The blind spot concerns consistency, the property that as training data grows without bound, the learned classifier’s error converges to the best achievable error, known as the Bayes error. Consistency is the most fundamental guarantee one can ask of a learning algorithm: without it, there is no assurance that more data actually helps. For classical statistical methods such as kernel estimators and nearest-neighbor rules, consistency was established decades ago. For greedy decision tree algorithms like CART, however, the question has remained stubbornly open, even as the algorithms conquered industry. The new paper shows why the question is so hard, and why the answer is worse than many theorists feared.

The authors demonstrate a counterexample in which the worst-case purity gain of any possible split is exactly zero, even when the training error is far from zero. The construction is a real-valued analogue of the XOR problem: features are distributed uniformly on the unit square, and the target function is the exclusive OR of whether each coordinate exceeds one half. Every axis-aligned cut leaves both children with the same class balance as the parent, so the impurity, measured by a standard function such as the Gini index, does not decrease at all. Because greedy algorithms stop or stall when purity gain vanishes, a zero gain typically leads to a non-convergent training error. In other words, there exist perfectly reasonable learning problems on which the greedy purity-maximization heuristic that underlies CART simply cannot make progress.

This obstruction is not a marginal curiosity. The paper proves a sharp characterization: under a product feature distribution, a greedy tree learner obtains no purity gain if and only if it has already reached zero error. The proof exploits the strong concavity of standard impurity functions, which lower bounds the purity gain of any split by a quantity proportional to the squared difference of conditional class means between the two children. When that difference is zero everywhere, the target function must be constant within every leaf, which is precisely the condition of zero error. The result connects decision tree learning to the classical theory of influences of variables on Boolean functions, a field pioneered by Kahn, Kalai and Linial in 1988, and shows that the impurity measure acts as a potential whose decrease is tied to the variance of the target.

The escape route proposed by the authors is disarmingly simple: restrict candidate split points to a fixed grid. Their earlier work introduced GridCART, a decision tree in which cuts are allowed only at the boundaries of a uniform grid over each feature dimension. This restriction guarantees that whenever the target function is grid-aligned, meaning it is piecewise constant on grid cells, some split always yields strictly positive purity gain. The paper quantifies this with a lower bound showing that the best purity gain over grid cuts is at least the squared variance of the target divided by a polynomial in the number of grid pieces. A potential argument then shows that the impurity of the tree decays inversely with depth, yielding a fitting error of order N cubed over K, where N is the number of grid pieces and K the tree depth.

Combining this fitting guarantee with classical results on histogram density estimation and histogram classifiers produces the headline theorem: GridCART is consistent with a rate of order n to the power of minus one over d plus two, where n is the sample size and d the dimension, provided the conditional probability function is Lipschitz smooth and the features follow a product distribution. The rate is achieved by choosing the grid width on the order of n to the minus one over d plus two, which balances the bias of approximating a smooth function on coarse cells against the variance of estimating cell probabilities from finite samples. The analysis extends beyond binary classification to multiclass problems, where the rate picks up a factor of the number of classes, and to regression under squared loss, where an Efron-Stein inequality replaces the Boolean influence machinery.

The truly new contribution of this article, however, is the extension of the grid-based construction to ensemble methods. The authors introduce GridForest, a random forest in which each tree is grown on grid-restricted cuts with the usual random selection of a subset of features at every node. Because the best feature is included in the sampled subset with probability m over d, where m is the subset size, the potential decrease slows by exactly that factor, and the consistency analysis carries through with an extra d over m in the bound. Under an additional margin condition of the kind introduced by Audibert and Tsybakov, which assumes the conditional probability rarely sits near the decision boundary of one half, GridForest achieves a fast rate of n to the minus alpha plus one over d plus two, matching the optimal rates known for plug-in classifiers. The proof carefully handles boundary cells, density estimation error, and the transfer of guarantees from the estimated to the true distribution.

The third algorithm, GridGBDT, applies the same grid restriction to gradient boosting decision trees, the family of methods behind celebrated systems such as XGBoost, LightGBM and CatBoost. In the square-loss specialization analyzed here, each boosting round fits a grid-based regression tree to the current residuals on the histogram estimate. The authors prove a contraction lemma showing that the square root of the squared error contracts by a factor of one minus the shrinkage at every round, plus a term governed by the per-round fitting accuracy. Unrolling this recursion over M rounds yields an excess error bound in which the optimization error decays exponentially in the number of rounds while the statistical terms match those of GridCART. Choosing the number of boosting rounds on the order of log n suffices, and under a margin condition the required tree depth can even be reduced.

