.

Importance-weighted variational inference approximates intractable densities by optimizing multi-sample Monte Carlo objectives whose tightness increases with the number of importance samples $N$. When reparameterization is unavailable, optimization relies on score-function (REINFORCE) gradient estimators. In this paper, we provide a comprehensive analysis of score-function gradient estimators for importance-weighted objectives by characterizing their asymptotic expectation and variance. We establish that standard variational inference for Monte Carlo objectives (VIMCO) gradient estimators suffer from a vanishing signal-to-noise ratio (SNR) as $N$ increases, which arises from a suboptimal control variate choice and can impair optimization. To address this issue, we derive a general mechanism for constructing VIMCO gradient estimators that avoid SNR collapse. We identify the High-Effective Sample Size optimal variance importance sampling estimator as belonging to this category of estimators and establish its $\sqrt{N}$ SNR scaling. This estimator does not, however, have zero variance at optimality and achieving its $\sqrt{N}$ SNR rate may require prohibitively large $N$ when the importance weights are degenerate. We overcome these limitations by introducing VIMCO gradient estimators that attain zero variance at optimality and support importance weight tempering. Experiments with discrete latent-variable and state-space models validate the theoretical predictions and illustrate the benefits of our estimators.

Stochastic non-convex non-concave optimization, formally characterized as Stochastic Variational Inequalities (SVIs), presents unique challenges due to rotational dynamics and the absence of a global merit function. While adaptive step-size methods (like Armijo line-search) have revolutionized convex minimization, their application to this setting is hindered by the Stochasticity Barrier: the noise in gradient estimation masks the true operator curvature, triggering erroneously large steps that destabilize convergence. In this work, we propose VR-SDA-A (Variance-Reduced Stochastic Descent-Ascent with Armijo), a novel algorithm that integrates recursive momentum (STORM) with a rigorous Same-Batch Curvature Verification mechanism. We introduce a theoretical framework based on a Lyapunov potential tracking the Operator Norm, proving that VR- SDA-A achieves an oracle complexity of O(epsilon -3) for finding an epsilon-stationary point in general Lipschitz continuous operators. This matches the optimal rate for non-convex minimization while uniquely enabling automated step-size adaptation in the saddle-point setting. We validate our approach on canonical rotational benchmarks and non-convex robust regression tasks, demonstrating that our method effectively suppresses limit cycles and accelerates convergence with reduced dependence on manual learning rate scheduling.

Maximum likelihood estimators (MLE) and control variate estimators (CVE) have been used in conjunction with known information across sketching algorithms and applications in machine learning. We prove that under certain conditions in an exponential family, an optimal CVE will achieve the same asymptotic variance as the MLE, giving a fixed point algorithm for the MLE. Experiments show the fixed point algorithm is faster and numerically stable compared to other root finding algorithms for the MLE for the bivariate Normal distribution, and we expect this to hold across distributions satisfying these conditions. We show how this algorithm leads to reproducibility for algorithms using MLE / CVE, and demonstrate how the algorithm leads to finding the MLE when the CV weights are known.

Federated learning (FL) is increasingly used in safety-critical and fairness-sensitive applications, where learning objectives must be optimized subject to functional constraints. This setting is challenging due to costly projection onto the resulting feasible set, stochastic constraint information, partial client participation, and limited communication at edge devices. We propose a federated stochastic switching framework based on primal-only updates, avoiding dual-variable tuning for functionally constrained federated optimization. For convex objectives, the framework further supports compression with error feedback to control compression-induced bias. We establish high-probability convergence and feasibility guarantees under stochastic gradients and partial client participation, and extend the analysis to weakly convex objectives through proximal-stationarity guarantees. Our bounds isolate the effects of stochastic gradients, constraint-evaluation noise, and partial participation, and show that direct switching can retain a nonvanishing constraint-estimation error even with exact local constraint evaluations. To overcome this limitation, we develop variance-reducing constraint estimators that yield vanishing feasibility and stationarity guarantees with fixed per-round client participation and constraint-evaluation batch sizes. Experiments on Neyman-Pearson classification, fairness-constrained learning on the ADULT dataset, and constrained deep reinforcement learning demonstrate the effectiveness of the proposed framework.

Uniform-noise discrete diffusion and flow models generate sequences non-autoregressively through iterative, context-dependent token replacements. However, these models are typically formulated as time-inhomogeneous continuous- or discrete-time Markov chains (CTMC/DTMC), sampled using independent Bernoulli change decisions per discretization step. This induces Poisson-binomial variance in per-position jump counts that grows with the number of required edits, leading to the common under-editing (residual noise) and over-editing (cascading substitutions) failure modes that degrade sample quality. We identify this sampler-induced variance as an orthogonal source of degradation, distinct from model-side errors and addressable purely at inference time. We propose Systematic Hazard Sampling (SHS), a training-free, drop-in, and hyperparameter-free inference principle for any sampler that admits a stay-vs.-replace decomposition. SHS models per-token edits as events driven by cumulative hazard (CTMC) or jump mass (DTMC) and triggers an edit whenever this quantity exceeds unit-spaced thresholds with a single random phase per position. For any fixed cumulative mass, this preserves the expected jump count while achieving the minimum conditional variance possible among unbiased integer estimators (at most 1/4), without altering per-jump destination sampling. Experiments on four uniform-noise discrete diffusion and flow language models spanning ~110M to ~3B parameters show that SHS consistently improves sample quality across numbers of function evaluations (NFE). Code is available at https://github.com/Jang-seunghwan/Systematic-Hazard-Sampling.

