arXiv subjects

Sebastian Kassing

Publications and source records attributed to Sebastian Kassing.

14 recordsLinked to original sources

arXiv2026

Strong Averaging Principle and Long-Time Dynamics for Fast-Slow SDEs with Increasing Time-Scale Separation and Degenerate Noise

We establish a strong averaging principle for fast-slow stochastic differential equations with a time-dependent scale-separation parameter $(\varepsilon_t)_{t \geq 0}$ satisfying $\varepsilon_t \to 0$ as $t \to \infty$. In contrast to approaches based on noise-induced smoothing or elliptic regularity, our approach relies on dissipativity of the frozen fast dynamics and therefore permits degenerate diffusion coefficients. We prove a maximal $L^p$-estimate between the slow variable and the averaged ODE at late times, with the classical strong convergence rate of order $1/2$. Under an additional decay condition on $(\varepsilon_t)_{t \ge 0}$, this estimate implies that the slow variable is almost surely an asymptotic pseudo-trajectory of the averaged ODE. As a consequence, we obtain criteria for the identification of possible limit points and for convergence toward asymptotically stable equilibria for the slow variable by analyzing the dynamical behavior of the averaged equation.

math.PR↗

arXiv2026

Optimal Asymptotic Rates for (Stochastic) Gradient Descent under the Local PL-Condition: A Geometric Approach

Stochastic gradient descent (SGD) has been studied extensively over the past decades due to its simplicity and broad applicability in machine learning. In this work, we analyze the local behavior of gradient descent and stochastic gradient descent for minimizing $C^2$-functions that satisfy the Polyak-Lojasiewicz (PL) inequality and under a multiplicative gradient noise model motivated by overparameterized neural networks. Using a geometric interpretation of the PL-condition, we prove a simple yet surprising fact: in this possibly non-convex setting, the asymptotic convergence rate of (S)GD matches the rate obtained for strongly convex quadratics.

math.OC↗

arXiv2026

An Approximate Ascent Approach To Prove Convergence of PPO

Proximal Policy Optimization (PPO) is among the most widely used deep reinforcement learning algorithms, yet its theoretical foundations remain incomplete. Most importantly, convergence and understanding of fundamental PPO advantages remain widely open. Under standard theory assumptions we show how PPO's policy update scheme (performing multiple epochs of minibatch updates on multi-use rollouts with a surrogate gradient) can be interpreted as approximated policy gradient ascent. We show how to control the bias accumulated by the surrogate gradients and use techniques from random reshuffling to prove a convergence theorem for PPO that sheds light on PPO's success. Additionally, we identify a previously overlooked issue in truncated Generalized Advantage Estimation commonly used in PPO. The geometric weighting scheme induces infinite mass collapse onto the longest $k$-step advantage estimator at episode boundaries. Empirical evaluations show that a simple weight correction can yield substantial improvements in environments with strong terminal signal, such as Lunar Lander.

cs.LG↗

arXiv2026

The Role of Target Update Frequencies in Q-Learning

The target network update frequency (TUF) is a central stabilization mechanism in (deep) Q-learning. However, their selection remains poorly understood and is often treated merely as another tunable hyperparameter rather than as a principled design decision. This work provides a theoretical analysis of target fixing in tabular Q-learning through the lens of approximate dynamic programming. We formulate periodic target updates as a nested optimization scheme in which each outer iteration applies an inexact Bellman optimality operator, approximated by a generic inner loop optimizer. Rigorous theory yields a finite-time convergence analysis for the asynchronous sampling setting, specializing to stochastic gradient descent in the inner loop. Our results deliver an explicit characterization of the bias-variance trade-off induced by the target update period, showing how to optimally set this critical hyperparameter. We prove that constant target update schedules are suboptimal, incurring a logarithmic overhead in sample complexity that is entirely avoidable with adaptive schedules. Our analysis shows that the optimal target update frequency increases geometrically over the course of the learning process.