Skeptics might worry that restricting splits to a grid sacrifices the adaptivity that makes conventional trees so effective in practice. The empirical evidence suggests otherwise. Across twenty datasets drawn from the OpenML repository, the public benchmarking platform for networked machine learning science, both GridForest and GridGBDT performed comparably to their conventional counterparts, random forests and standard gradient boosting decision trees. The grid-based ensembles matched predictive accuracy while carrying something their competitors lack: provable convergence guarantees. The code and experimental results are slated for release alongside the publication, and the datasets themselves are freely available, making the claims straightforward to verify and build upon.

The significance of this work extends beyond one algorithm family. It closes a conceptual gap that has lingered since the 1980s, showing that the greedy purity-gain heuristic is not merely hard to analyze but provably broken in the worst case, and that a minimal, computationally cheap modification restores full theoretical soundness. Grid-based splitting adds essentially no overhead, since candidate cut points are fixed in advance rather than searched over continuous ranges. As machine learning increasingly demands algorithms whose reliability can be certified, the message of this research is resonant: sometimes the path to trustworthy artificial intelligence runs not through exotic new architectures, but through a careful re-examination of the humble tools already running the world, and a grid drawn at exactly the right spacing.

Subject of Research: Theoretical consistency guarantees for grid-based decision tree, random forest, and gradient boosting learning algorithms

Article Title: Grid-based Decision Tree Learning

Article References: Zheng, Q.-C., Zhang, S.-Q., Lyu, S.-H., Jiang, Y., & Zhou, Z.-H. (2026). Grid-based Decision Tree Learning. Machine Learning, 115(10), Article 242. https://doi.org/10.1007/s10994-026-07165-0

Image Credits: AI Generated

DOI: 10.1007/s10994-026-07165-0

Keywords: decision trees, CART, consistency, random forests, gradient boosting, statistical learning theory, purity gain, ensemble learning, histogram classifiers, machine learning theory, OpenML, XGBoost

The research team, led by José Alejandro Fernández Fernández with Renata Kubus and Inés Martín de Santos, drew its data from the Refinitiv Eikon database and constructed a balanced panel of 666 unique firms, yielding 2,612 estimation observations after excluding a small number of records with missing ESG outcomes. Rather than relying on conventional linear econometrics, which the authors argue can obscure threshold-dependent and nonlinear relationships, they deployed random forests built from 500 regression trees, each constrained to a maximum depth of eight, alongside regression-tree benchmarks, permutation importance, SHAP values, and partial dependence plots.

The methodological care is notable. All preprocessing, including winsorization of extreme values at the first and ninety-ninth percentiles, median imputation of missing predictors, and a logarithmic transformation of total assets, was fitted strictly within each training fold and then applied to validation data, preventing information leakage. The train-test split was grouped by firm, so no company’s observations appeared in both partitions. A separate temporal validation trained the models on 2019 to 2021 and tested them on 2022, and out-of-bag estimates provided an additional ensemble-based diagnostic.

Predictive performance varied meaningfully across industries. Financials proved the most predictable sector, achieving a grouped cross-validated R-squared of 0.415 and a temporal test score of 0.417, with out-of-bag performance reaching 0.571. Utilities followed with a temporal R-squared of 0.363, while Industrials and Basic Materials landed in the mid-range. Energy showed the least stable results, with a negative single grouped holdout but positive five-fold and temporal scores, a dispersion the authors attribute to the sector’s smaller sample and warrant cautious interpretation.

When the researchers jointly permuted six analytical blocks of predictors on the 2022 holdout, the size signal proved overwhelming. Disrupting the size block degraded test performance by a mean R-squared decline of 0.650 in Financials, 0.566 in Utilities, 0.488 in Energy, and 0.487 in Industrials, with Basic Materials at 0.348. Leverage ranked second in four of the five sectors, while profitability took that position in Financials. This block-level evidence matters because correlated accounting ratios can redistribute individual importance scores; by permuting whole blocks at once, the team confirmed that the dominance of firm size is not an artifact of importance being split among overlapping variables.

Beyond the shared size effect, the sectors diverged sharply. In Energy and Utilities, the predictive structure was comparatively concentrated, centered on firm scale and capital expenditure intensity. Intriguingly, capital expenditure showed predominantly negative nonlinear associations with predicted ESG scores beyond moderate thresholds in both sectors, suggesting that spending more on plant and equipment does not automatically translate into stronger sustainability ratings. The authors caution that this pattern may reflect differences in the orientation and efficiency of capital allocation rather than investment volume itself, though the observational design cannot establish the mechanism.

