In this work, we introduce basic inequalities for first-order iterative optimization algorithms, forming a simple yet versatile framework which connects implicit and explicit regularization. Building on related comparison inequalities for optimization iterates that already exist in the literature, we extend and unify these arguments to produce a general framework, which can be used as a tool for statistical analysis. In more detail, let f denote the objective function to be optimized. Given a first-order iterative algorithm initialized at θ0, with current iterate θT, the basic inequality upper bounds f(θT)−f(z) for any reference point z in terms of the accumulated step sizes, and the distances between θ0, θT, and z. These distances are measured in a geometry inherent to the optimization algorithm, which then translates into a notion of regularization being applied across the path of iterates. In addition to refining existing results on gradient descent, we provide new results for mirror descent and other first-order methods. We then show how to use these basic inequalities to derive elementary yet useful bounds on the prediction risk of early-stopped gradient descent and exponentiated gradient descent iterates in generalized linear models. We also supplement these findings with numerical experiments.
A fundamental problem in machine learning is understanding the effect of early stopping on the parameters obtained and the generalization capabilities of the model. Even for linear models, the effect is not fully understood for arbitrary learning rates and data. In this paper, we analyze the dynamics of discrete full batch gradient descent for linear regression. With minimal distributional assumptions, we characterize the trajectory of the parameters and the expected excess risk. Using this characterization, we show that when training with any learning rate schedule and finite time horizon, the early stopped solution is equivalent to the minimum norm solution for a generalized ridge regression problem. We also prove that early stopping is beneficial for generic data with arbitrary spectrum and for a wide variety of learning rate schedules. We provide an estimate for the optimal stopping time and empirically demonstrate the accuracy of our estimate.
In this paper we prove some concentration inequalities for two types of error probabilities in the Empirical Risk Principle (ERP) in statistical learning, which provide a lower bound and an upper bound for the minimal risk (in terms of the minimal empirical risk) with non-asymptotic high confidence. The usual boundedness condition of the empirical risk function is relaxed to the Gaussian or exponential integrability condition. The confidence of the lower bound of the minimal risk is shown to be independent of the number of training parameters and the dimension of the input vectors, allowing one to detect the deficiency of a learning machine efficiently; and the confidence of the upper bound of the minimal risk is proved to be high provided that the sample size n is much greater than the box dimension of the parameter set Θ in the Orlicz metric dψ1 associated with the risk functions. Our work is based on Talagrand's concentration inequalities (the sharp versions by Bousquet and Klein-Rio), transport-entropy inequalities and the recent progress in the theory of empirical processes and statistical learning.
Policy-gradient methods usually optimize expected return, but many real world applications care about distributional properties of returns: tail risk, outlier robustness, or best-of-K discovery. We introduce OrderGrad, a family of likelihood-ratio and reparameterization gradient estimators for order-statistic objectives. OrderGrad optimizes finite-sample L-statistics, i.e., weighted averages of sorted rewards or costs, recovering objectives such as VaR, CVaR, trimmed means, medians, and top-m/best-of-K criteria by changing only the rank weights. For any fixed sample size and rank-weight vector, OrderGrad provides an unbiased gradient estimator for the corresponding order-statistic objective. The method is implemented as a simple reward transformation that can then be used in an otherwise standard policy-gradient or reparameterized update. We study the resulting estimator's variance behavior and evaluate it on tasks where mean optimization is mismatched to the deployment objective, including LLM math post-training and other tasks. OrderGrad provides a unified, plug-and-play route to risk-averse, robust, and exploratory learning. Code: https://github.com/paavo5/ordergrad