cs.LG↗

arXiv2025

ODE approximation for the Adam algorithm: General and overparametrized setting

The Adam optimizer is currently presumably the most popular optimization method in deep learning. In this article we develop an ODE based method to study the Adam optimizer in a fast-slow scaling regime. For fixed momentum parameters and vanishing step-sizes, we show that the Adam algorithm is an asymptotic pseudo-trajectory of the flow of a particular vector field, which is referred to as the Adam vector field. Leveraging properties of asymptotic pseudo-trajectories, we establish convergence results for the Adam algorithm. In particular, in a very general setting we show that if the Adam algorithm converges, then the limit must be a zero of the Adam vector field, rather than a local minimizer or critical point of the objective function. In contrast, in the overparametrized empirical risk minimization setting, the Adam algorithm is able to locally find the set of minima. Specifically, we show that in a neighborhood of the global minima, the objective function serves as a Lyapunov function for the flow induced by the Adam vector field. As a consequence, if the Adam algorithm enters a neighborhood of the global minima infinitely often, it converges to the set of global minima.

math.OC↗

arXiv2025

Controlling the Flow: Stability and Convergence for Stochastic Gradient Descent with Decaying Regularization

The present article studies the minimization of convex, L-smooth functions defined on a separable real Hilbert space. We analyze regularized stochastic gradient descent (reg-SGD), a variant of stochastic gradient descent that uses a Tikhonov regularization with time-dependent, vanishing regularization parameter. We prove strong convergence of reg-SGD to the minimum-norm solution of the original problem without additional boundedness assumptions. Moreover, we quantify the rate of convergence and optimize the interplay between step-sizes and regularization decay. Our analysis reveals how vanishing Tikhonov regularization controls the flow of SGD and yields stable learning dynamics, offering new insights into the design of iterative algorithms for convex problems, including those that arise in ill-posed inverse problems. We validate our theoretical findings through numerical experiments on image reconstruction and ODE-based inverse problems.

math.OC↗

arXiv2024

Polyak's Heavy Ball Method Achieves Accelerated Local Rate of Convergence under Polyak-Lojasiewicz Inequality

In this work, we analyze the convergence of Polyak's heavy ball method in both continuous and discrete time for non-convex $C^4$-objective functions satisfying the Polyak-Lojasiewicz inequality. Under this weak assumption, we recover the asymptotic convergence rates originally derived by Polyak in [Polyak, U.S.S.R. Comput. Math. and Math. Phys., 1964] for strongly convex objectives. Our results demonstrate that the heavy ball method exhibits asymptotic local acceleration on this class of functions. In particular, in the discrete time setting, we prove local convergence of the iterates to a minimum once the method enters a sufficiently small neighborhood of the set of minima, for a broad range of hyperparameters, including aggressive choices for the momentum parameter and the step-size for which global convergence is known to fail. Instead of the usually employed Lyapunov-type arguments, our approach leverages a new differential geometric perspective of the Polyak-Lojasiewicz inequality proposed in [Rebjock and Boumal, Math. Program., 2025].

math.OC↗

arXiv2024

Stochastic Modified Flows for Riemannian Stochastic Gradient Descent

We give quantitative estimates for the rate of convergence of Riemannian stochastic gradient descent (RSGD) to Riemannian gradient flow and to a diffusion process, the so-called Riemannian stochastic modified flow (RSMF). Using tools from stochastic differential geometry we show that, in the small learning rate regime, RSGD can be approximated by the solution to the RSMF driven by an infinite-dimensional Wiener process. The RSMF accounts for the random fluctuations of RSGD and, thereby, increases the order of approximation compared to the deterministic Riemannian gradient flow. The RSGD is build using the concept of a retraction map, that is, a cost efficient approximation of the exponential map, and we prove quantitative bounds for the weak error of the diffusion approximation under assumptions on the retraction map, the geometry of the manifold, and the random estimators of the gradient.