Industrials and Basic Materials displayed the most heterogeneous and nonlinear configurations. In Industrials, the assets-to-equity ratio emerged as the second most relevant predictor, with free operating cash flow showing negative associations beyond levels near 0.14 and debt service contributing threshold-dependent effects across firm-size regions. Basic Materials exhibited a U-shaped relationship between EBITDA-to-equity and predicted ESG scores among smaller firms, with the direction changing around values close to 0.28. Financials, by contrast, presented the most concentrated and stable structure of all, with size dominating and profitability indicators such as EBITDA-to-equity and EBITDA-to-total-assets providing secondary contributions, while debt variables played a comparatively minor role.

The study also documents a broader convergence story in the raw ESG data. Average scores rose across all five sectors between 2019 and 2022, with Energy posting the strongest growth at 39 percent, followed by Basic Materials at 27 percent and Industrials at 24 percent, while Utilities and Financials improved more modestly at 18 and 17 percent. Meanwhile, the coefficient of variation declined across sectors, indicating that ESG practices are converging, even as Financials and Energy retain higher internal dispersion, pointing to uneven adoption within those industries.

The policy implications are interpretive rather than causal, as the authors emphasize, but they are consequential. The dominance of firm size suggests that ESG integration may be structurally easier for large corporations with sophisticated governance and reporting infrastructure, implying that smaller and medium-sized firms face proportionally higher compliance costs and could benefit from more proportional disclosure requirements. The sectoral heterogeneity in how leverage, profitability, and investment relate to ESG scores further argues against one-size-fits-all regulation, particularly for capital-intensive industries where financing structure matters most. And the finding that higher investment intensity sometimes accompanies weaker ESG outcomes underscores the need to distinguish investment quality from quantity when designing sustainability incentives.

Perhaps the most sobering takeaway for investors is that market valuation signals, captured through the historic earnings-to-price ratio, showed weak and often negative alignment with ESG performance in several sectors, hinting that sustainability information may not be fully priced into American equities. The authors argue that strengthening ESG transparency, comparability, and interpretability could improve market efficiency and channel capital more effectively. What the study ultimately delivers is a nuanced map: ESG performance carries a common scale-related backbone, but the financial characteristics that matter beyond scale depend fundamentally on the economic structure of each industry, a conclusion that resists the temptation of a single homogeneous financial profile for sustainability success.

Subject of Research: Machine learning analysis of the relationship between financial health indicators and ESG scores in US firms

Article Title: “Machine learning analysis of ESG score and financial health in the USA”

Article References: “Machine learning analysis of ESG score and financial health in the USA”. (n.d.). Original publication

Image Credits: AI Generated

DOI: Not provided

Keywords: ESG, machine learning, random forests, SHAP values, S&P 1500, firm size, financial health, sustainable finance, corporate governance, sectoral analysis, explainable AI, Refinitiv Eikon

The study, published in the journal Discover Conservation, is a systematic review of more than three decades of bat bioacoustics research. Researchers Luna Pino-Aedo and Andrés Muñoz-Sáez of the Universidad de Chile searched the ISI Web of Science database for studies published between 1990 and 2022, the period since passive acoustic monitoring of terrestrial ecosystems began in earnest. Their initial queries returned 531 candidate articles, and after screening each one at full text, 207 studies met the criteria: they analyzed bat echolocation calls and specified exactly how those calls were analyzed. From this body of work the authors catalogued 34 distinct echolocation analysis methods, spanning commercial software packages and a growing arsenal of supervised machine learning techniques.

The review reveals a field that has expanded explosively. Most of the included studies appeared after 2010, a trend the authors attribute to the falling cost and rising availability of automated ultrasonic recorders, with a notable peak in publications in 2021 that may reflect pandemic-era restrictions pushing researchers toward data analysis rather than fieldwork. Geographically, the science is heavily skewed: the United States accounted for 36.28 percent of the articles, followed by China with 12.56 percent. That concentration matters, because reference libraries of bat calls and the classifiers trained on them are richest for Nearctic and Palearctic species, leaving tropical hotspots of bat diversity comparatively underserved.

Echolocation itself is a remarkable biological technology. Bats generate ultrasonic pulses in the larynx and emit them through the mouth or nose; the returning echoes build an acoustic image of the surroundings that the animals use to navigate, forage and communicate. Because each species tends to produce calls with characteristic frequencies, durations and shapes, researchers realized decades ago that recordings could be used to identify species without ever catching them. Bat detectors from manufacturers such as Pettersson, Titley Scientific, Wildlife Acoustics and Avisoft dominated the reviewed studies, capturing calls at high sampling frequencies for later analysis through spectrograms and oscillograms.

But turning a recording into a species name is where things get complicated. Among the 118 studies whose primary goal was species identification, only 29 relied on fully automated identification; 55 used manual identification by a specialist, and 34 used a mixed approach in which experts verified or complemented machine output. The most frequently used software packages were Kaleidoscope, Avisoft, Analook, BatSound and SonoBat, often made by the same companies that build the detectors. While all of these tools visualize calls and filter noise, only some, notably Kaleidoscope and SonoBat, were actually used for automated species classification; the rest served mainly for manual inspection and parameter measurement.

