SAUSS: Stochastic Approximation with Unbiased Simulated Scores for Limited Dependent Variable Models
Authors: Sokbae Lee, Yuan Liao, Myung Hwan Seo, Youngki Shin
Organizations: Department of Economics, Columbia University, New York, NY 10027, USA. · Department of Economics, University of Iowa, Iowa City, IA 52242, USA. · Department of Economics, The Hong Kong University of Science and Technology, Hong Kong. · Department of Economics, McMaster University, Hamilton, ON L8S 4L8, Canada.
Simulated maximum likelihood is widely used in latent-variable models where likelihood evaluation requires intractable integrals or probabilities. With a fixed number of simulation draws, its estimating equation is generally biased. We develop Stochastic Approximation with Unbiased Simulated Scores (SAUSS), combining score simulation with stochastic gradient descent. The Hajivassiliou--McFadden identity, an instance of Fisher's identity, expresses the likelihood score as the conditional expectation of the complete-data score given the observed data. Exact conditional draws yield an unbiased simulated score for any fixed number of draws. The estimator averages stochastic gradient iterates computed using mini-batches of fixed size regardless of sample size, avoiding repeated full-sample likelihood evaluations. Under local regularity and initialization conditions, we establish asymptotic normality and a functional central limit theorem for the partial-sum process of the iterates under multi-pass sampling from an observed sample. These results support plug-in and random-scaling confidence intervals that remain asymptotically valid when algorithmic uncertainty is non-negligible relative to sampling uncertainty. We develop multinomial probit as a leading application. In simulations and two applications, SAUSS returns estimates in every reported run, including cases where the simulated maximum likelihood implementation returns none. When both return estimates, SAUSS gives comparable estimates in less time.
Figures & tables
Ytb∗∣{Ytb∗∈Dtb}
Algorithm 1 SAUSS
Panel A: Regression coefficients
Method
Child skills
Family outcomes
Maternal earnings (1,000 EUR)
Family’s opinion
Friends’ opinion
Logit
0.6061)
1.2351)
0.0041)
0.2326)
0.2122)
(0.1617)
(0.1463)
(0.0017)
(0.0255)
(0.0254)
GHK SML
0.4553)
1.1614)
0.0033)
0.2689)
0.2616)
(0.1684)
(0.1876)
(0.0014)
(0.0365)
(0.0388)
SAUSS
0.4673)
1.2376)
0.0038)
0.2814)
0.2817)
Table 1: Logit and multinomial probit estimates, maternal labor-supply intentions
Langevin-based Thompson Sampling (TS) methods for multi-armed bandits use likelihood gradients to approximate posterior samples when exact sampling is intractable. As new rewards arrive, the posterior changes, and these methods run an inner Markov chain to approximate the updated target. To simplify this repeated sampling procedure, we introduce TS-SA, which incorporates stochastic approximation (SA) within the TS framework. In each round, TS-SA takes one Langevin step using the newly observed reward and applies an SA update to combine the noisy proposal with the current parameter estimate. By retaining this estimate across rounds, TS-SA maintains one continuing Markov chain for each arm, preserving the influence of past observations without storing or sampling them again. Within this recursion, the proposal step size remains fixed across rounds, while the SA weights decrease as observations accumulate to control the effect of noise on the estimate. This continuing-chain formulation simplifies the theoretical analysis, allowing us to establish a near-optimal worst-case regret bound of O(KT) under log-concavity and smoothness assumptions with one update per new reward. Our experiments further demonstrate that TS-SA, using a single-step Langevin update after an initial warm-up, achieves strong performance against standard baselines.
Weixin Wang, Haoyang Zheng, Guang Lin +2
Duke University · Purdue University · Morgan Stanley
Neural likelihood surrogates (e.g., Neural Ratio Estimation) for stochastic process models are commonly trained via probabilistic classification on simulated data, which forces a tradeoff between surrogate quality and training costs. For structured models where the exact score ∇θlogp(x∣θ) is available, this information can be incorporated into training by augmenting the cross-entropy loss with a score-matching term. However, the optimal weighting of the two losses is not known a priori, and selecting it by hand requires expensive tuning that undercuts the computational savings. We propose an adaptive, tuning-free algorithm that sets the score loss weights during training based on loss gradients, adding minimal overhead to standard classifier training. We evaluate our approach on case studies involving network dynamics and spatial processes, demonstrating that it improves surrogate quality at a drastically lower computational cost than generating more training data. Notably, in some cases, our approach achieves downstream inference performance equivalent to a 10x increase in training data with less than a 1.1x increase in training time.
Alexander Shen, Mikael Kuusela
Department of Statistics and Data Science Carnegie Mellon University Pittsburgh, PA 15213
For constant-stepsize stochastic approximation (SA), the iterates converge in distribution to a stationary law that depends on the stepsize α. Steady-state convergence (SSC) concerns the limit of the scaled stationary distribution as α↓0. Existing SSC theory requires i.i.d. or additive noise and global differentiability of the mean operator, and yields suboptimal rates. We develop a unified SSC theory for constant-stepsize contractive SA driven by Markovian, multiplicative noise, covering both locally differentiable and locally nondifferentiable mean operators. A key methodological contribution is a multi-step universality framework that progressively reduces the original stochastic recursion to tractable auxiliary dynamics while preserving its steady-state limit. Under local quadratic linearization at the fixed point, we obtain a Gaussian approximation of the scaled steady state at the optimal rate O(α) in Wasserstein-2 distance, which further gives finite-time Gaussian approximations for the raw iterates. In the locally nondifferentiable regime, we establish a general SSC result and show that the leading-order asymptotic bias can be of order α, in contrast to the α-order bias in the smooth regime. We apply the theory to Markovian linear SA and asynchronous Q-learning, neither of which is covered by prior results. We further propose a bias-reduction scheme for Q-learning that requires no knowledge of the local smoothness regime, validated by numerical experiments.
Yixuan Zhang, Qiaomin Xie
Department of Industrial & Systems Engineering, University of Wisconsin–Madison