Many recent works have studied the eigenvalue spectrum of the Conjugate Kernel (CK) defined by the nonlinear feature map of a feedforward neural network. However, existing results only establish weak convergence of the empirical eigenvalue distribution, and fall short of providing precise quantitative characterizations of the “spike” eigenvalues and eigenvectors that often capture the low-dimensional signal structure of the learning problem. In this work, we characterize these signal eigenvalues and eigenvectors for a nonlinear version of the spiked covariance model, including the CK as a special case. Using this general result, we give a quantitative description of how spiked eigenstructure in the input data propagates through the hidden layers of a neural network with random weights. As a second application, we study a simple regime of representation learning where the weight matrix develops a rank-one signal component over training and characterize the alignment of the target function with the spike eigenvector of the CK on test data.

We study estimation in the linear model$y=A \beta ^{\star} +\epsilon $, in a Bayesian setting where$ \beta ^{\star} $has an entrywise i.i.d. prior and the design$A$is rotationally-invariant in law. In the large system limit as dimension and sample size increase proportionally, a set of related conjectures have been postulated for the asymptotic mutual information, Bayes-optimal mean squared error, and TAP mean-field equations that characterize the Bayes posterior mean of$ \beta ^{\star} $. In this work, we prove these conjectures for a general class of signal priors and for arbitrary rotationally-invariant designs$A$, under a “high-temperature” condition that restricts the range of eigenvalues of$A^{\top} A$and encompasses regimes of sufficiently low signal-to-noise ratio. Our proof uses a conditional second-moment method argument, where we condition on the iterates of a version of the Vector AMP algorithm for solving the TAP mean-field equations.

Uniswap v3 is the largest decentralized exchange for digital currencies. A novelty of its design is that it allows a liquidity provider (LP) to allocate liquidity to one or more closed intervals of the price of an asset instead of the full range of possible prices. An LP earns fee rewards proportional to the amount of its liquidity allocation when prices move in this interval. This induces the problem of strategic liquidity provision: smaller intervals result in higher concentration of liquidity and correspondingly larger fees when the price remains in the interval, but with higher risk as prices may exit the interval leaving the LP with no fee rewards. Although reallocating liquidity to new intervals can mitigate this loss, it comes at a cost, as LPs must expend gas fees to do so. We formalize the dynamic liquidity provision problem and focus on a general class of strategies for which we provide a neural network-based optimization framework for maximizing LP earnings. We model a single LP that faces an exogenous sequence of price changes that arise from arbitrage and non-arbitrage trades in the decentralized exchange. We present experimental results informed by historical price data that demonstrate large improvements in LP earnings over existing allocation strategy baselines. Moreover we provide insight into qualitative differences in optimal LP behaviour in different economic environments.

We study estimation in the linear model y = Aβ⋆+ ϵ, in a Bayesian setting where β⋆has an entrywise i.i.d. prior and the design A is rotationally-invariant in law. In the large system limit as dimension and sample size increase proportionally, a set of related conjectures have been postulated for the asymptotic mutual information, Bayes-optimal mean squared error, and TAP mean-field equations that characterize the Bayes posterior mean of β⋆. In this work, we prove these conjectures for a general class of signal priors and for arbitrary rotationally-invariant designs A, under a "high-temperature" condition that restricts the range of eigenvalues of A⊤A. Our proof uses a conditional second-moment method argument, where we condition on the iterates of a version of the Vector AMP algorithm for solving the TAP mean-field equations.

We study estimation of a gradient-sparse parameter vector $\boldsymbol{\theta}^* \in \mathbb{R}^p$, having strong gradient-sparsity $s^*:=\|\nabla_G \boldsymbol{\theta}^*\|_0$ on an underlying graph $G$. Given observations $Z_1,\ldots,Z_n$ and a smooth, convex loss function $\mathcal{L}$ for which $\boldsymbol{\theta}^*$ minimizes the population risk $\mathbb{E}[\mathcal{L}(\boldsymbol{\theta};Z_1,\ldots,Z_n)]$, we propose to estimate $\boldsymbol{\theta}^*$ by a projected gradient descent algorithm that iteratively and approximately projects gradient steps onto spaces of vectors having small gradient-sparsity over low-degree spanning trees of $G$. We show that, under suitable restricted strong convexity and smoothness assumptions for the loss, the resulting estimator achieves the squared-error risk $\frac{s^*}{n} \log (1+\frac{p}{s^*})$ up to a multiplicative constant that is independent of $G$. In contrast, previous polynomial-time algorithms have only been shown to achieve this guarantee in more specialized settings, or under additional assumptions for $G$ and/or the sparsity pattern of $\nabla_G \boldsymbol{\theta}^*$. As applications of our general framework, we apply our results to the examples of linear models and generalized linear models with random design.