Alongside commercial software, the review documented a parallel rise of supervised learning methods implemented in platforms such as Python and RStudio. Discriminant function analysis, a statistical technique that models predefined classes from numerical acoustic features, was among the most common, followed by artificial neural networks, random forests and support vector machines. Each brings distinct trade-offs. In a benchmark study by Armitage and Ober, artificial neural networks achieved the highest training accuracy at 86.1 percent of calls correctly identified but demanded the most computational resources and training time. Random forests delivered the best overall combination of accuracy, sensitivity and specificity at 85.5 percent while remaining fast to train. Discriminant function analysis was quickest but least accurate on average at 75.9 percent. In Uruguay, Botto Nuñez and colleagues found random forests outperformed both support vector machines and neural networks, correctly identifying 92.9 percent of calls with lower variance than either alternative.

Head-to-head comparisons under controlled conditions expose the gap between machine learning and commercial packages most starkly. Tabak and colleagues built a deep learning model based on convolutional neural networks that achieved an average accuracy of 91 percent, while Kaleidoscope Pro and BCID managed only 65 percent and 61 percent respectively on the same task. Earlier discriminant function and neural network approaches also outperformed the modern software evaluated by Rydell and colleagues, whose study found that the best automated program still fell short of the top human expert, though the software did reliably classify calls at the genus level and distinguished species with highly distinctive calls. Brabant and colleagues, testing four programs on nine European species, found BatIdent achieved the highest proportion of correct species-level identifications but also produced more misclassifications, while Kaleidoscope left the largest share of calls unidentified. Sensitivity settings materially changed results: Kaleidoscope’s accuracy rose under more sensitive configurations, whereas SonoChiro stayed relatively stable across settings.

Perhaps most disquieting is what happens when there is no ground truth at all. Studies by Lemen and by Nocera that assessed agreement between software packages without a reference identification found concordance rates of only around 40 percent, varying by species and sampling period. Agreement between programs, the authors conclude, is not a reliable proxy for accuracy. Even software version matters in unpredictable ways: Goodwin and Gillam found the best performance from Kaleidoscope Pro version 4.3.0 and SonoBat version 3.2.2, meaning newer releases were not necessarily better classifiers. Accuracy reporting itself remains rare, with only 28 of the 207 studies quantifying how often their methods classified calls correctly.

The review also catalogues the tangled web of factors that influence identification beyond the algorithms themselves. Call type matters, since search, approach, feeding and social calls differ within a single species, and many species produce calls so similar that even experts struggle. Recording context matters, whether calls come from free-flying animals or hand-released individuals. Weather matters, with temperature shifting peak frequencies. So do geography, age, morphology and even the roost a population occupies, all of which inject intraspecific variation that classifiers trained on narrow libraries cannot handle. Habitat structure and detector placement also shape outcomes: models trained on calls recorded at structurally complex forest edges outperformed those trained in open spaces, and recording angle significantly affected correct classification.

The authors’ prescription is a hybrid future. Automated classifiers should be treated as accelerators that triage enormous audio volumes and flag candidate events for expert review, not as stand-alone oracles, and they should be deployed unsupervised only for taxa with highly distinctive calls. Capture-based methods such as mist-netting remain indispensable, both for species with highly directional sound beams that evade standard detectors and for building verified reference datasets; combined acoustic-and-capture designs consistently detect more species than either method alone. The researchers call for open, regionally representative call libraries with detailed metadata on habitat and recording context, versioned benchmark datasets spanning detectors and ecoregions, and transparent reporting of complete analysis pipelines. With bat populations declining worldwide amid accelerating biodiversity loss, the stakes of getting this right could hardly be higher: misidentified calls can translate directly into misinformed conservation decisions for one of the planet’s most important, and most imperiled, groups of mammals.

Subject of Research: Systematic review of bat echolocation call analysis methods for species identification and conservation monitoring

Article Title: A systematic review of bat echolocation analysis methods and their implications for monitoring and conservation

Article References: Pino-Aedo, L., & Muñoz-Sáez, A. (2026). A systematic review of bat echolocation analysis methods and their implications for monitoring and conservation. Discover Conservation, 3(1), Article 17. https://doi.org/10.1007/s44353-026-00087-x

Image Credits: AI Generated

DOI: 10.1007/s44353-026-00087-x

Keywords: bats, echolocation, bioacoustics, machine learning, acoustic monitoring, species identification, conservation, neural networks, random forests, Kaleidoscope, passive acoustic monitoring, biodiversity