Neural Information Processing Systems http://nips.cc/

Figure 1: Prediction risk of ridge regression on trained CK features (erf) after one feature learning step.

X>y, where Σw is the weighting matrix.

We study the following learning problem with dependent data: Given a trajectory of length $n$ from a stationary Markov chain with $k$ states, the goal is to predict the distribution of the next state.

Decision trees are powerful machine learning algorithms, widely used in fields such as economics and medicine for their simplicity and interpretability. However, decision trees such as CART are prone to overfitting, especially when grown deep or the sample size is small. Conventional methods to reduce overfitting include pre-pruning and post-pruning, which constrain the growth of uninformative branches. In this paper, we propose a complementary approach by introducing a covariance-driven splitting criterion for regression trees (CovRT). This method is more robust to overfitting than the empirical risk minimization criterion used in CART, as it produces more balanced and stable splits and more effectively identifies covariates with true signals. We establish an oracle inequality of CovRT and prove that its predictive accuracy is comparable to that of CART in high-dimensional settings. We find that CovRT achieves superior prediction accuracy compared to CART in both simulations and real-world tasks.

We introduce \textit{basic inequalities} for first-order iterative optimization algorithms, forming a simple and versatile framework that connects implicit and explicit regularization. While related inequalities appear in the literature, we isolate and highlight a specific form and develop it as a well-rounded tool for statistical analysis. Let $f$ denote the objective function to be optimized. Given a first-order iterative algorithm initialized at $θ_0$ with current iterate $θ_T$, the basic inequality upper bounds $f(θ_T)-f(z)$ for any reference point $z$ in terms of the accumulated step sizes and the distances between $θ_0$, $θ_T$, and $z$. The bound translates the number of iterations into an effective regularization coefficient in the loss function. We demonstrate this framework through analyses of training dynamics and prediction risk bounds. In addition to revisiting and refining known results on gradient descent, we provide new results for mirror descent with Bregman divergence projection, for generalized linear models trained by gradient descent and exponentiated gradient descent, and for randomized predictors. We illustrate and supplement these theoretical findings with experiments on generalized linear models.