cs.LG↗

arXiv2023

On the existence of optimal shallow feedforward networks with ReLU activation

We prove existence of global minima in the loss landscape for the approximation of continuous target functions using shallow feedforward artificial neural networks with ReLU activation. This property is one of the fundamental artifacts separating ReLU from other commonly used activation functions. We propose a kind of closure of the search space so that in the extended space minimizers exist. In a second step, we show under mild assumptions that the newly added functions in the extension perform worse than appropriate representable ReLU networks. This then implies that the optimal response in the extended target space is indeed the response of a ReLU network.

cs.LG↗

arXiv2023

On the existence of minimizers in shallow residual ReLU neural network optimization landscapes

In this article, we show existence of minimizers in the loss landscape for residual artificial neural networks (ANNs) with multi-dimensional input layer and one hidden layer with ReLU activation. Our work contrasts earlier results in [D. Gallon, A. Jentzen, and F. Lindner, preprint, arXiv:2211.15641, 2022] and [P. Petersen, M. Raslan, and F. Voigtlaender, Found. Comput. Math., 21 (2021), pp. 375-444] which showed that in many situations minimizers do not exist for common smooth activation functions even in the case where the target functions are polynomials. The proof of the existence property makes use of a closure of the search space containing all functions generated by ANNs and additional discontinuous generalized responses. As we will show, the additional generalized responses in this larger space are suboptimal so that the minimum is attained in the original function class.

math.OC↗

arXiv2023

Stochastic Modified Flows, Mean-Field Limits and Dynamics of Stochastic Gradient Descent

We propose new limiting dynamics for stochastic gradient descent in the small learning rate regime called stochastic modified flows. These SDEs are driven by a cylindrical Brownian motion and improve the so-called stochastic modified equations by having regular diffusion coefficients and by matching the multi-point statistics. As a second contribution, we introduce distribution dependent stochastic modified flows which we prove to describe the fluctuating limiting dynamics of stochastic gradient descent in the small learning rate - infinite width scaling regime.

math.PR↗

arXiv2023

Exponential convergence rates for momentum stochastic gradient descent in the overparametrized setting

We prove explicit bounds on the exponential rate of convergence for the momentum stochastic gradient descent scheme (MSGD) for arbitrary, fixed hyperparameters (learning rate, friction parameter) and its continuous-in-time counterpart in the context of non-convex optimization. In the small step-size regime and in the case of flat minima or large noise intensities, these bounds prove faster convergence of MSGD compared to plain stochastic gradient descent (SGD). The results are shown for objective functions satisfying a local Polyak-Lojasiewicz inequality and under assumptions on the variance of MSGD that are satisfied in overparametrized settings. Moreover, we analyze the optimal choice of the friction parameter and show that the MSGD process almost surely converges to a local minimum.

math.OC↗

arXiv2021

Convergence of stochastic gradient descent schemes for Lojasiewicz-landscapes

In this article, we consider convergence of stochastic gradient descent schemes (SGD), including momentum stochastic gradient descent (MSGD), under weak assumptions on the underlying landscape. More explicitly, we show that on the event that the SGD stays bounded we have convergence of the SGD if there is only a countable number of critical points or if the objective function satisfies Lojasiewicz-inequalities around all critical levels as all analytic functions do. In particular, we show that for neural networks with analytic activation function such as softplus, sigmoid and the hyperbolic tangent, SGD converges on the event of staying bounded, if the random variables modelling the signal and response in the training are compactly supported.

cs.LG↗

arXiv2019

Central limit theorems for stochastic gradient descent with averaging for stable manifolds

In this article we establish new central limit theorems for Ruppert-Polyak averaged stochastic gradient descent schemes. Compared to previous work we do not assume that convergence occurs to an isolated attractor but instead allow convergence to a stable manifold. On the stable manifold the target function is constant and the oscillations in the tangential direction may be significantly larger than the ones in the normal direction. As we show, one still recovers a central limit theorem with the same rates as in the case of isolated attractors. Here we consider step-sizes $\gamma_n=n^{-\gamma}$ with $\gamma\in(\frac34,1)$, typically.

math.PR↗