Felix Benning

Felix Benning, Leif Döring

We prove that all 'gradient span algorithms' have asymptotically deterministic behavior on scaled Gaussian random functions as the dimension tends to infinity. In particular, this result explains the counterintuitive phenomenon that different training runs of many large machine learning models result in approximately equal cost curves despite random initialization on a complicated non-convex landscape. The distributional assumption of (non-stationary) isotropic Gaussian random functions we use is sufficiently general to serve as realistic model for machine learning training but also encompass spin glasses and random quadratic functions.

Felix Benning, Steffen Dereich

Whether or not a local minimum of a cost function has a strongly convex neighborhood greatly influences the asymptotic convergence rate of optimizers. In this article, we rigorously analyze the prevalence of this property for the mean squared error induced by shallow, 1-hidden layer neural networks with analytic activation functions when applied to regression problems. The parameter space is divided into two domains: the 'efficient domain' (all parameters for which the respective realization function cannot be generated by a network having a smaller number of neurons) and the 'redundant domain' (the remaining parameters). In almost all regression problems on the efficient domain the optimization landscape only features local minima that are strongly convex. Formally, we will show that for certain randomly picked regression problems the optimization landscape is almost surely a Morse function on the efficient domain. The redundant domain has significantly smaller dimension than the efficient domain and on this domain, potential local minima are never isolated.

Felix Benning, Leif Döring

Classical worst-case optimization theory neither explains the success of optimization in machine learning, nor does it help with step size selection. In this paper we demonstrate the viability and advantages of replacing the classical 'convex function' framework with a 'random function' framework. With complexity $\mathcal{O}(n^3d^3)$, where $n$ is the number of steps and $d$ the number of dimensions, Bayesian optimization with gradients has not been viable in large dimension so far. By bridging the gap between Bayesian optimization (i.e. random function optimization theory) and classical optimization we establish viability. Specifically, we use a 'stochastic Taylor approximation' to rediscover gradient descent, which is scalable in high dimension due to $\mathcal{O}(nd)$ complexity. This rediscovery yields a specific step size schedule we call Random Function Descent (RFD). The advantage of this random function framework is that RFD is scale invariant and that it provides a theoretical foundation for common step size heuristics such as gradient clipping and gradual learning rate warmup.

Felix Benning

This thesis reviews numerical optimization methods with machine learning problems in mind. Since machine learning models are highly parametrized, we focus on methods suited for high dimensional optimization. We build intuition on quadratic models to figure out which methods are suited for non-convex optimization, and develop convergence proofs on convex functions for this selection of methods. With this theoretical foundation for stochastic gradient descent and momentum methods, we try to explain why the methods used commonly in the machine learning field are so successful. Besides explaining successful heuristics, the last chapter also provides a less extensive review of more theoretical methods, which are not quite as popular in practice. So in some sense this work attempts to answer the question: Why are the default Tensorflow optimizers included in the defaults?