Pratik Worah
The focus of my research is on machine learning and mathematics (specifically randomized algorithms and probability theory). I have applied tools from these areas to problems in economics, optimization, and biology.
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Given a training data-set $\mathcal{S}$, and a reference data-set $\mathcal{T}$, we design a simple and efficient algorithm to reweigh the loss function such that the limiting distribution of the neural network weights that result from training on $\mathcal{S}$ approaches the limiting distribution that would have resulted by training on $\mathcal{T}$. Such reweighing can be used to correct for Train-Test distribution shift, when we don't have access to the labels of $\mathcal{T}$. It can also be used to perform (soft) multi-criteria optimization on neural nets, when we have access to the labels of $\mathcal{T}$, but $\mathcal{S}$ and $\mathcal{T}$ have few common points.
As a motivating application, we train a graph neural net to recognize small molecule binders to MNK2 (a MAP Kinase, responsible for cell signaling) which are non-binders to MNK1 (a very similar protein), even in the absence of training data common to both data-sets. We are able to tune the reweighing parameters so that overall change in holdout loss is negligible, but the selectivity, i.e., the fraction of top 100 MNK2 binders that are MNK1 non-binders, increases from 54\% to 95\%, as a result of our reweighing.
We expect the algorithm to be applicable in other settings as well, since we prove that when the metric entropy of the input data-sets is bounded, our random sampling based greedy algorithm outputs a close to optimal reweighing, i.e., the two invariant distributions of network weights will be provably close in total variation distance.
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The Landscape of Nonconvex-Nonconcave Minimax Optimization
Ben Grimmer
Haihao (Sean) Lu
Mathematical Programming (Springer Nature), na (2023), na
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Minimax optimization has become a central tool for modern machine learning with applications in robust optimization, game theory and training GANs. These applications are often nonconvex-nonconcave, but the existing theory is unable to identify and deal with the fundamental difficulties posed by nonconvex-nonconcave structures.
We break this historical barrier by identifying three regions of nonconvex-nonconcave bilinear minimax problems and characterizing their different solution paths. For problems where the interaction between the agents is sufficiently strong, we derive global linear convergence guarantees. Conversely when the interaction between the agents is fairly weak, we derive local linear convergence guarantees. Between these two settings, we characterize the types of cycles that can occur, preventing the convergence of the solution path.
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Learning Rate Schedules in the Presence of Distribution Shift
Adel Javanmard
Proceedings of the 40th International Conference on Machine Learning (2023), pp. 9523-9546
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We design learning rate schedules that minimize regret for SGD-based online learning in the presence of a changing data distribution. We fully characterize the optimal learning rate schedule for online linear regression via a novel analysis with stochastic differential equations. For general convex loss functions, we propose new learning rate schedules that are robust to distribution shift, and we give upper and lower bounds for the regret that only differ by constants. For non-convex loss functions, we define a notion of regret based on the gradient norm of the estimated models and propose a learning schedule that minimizes an upper bound on the total expected regret. Intuitively, one expects changing loss landscapes to require more exploration, and we confirm that optimal learning rate schedules typically increase in the presence of distribution shift. Finally, we provide experiments for high-dimensional regression models and neural networks to illustrate these learning rate schedules and their cumulative regret.
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In the Learning to Price setting, a seller posts prices over time with the goal of maximizing revenue while learning the buyer's valuation. This problem is very well understood when values are stationary (fixed or iid). Here we study the problem where the buyer's value is a moving target, i.e., they change over time either by a stochastic process or adversarially with bounded variation. In either case, we provide matching upper and lower bounds on the optimal revenue loss. Since the target is moving, any information learned soon becomes out-dated, which forces the algorithms to keep switching between exploring and exploiting phases.
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Neural network configurations with random weights play an important role in the analysis of deep learning. They define the initial loss landscape and are closely related to kernel and random feature methods. Despite the fact that these networks are built out of random matrices, the vast and powerful machinery of random matrix theory has so far found limited success in studying them. A main obstacle in this direction is that neural networks are nonlinear, which prevents the straightforward utilization of many of the existing mathematical results. In this work, we open the door for direct applications of random matrix theory to deep learning by demonstrating that the entrywise nonlinearities typically applied in neural networks can be incorporated into a standard method of proof in random matrix theory known as the moments method. The test case for our study is the Gram matrix $Y^TY$, $Y=\act(WX)$, where $W$ is a random weight matrix, $X$ is a random data matrix, and $\act$ is a pointwise nonlinear activation function. We derive an explicit representation for the trace of the resolvent of this matrix, which defines its limiting spectral distribution. We apply these results to the computation of the asymptotic performance of single-layer random feature methods on a memorization task and to the analysis of the eigenvalues of the data covariance matrix as it propagates through a neural network. As a byproduct of our analysis, we identify an intriguing new class of activation functions with favorable properties.
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An important factor contributing to the success of deep learning has been the remarkable ability to optimize large neural networks using simple first-order optimization algorithms like stochastic gradient descent. While the efficiency of such methods depends crucially on the local curvature of the loss surface, very little is actually known about how this geometry depends on network architecture and hyperparameters. In this work, we extend a recently-developed framework for studying spectra of nonlinear random matrices to characterize an important measure of curvature, namely the eigenvalues of the Fisher information matrix. We focus on a single-hidden-layer neural network with Gaussian data and weights and provide an exact expression for the spectrum in the limit of infinite width. We find that linear networks suffer worse conditioning than nonlinear networks and that nonlinear networks are generically non-degenerate. We also predict and demonstrate empirically that by adjusting the nonlinearity, the spectrum can be tuned so as to improve the efficiency of first-order optimization methods.
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