Stability and Generalization in Looped Transformers

Looped transformers promise test-time compute scaling by spending more iterations on harder problems, but it remains unclear which architectural choices let them extrapolate to harder problems at test time rather than memorize training-specific solutions. We introduce a fixed-point based framework for analyzing looped architectures along three axes of stability -- reachability, input-dependence, and geometry -- and use it to characterize when fixed-point iteration yields meaningful predictions. Theoretically, we prove that looped networks without recall have countable fixed points and cannot achieve strong input-dependence at any spectral regime, while recall combined with outer normalization reliably produces a regime in which fixed points are simultaneously reachable, locally smooth in the input, and supported by stable backpropagation. Empirically, we train single-layer looped transformers on chess, sudoku, and prefix-sums and find that downstream performance tracks the framework's predictions across tasks and architectural configurations. We additionally introduce internal recall, a novel recall placement variant, and show that it becomes competitive with -- and on sudoku, substantially better than -- standard recall placement once outer normalization is applied.

A Case for Library-Level k-Means Binning in Histogram Gradient-Boosted Trees

Modern gradient-boosted decision trees (GBDTs) accelerate split finding with histogram-based binning, which reduces complexity from O(N) to O(B) given a fixed bin budget B. However, the predominant quantile binning strategy-designed to distribute data points evenly among bins-may overlook critical boundary values that could enhance predictive performance. In this work, we propose replacing quantile binning with a k-means discretizer initialized with quantile bins. We test this swap on 33 OpenML tasks plus synthetics that control for modality, skew, and bin budget. Across 18 regression datasets, k-means shows no statistically significant losses at the 5% level and wins in four cases-most strikingly a 55% MSE drop on one particularly skewed dataset-even though k-means' mean reciprocal rank (MRR) is slightly lower (0.65 vs 0.72). On the 15 classification datasets the two methods are statistically tied (MRR 0.70 vs 0.68) with gaps $\leq$0.2 pp. Synthetic experiments confirm consistently large MSE gains-typically >20% and rising to 90% as outlier magnitude increases or bin budget drops. We find that k-means keeps error on par with exact splitting when extra cuts add little value, yet still recovers key split points that quantile overlooks. As such, we advocate for a built-in bin_method=k-means flag, especially in regression tasks and in tight-budget settings such as the 32-64-bin GPU regime-because it is a "safe default" with large upside, yet adds only a one-off, cacheable overhead ($\approx$ 2s to bin 10M rows on one core).