Graph matching, also known as network alignment, aims at recovering the latent vertex correspondence between two unlabeled, edge-correlated weighted graphs. To tackle this task, we propose a spectral method, GRAph Matching by Pairwise eigen-Alignments (GRAMPA), which first constructs a similarity matrix as a weighted sum of outer products between all pairs of eigenvectors of the two graphs, and then outputs a matching by a simple rounding procedure. For a universality class of correlated Wigner models, GRAMPA achieves exact recovery of the latent matching between two graphs with edge correlation $1 - 1/\mathrm{polylog}(n)$ and average degree at least $\mathrm{polylog}(n)$. This matches the state-of-the-art guarantees for polynomial-time algorithms established for correlated Erdős-Rényi graphs, and significantly improves over existing spectral methods. The superiority of GRAMPA is also demonstrated on a variety of synthetic and real datasets, in terms of both statistical accuracy and computational efficiency.

We study the eigenvalue distributions of the Conjugate Kernel and Neural Tangent Kernel associated to multi-layer feedforward neural networks. In an asymptotic regime where network width is increasing linearly in sample size, under random initialization of the weights, and for input samples satisfying a notion of approximate pairwise orthogonality, we show that the eigenvalue distributions of the CK and NTK converge to deterministic limits. The limit for the CK is described by iterating the Marcenko-Pastur map across the hidden layers. The limit for the NTK is equivalent to that of a linear combination of the CK matrices across layers, and may be described by recursive fixed-point equations that extend this Marcenko-Pastur map. We demonstrate the agreement of these asymptotic predictions with the observed spectra for both synthetic and CIFAR-10 training data, and we perform a small simulation to investigate the evolutions of these spectra over training.

In this paper we propose a hybrid architecture of actor-critic algorithms for reinforcement learning in parameterized action space, which consists of multiple parallel sub-actor networks to decompose the structured action space into simpler action spaces along with a critic network to guide the training of all sub-actor networks. While this paper is mainly focused on parameterized action space, the proposed architecture, which we call hybrid actor-critic, can be extended for more general action spaces which has a hierarchical structure. We present an instance of the hybrid actor-critic architecture based on proximal policy optimization (PPO), which we refer to as hybrid proximal policy optimization (H-PPO). Our experiments test H-PPO on a collection of tasks with parameterized action space, where H-PPO demonstrates superior performance over previous methods of parameterized action reinforcement learning.

We investigate a sequential optimization procedure to minimize the empirical risk functional $f_{\hat\theta}(x) = \frac{1}{2}\|G_{\hat\theta}(x) - y\|^2$ for certain families of deep networks $G_{\theta}(x)$. The approach is to optimize a sequence of objective functions that use network parameters obtained during different stages of the training process. When initialized with random parameters $\theta_0$, we show that the objective $f_{\theta_0}(x)$ is ``nice'' and easy to optimize with gradient descent. As learning is carried out, we obtain a sequence of generative networks $x \mapsto G_{\theta_t}(x)$ and associated risk functions $f_{\theta_t}(x)$, where $t$ indicates a stage of stochastic gradient descent during training. Since the parameters of the network do not change by very much in each step, the surface evolves slowly and can be incrementally optimized. The algorithm is formalized and analyzed for a family of expansive networks. We call the procedure {\it surfing} since it rides along the peak of the evolving (negative) empirical risk function, starting from a smooth surface at the beginning of learning and ending with a wavy nonconvex surface after learning is complete. Experiments show how surfing can be used to find the global optimum and for compressed sensing even when direct gradient descent on the final learned network fails.

Several probabilistic models from high-dimensional statistics and machine learning reveal an intriguing and yet poorly understood dichotomy. Either simple local algorithms succeed in estimating the object of interest, or even sophisticated semi-definite programming (SDP) relaxations fail. In order to explore this phenomenon, we study a classical SDP relaxation of the minimum graph bisection problem, when applied to Erdos-Renyi random graphs with bounded average degree d > 1, and obtain several types of results. First, we use a dual witness construction (using the so-called non-backtracking matrix of the graph) to upper bound the SDP value. Second, we prove that a simple local algorithm approximately solves the SDP to within a factor 2d^2/(2d^2 + d - 1) of the upper bound. In particular, the local algorithm is at most 8/9 suboptimal, and 1 + O(d^-1) suboptimal for large degree.