We study concentration inequalities for structured weighted sums of random data, including (i) tensor inner products and (ii) sequential matrix sums. We are interested in tail bounds and concentration inequalities for those structured weighted sums under exchangeability, extending beyond the classical framework of independent terms. We develop Hoeffding and Bernstein bounds provided with structure-dependent exchangeability. Along the way, we recover known results in weighted sum of exchangeable random variables and i.i.d. sums of random matrices to the optimal constants. Notably, we develop a sharper concentration bound for combinatorial sum of matrix arrays than the results previously derived from Chatterjee's method of exchangeable pairs. For applications, the richer structures provide us with novel analytical tools for estimating the average effect of multi-factor response models and studying fixed-design sketching methods in federated averaging. We apply our results to these problems, and find that our theoretical predictions are corroborated by numerical evidence.

Several optimism-based stochastic bandit algorithms -- including UCB, UCB-V, linear UCB, and finite-arm GP-UCB -- achieve logarithmic regret using proofs that, despite superficial differences, follow essentially the same structure. This note isolates the minimal ingredients behind these analyses: a single high-probability concentration condition on the estimators, after which logarithmic regret follows from two short deterministic lemmas describing radius collapse and optimism-forced deviations. The framework yields unified, near-minimal proofs for these classical algorithms and extends naturally to many contemporary bandit variants.

This thesis develops methods for causal inference and causal representation learning (CRL) in high-dimensional, time-varying data. The first contribution introduces the Causal Dynamic Variational Autoencoder (CDVAE), a model for estimating Individual Treatment Effects (ITEs) by capturing unobserved heterogeneity in treatment response driven by latent risk factors that affect only outcomes. CDVAE comes with theoretical guarantees on valid latent adjustment and generalization bounds for ITE error. Experiments on synthetic and real datasets show that CDVAE outperforms baselines, and that state-of-the-art models greatly improve when augmented with its latent substitutes, approaching oracle performance without access to true adjustment variables. The second contribution proposes an efficient framework for long-term counterfactual regression based on RNNs enhanced with Contrastive Predictive Coding (CPC) and InfoMax. It captures long-range dependencies under time-varying confounding while avoiding the computational cost of transformers, achieving state-of-the-art results and introducing CPC into causal inference. The third contribution advances CRL by addressing how latent causes manifest in observed variables. We introduce a model-agnostic interpretability layer based on the geometry of the decoder Jacobian. A sparse self-expression prior induces modular, possibly overlapping groups of observed features aligned with shared latent influences. We provide recovery guarantees in both disjoint and overlapping settings and show that meaningful latent-to-observed structure can be recovered without anchor features or single-parent assumptions. Scalable Jacobian-based regularization techniques are also developed.

Time series forecasting relies on predicting future values from historical data, yet most state-of-the-art approaches-including transformer and multilayer perceptron-based models-optimize using Mean Squared Error (MSE), which has two fundamental weaknesses: its point-wise error computation fails to capture temporal relationships, and it does not account for inherent noise in the data. To overcome these limitations, we introduce the Residual-Informed Loss (RI-Loss), a novel objective function based on the Hilbert-Schmidt Independence Criterion (HSIC). RI-Loss explicitly models noise structure by enforcing dependence between the residual sequence and a random time series, enabling more robust, noise-aware representations. Theoretically, we derive the first non-asymptotic HSIC bound with explicit double-sample complexity terms, achieving optimal convergence rates through Bernstein-type concentration inequalities and Rademacher complexity analysis. This provides rigorous guarantees for RI-Loss optimization while precisely quantifying kernel space interactions. Empirically, experiments across eight real-world benchmarks and five leading forecasting models demonstrate improvements in predictive performance, validating the effectiveness of our approach. The code is publicly available at: https://github.com/shang-xl/RI-Loss.

While practitioners often employ variable importance methods that rely on this impurity-based information, these methods remain poorly characterized from a theoretical perspective.

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In this section, we fill in the missing details for proving Theorem 1, including a statement of the concentration bound used to establish Lemma 2, and a proof for Lemma 3. We first provide some

This paper presents a tractable algorithm for estimating an unknown Lipschitz function from noisy observations and establishes an upper bound on its convergence rate. The approach extends max-affine methods from convex shape-restricted regression to the more general Lipschitz setting. A key component is a nonlinear feature expansion that maps max-affine functions into a subclass of delta-convex functions, which act as universal ap-proximators of Lipschitz functions while preserving their Lipschitz constants. Leveraging this property, the estimator attains the minimax convergence rate (up to logarithmic factors) with respect to the intrinsic dimension of the data under squared loss and subgaussian distributions in the random design setting. The algorithm integrates adaptive partitioning to capture intrinsic dimension, a penalty-based regularization mechanism that removes the need to know the true Lipschitz constant, and a two-stage optimization procedure combining a convex initialization with local refinement. The framework is also straightforward to adapt to convex shape-restricted regression. Experiments demonstrate competitive performance relative to other theoretically justified methods, including nearest-neighbor and kernel-based regressors.