In this paper, we analyze the two time-scale stochastic approximation (TTSSA) algorithm introduced in Borkar (1997) using a martingale approach. This approach leads to simple sufficient conditions for the iterations to be bounded almost surely, as well as estimates on the rate of convergence of the mean-squared error of the TTSSA algorithm to zero. Our theory is applicable to nonlinear equations, in contrast to many papers in the TTSSA literature which assume that the equations are linear. The convergence of TTSSA is proved in the "almost sure" sense, in contrast to earlier papers on TTSSA that establish convergence in distribution, convergence in the mean, and the like. Moreover, in this paper we establish different rates of convergence for the fast and the slow subsystems, perhaps for the first time. Finally, all of the above results to continue to hold in the case where the two measurement errors have nonzero conditional mean, and/or have conditional variances that grow without bound as the iterations proceed. This is in contrast to previous papers which assumed that the errors form a martingale difference sequence with uniformly bounded conditional variance. It is shown that when the measurement errors have zero conditional mean and the conditional variance remains bounded, the mean-squared error of the iterations converges to zero at a rate of $o(t^{-η})$ for all $η\in (0,1)$. This improves upon the rate of $O(t^{-2/3})$ proved in Doan (2023) (which is the best bound available to date). Our bound is virtually the same as the rate of $O(t^{-1})$ proved in Doan (2024), but for a Polyak-Ruppert averaged version of TTSSA, and not directly. Rates of convergence are also established for the case where the errors have nonzero conditional mean and/or unbounded conditional variance.

It also does not enjoy SNIS's inherent stability and boundedness.

Many recent reasoning gains in large language models can be explained as distribution sharpening: biasing generation toward high-likelihood trajectories already supported by the pretrained model, rather than modifying its weights. A natural formalization is the sequence-level power distribution $π_α(y\mid x)\propto p_θ(y\mid x)^α$ ($α>1$), which concentrates mass on whole sequences instead of adjusting token-level temperature. Prior work shows that Metropolis--Hastings (MH) sampling from this distribution recovers strong reasoning performance, but at order-of-magnitude inference slowdowns. We introduce Power-SMC, a training-free Sequential Monte Carlo scheme that targets the same objective while remaining close to standard decoding latency. Power-SMC advances a small particle set in parallel, corrects importance weights token-by-token, and resamples when necessary, all within a single GPU-friendly batched decode. We prove that temperature $τ=1/α$ is the unique prefix-only proposal minimizing incremental weight variance, interpret residual instability via prefix-conditioned Rényi entropies, and introduce an exponent-bridging schedule that improves particle stability without altering the target. On MATH500, Power-SMC matches or exceeds MH power sampling while reducing latency from $16$--$28\times$ to $1.4$--$3.3\times$ over baseline decoding.

A fundamental problem of causal discovery is cause-effect inference, to learn the correct causal direction between two random variables. Significant progress has been made through modelling the effect as a function of its cause and a noise term, which allows us to leverage assumptions about the generating function class. The recently introduced heteroscedastic location-scale noise functional models (LSNMs) combine expressive power with identifiability guarantees. LSNM model selection based on maximizing likelihood achieves state-of-the-art accuracy, when the noise distributions are correctly specified. However, through an extensive empirical evaluation, we demonstrate that the accuracy deteriorates sharply when the form of the noise distribution is misspecified by the user. Our analysis shows that the failure occurs mainly when the conditional variance in the anti-causal direction is smaller than that in the causal direction. As an alternative, we find that causal model selection through residual independence testing is much more robust to noise misspecification and misleading conditional variance.

In this study, a scalable online kernel learning framework is proposed for estimating bidirectional causal effects in systems characterized by mutual dependence and heteroskedasticity. Traditional causal inference often focuses on unidirectional effects, overlooking the common bidirectional relationships in real-world phenomena. Building on heteroskedasticity-based identification, the proposed method integrates a quasi-maximum likelihood estimator for simultaneous equation models with large scale online kernel learning. It employs random Fourier feature approximations to flexibly model nonlinear conditional means and variances, while an adaptive online gradient descent algorithm ensures computational efficiency for streaming and high-dimensional data. Results from extensive simulations demonstrate that the proposed method achieves superior accuracy and stability than single equation and polynomial approximation baselines, exhibiting lower bias and root mean squared error across various data-generating processes. These results confirm that the proposed approach effectively captures complex bidirectional causal effects with near-linear computational scaling. By combining econometric identification with modern machine learning techniques, the proposed framework offers a practical, scalable, and theoretically grounded solution for large scale causal inference in natural/social science, policy making, business, and industrial applications.

AI has the potential to transform scientific discovery by analyzing vast datasets with little human effort. However, current workflows often do not provide the accuracy or statistical guarantees that are needed. We introduce active measurement, a human-in-the-loop AI framework for scientific measurement. An AI model is used to predict measurements for individual units, which are then sampled for human labeling using importance sampling. With each new set of human labels, the AI model is improved and an unbiased Monte Carlo estimate of the total measurement is refined. Active measurement can provide precise estimates even with an imperfect AI model, and requires little human effort when the AI model is very accurate. We derive novel estimators, weighting schemes, and confidence intervals, and show that active measurement reduces estimation error compared to alternatives in several measurement tasks.

We study the problem of efficiently estimating the mean of a $k$-class random variable, $Y$, using a limited number of labels, $N$, in settings where the analyst has access to auxiliary information (i.e.: covariates) $X$ that may be informative about $Y$. We propose an active learning algorithm ("PartiBandits") to estimate $\mathbb{E}[Y]$. The algorithm yields an estimate, $\widehatμ_{\text{PB}}$, such that $\left( \widehatμ_{\text{PB}} - \mathbb{E}[Y]\right)^2$ is $\tilde{\mathcal{O}}\left( \frac{ν+ \exp(c \cdot (-N/\log(N))) }{N} \right)$, where $c > 0$ is a constant and $ν$ is the risk of the Bayes-optimal classifier. PartiBandits is essentially a two-stage algorithm. In the first stage, it learns a partition of the unlabeled data that shrinks the average conditional variance of $Y$. In the second stage it uses a UCB-style subroutine ("WarmStart-UCB") to request labels from each stratum round-by-round. Both the main algorithm's and the subroutine's convergence rates are minimax optimal in classical settings. PartiBandits bridges the UCB and disagreement-based approaches to active learning despite these two approaches being designed to tackle very different tasks. We illustrate our methods through simulation using nationwide electronic health records. Our methods can be implemented using the PartiBandits package in R.

We study the problem of learning the topology of a directed Gaussian Graphical Model under the equal-variance assumption, where the graph has $n$ nodes and maximum in-degree $d$. Prior work has established that $O(d \log n)$ samples are sufficient for this task. However, an important factor that is often overlooked in these analyses is the dependence on the condition number of the covariance matrix of the model. Indeed, all algorithms from prior work require a number of samples that grows polynomially with this condition number. In many cases this is unsatisfactory, since the condition number could grow polynomially with $n$, rendering these prior approaches impractical in high-dimensional settings. In this work, we provide an algorithm that recovers the underlying graph and prove that the number of samples required is independent of the condition number. Furthermore, we establish lower bounds that nearly match the upper bound up to a $d$-factor, thus providing an almost tight characterization of the true sample complexity of the problem. Moreover, under a further assumption that all the variances of the variables are bounded, we design a polynomial-time algorithm that recovers the underlying graph, at the cost of an additional polynomial dependence of the sample complexity on $d$. We complement our theoretical findings with simulations on synthetic datasets that confirm our predictions.