Sampling from unnormalized probability densities is a pervasive challenge across the computational and physical sciences. Diffusion models provide a powerful generative framework for this task, but their success relies on accurately estimating the score of the perturbed target distribution. Current approaches face a dichotomy between two standard estimation methods: the Denoising Score Identity (DSI) requires data samples and exhibits high variance at low noise levels, whereas the Target Score Identity (TSI) relies on the energy function and suffers from diverging variance at high noise levels. In this work, we reconcile both approaches by introducing the Control Variate Score Identity (CVSI), an unbiased estimator with an analytically optimal, state- and time-dependent control coefficient that theoretically minimizes variance over the entire diffusion process. CVSI serves as a robust plug-in estimator that significantly enhances performance and efficiency in data-free sampler learning and training-free diffusion sampling. These gains scale to complex, high-dimensional energy-based models.

Performative prediction (PP) is an algorithmic framework for optimizing machine learning (ML) models where the model's deployment affects the distribution of the data it is trained on. Compared to traditional ML with fixed data, designing algorithms in PP converging to a stable point -- known as a stationary performative stable (SPS) solution -- is more challenging than the counterpart in conventional ML tasks due to the model-induced distribution shifts. While considerable efforts have been made to find SPS solutions using methods such as repeated gradient descent (RGD) and greedy stochastic gradient descent (SGD-GD), most prior studies assumed a strongly convex loss until a recent work established $O(1/\sqrt{T})$ convergence of SGD-GD to SPS solutions under smooth, non-convex losses. However, this latest progress is still based on the restricted bounded variance assumption in stochastic gradient estimates and yields convergence bounds with a non-vanishing error neighborhood that scales with the variance. This limitation motivates us to improve convergence rates and reduce error in stochastic optimization for PP, particularly in non-convex settings. Thus, we propose a new algorithm called stochastic performative prediction with variance reduction (SPRINT) and establish its convergence to an SPS solution at a rate of $O(1/T)$. Notably, the resulting error neighborhood is independent of the variance of the stochastic gradients. Experiments on multiple real datasets with non-convex models demonstrate that SPRINT outperforms SGD-GD in both convergence rate and stability.

We develop sampling methods, which consist of Gaussian invariant versions of random walk Metropolis (RWM), Metropolis adjusted Langevin algorithm (MALA) and second order Hessian or Manifold MALA. Unlike standard RWM and MALA, we show that Gaussian invariant sampling can lead to ergodic estimators with improved statistical efficiency. This is due to a remarkable property of Gaussian invariance that allows us to obtain exact analytical solutions to the Poisson equation for Gaussian targets. These solutions can be used to construct efficient and easy to use control variates for variance reduction of estimators under any intractable target. We demonstrate the new samplers and estimators in several examples, including high dimensional targets in latent Gaussian models where we compare against several advanced methods and obtain state-of-the-art results. We also provide theoretical results regarding geometric ergodicity, and an optimal scaling analysis that shows the dependence of the optimal acceptance rate on the Gaussianity of the target.

Assume that we would like to estimate the expected value of a function $F$ with respect to an intractable density $π$, which is specified up to some unknown normalising constant. We prove that if $π$ is close enough under KL divergence to another density $q$, an independent Metropolis sampler estimator that obtains samples from $π$ with proposal density $q$, enriched with a variance reduction computational strategy based on control variates, achieves smaller asymptotic variance than i.i.d. sampling from $π$. The control variates construction requires no extra computational effort but assumes that the expected value of $F$ under $q$ is analytically available. We illustrate this result by calculating the marginal likelihood in a linear regression model with prior-likelihood conflict and a non-conjugate prior. Furthermore, we propose an adaptive independent Metropolis algorithm that adapts the proposal density such that its KL divergence with the target is being reduced. We demonstrate its applicability in a Bayesian logistic and Gaussian process regression problems and we rigorously justify our asymptotic arguments under easily verifiable and essentially minimal conditions.

We study stochastic Cubic Newton methods for solving general possibly non-convex minimization problems. We propose a new framework, which we call the helper framework, that provides a unified view of the stochastic and variance-reduced second-order algorithms equipped with global complexity guarantees. It can also be applied to learning with auxiliary information. Our helper framework offers the algorithm designer high flexibility for constructing and analyzing the stochastic Cubic Newton methods, allowing arbitrary size batches, and the use of noisy and possibly biased estimates of the gradients and Hessians, incorporating both the variance reduction and the lazy Hessian updates. We recover the best-known complexities for the stochastic and variance-reduced Cubic Newton, under weak assumptions on the noise. A direct consequence of our theory is the new lazy stochastic second-order method, which significantly improves the arithmetic complexity for large dimension problems. We also establish complexity bounds for the classes of gradient-dominated objectives, that include convex and strongly convex problems. For Auxiliary Learning, we show that using a helper (auxiliary function) can outperform training alone if a given similarity measure is small.