Alexander Strang

Shishir Adhikari, Alkan Kabakçıoğlu, Alexander Strang et al.

The algorithms used to train neural networks, like stochastic gradient descent (SGD), have close parallels to natural processes that navigate a high-dimensional parameter space -- for example protein folding or evolution. Our study uses a Fokker-Planck approach, adapted from statistical physics, to explore these parallels in a single, unified framework. We focus in particular on the stationary state of the system in the long-time limit, which in conventional SGD is out of equilibrium, exhibiting persistent currents in the space of network parameters. As in its physical analogues, the current is associated with an entropy production rate for any given training trajectory. The stationary distribution of these rates obeys the integral and detailed fluctuation theorems -- nonequilibrium generalizations of the second law of thermodynamics. We validate these relations in two numerical examples, a nonlinear regression network and MNIST digit classification. While the fluctuation theorems are universal, there are other aspects of the stationary state that are highly sensitive to the training details. Surprisingly, the effective loss landscape and diffusion matrix that determine the shape of the stationary distribution vary depending on the simple choice of minibatching done with or without replacement. We can take advantage of this nonequilibrium sensitivity to engineer an equilibrium stationary state for a particular application: sampling from a posterior distribution of network weights in Bayesian machine learning. We propose a new variation of stochastic gradient Langevin dynamics (SGLD) that harnesses without replacement minibatching. In an example system where the posterior is exactly known, this SGWORLD algorithm outperforms SGLD, converging to the posterior orders of magnitude faster as a function of the learning rate.

Alexander Strang, David SeWell, Alexander Kim et al.

How are the advantage relations between a set of agents playing a game organized and how do they reflect the structure of the game? In this paper, we illustrate "Principal Trade-off Analysis" (PTA), a decomposition method that embeds games into a low-dimensional feature space. We argue that the embeddings are more revealing than previously demonstrated by developing an analogy to Principal Component Analysis (PCA). PTA represents an arbitrary two-player zero-sum game as the weighted sum of pairs of orthogonal 2D feature planes. We show that the feature planes represent unique strategic trade-offs and truncation of the sequence provides insightful model reduction. We demonstrate the validity of PTA on a quartet of games (Kuhn poker, RPS+2, Blotto, and Pokemon). In Kuhn poker, PTA clearly identifies the trade-off between bluffing and calling. In Blotto, PTA identifies game symmetries, and specifies strategic trade-offs associated with distinct win conditions. These symmetries reveal limitations of PTA unaddressed in previous work. For Pokemon, PTA recovers clusters that naturally correspond to Pokemon types, correctly identifies the designed trade-off between those types, and discovers a rock-paper-scissor (RPS) cycle in the Pokemon generation type - all absent any specific information except game outcomes.

Guillaume Corlouer, Avi Semler, Alexander Strang et al.

Deep linear networks (DLNs) are used as an analytically tractable model of the training dynamics of deep neural networks. While gradient descent in DLNs is known to exhibit saddle-to-saddle dynamics, the impact of stochastic gradient descent (SGD) noise on this regime remains poorly understood. We investigate the dynamics of SGD during training of DLNs in the saddle-to-saddle regime. We model the training dynamics as stochastic Langevin dynamics with anisotropic, state-dependent noise. Under the assumption of aligned and balanced weights, we derive an exact decomposition of the dynamics into a system of one-dimensional per-mode stochastic differential equations. This establishes that the maximal diffusion along a mode precedes the corresponding feature being completely learned. We also derive the stationary distribution of SGD for each mode: in the absence of label noise, its marginal distribution along specific features coincides with the stationary distribution of gradient flow, while in the presence of label noise it approximates a Boltzmann distribution. Finally, we confirm experimentally that the theoretical results hold qualitatively even without aligned or balanced weights. These results establish that SGD noise encodes information about the progression of feature learning but does not fundamentally alter the saddle-to-saddle dynamics.

Shiv Agrawal, Hwanwoo Kim, Daniel Sanz-Alonso et al.

Hierarchical models with gamma hyperpriors provide a flexible, sparse-promoting framework to bridge $L^1$ and $L^2$ regularizations in Bayesian formulations to inverse problems. Despite the Bayesian motivation for these models, existing methodologies are limited to \textit{maximum a posteriori} estimation. The potential to perform uncertainty quantification has not yet been realized. This paper introduces a variational iterative alternating scheme for hierarchical inverse problems with gamma hyperpriors. The proposed variational inference approach yields accurate reconstruction, provides meaningful uncertainty quantification, and is easy to implement. In addition, it lends itself naturally to conduct model selection for the choice of hyperparameters. We illustrate the performance of our methodology in several computed examples, including a deconvolution problem and sparse identification of dynamical systems from time series data.