When balancing the practical tradeoffs of iterative methods for large-scale optimization, the learning rate schedule remains notoriously difficult to understand and expensive to tune. We demonstrate the presence of these subtleties even in the innocuous case when the objective is a convex quadratic. We reinterpret an iterative algorithm from the numerical analysis literature as what we call the Chebyshev learning rate schedule for accelerating vanilla gradient descent, and show that the problem of mitigating instability leads to a fractal ordering of step sizes. We provide some experiments and discussion to challenge current understandings of the "edge of stability" in deep learning: even in simple settings, provable acceleration can be obtained by making negative local progress on the objective.

An exciting recent development is the uptake of deep learning in many scientific fields, where the objective is seeking novel scientific insights and discoveries. To interpret a learning outcome, researchers perform hypothesis testing for explainable features to advance scientific domain knowledge. In such a situation, testing for a blackbox learner poses a severe challenge because of intractable models, unknown limiting distributions of parameter estimates, and high computational constraints. In this article, we derive two consistent tests for the feature relevance of a blackbox learner. The first one evaluates a loss difference with perturbation on an inference sample, which is independent of an estimation sample used for parameter estimation in model fitting. The second further splits the inference sample into two but does not require data perturbation. Also, we develop their combined versions by aggregating the order statistics of the $p$-values based on repeated sample splitting. To estimate the splitting ratio and the perturbation size, we develop adaptive splitting schemes for suitably controlling the Type \rom{1} error subject to computational constraints. By deflating the \textit{bias-sd-ratio}, we establish asymptotic null distributions of the test statistics and their consistency in terms of statistical power. Our theoretical power analysis and simulations indicate that the one-split test is more powerful than the two-split test, though the latter is easier to apply for large datasets. Moreover, the combined tests are more stable while compensating for a power loss by repeated sample splitting. Numerically, we demonstrate the utility of the proposed tests on two benchmark examples. Accompanying this paper is our Python library {\tt dnn-inference} https://dnn-inference.readthedocs.io/en/latest/ that implements the proposed tests.

Strong empirical evidence that one machine-learning algorithm A outperforms another one B ideally calls for multiple trials optimizing the learning pipeline over sources of variation such as data sampling, data augmentation, parameter initialization, and hyperparameters choices. This is prohibitively expensive, and corners are cut to reach conclusions. We model the whole benchmarking process, revealing that variance due to data sampling, parameter initialization and hyperparameter choice impact markedly the results. We analyze the predominant comparison methods used today in the light of this variance. We show a counter-intuitive result that adding more sources of variation to an imperfect estimator approaches better the ideal estimator at a 51 times reduction in compute cost. Building on these results, we study the error rate of detecting improvements, on five different deep-learning tasks/architectures. This study leads us to propose recommendations for performance comparisons.

Gaussian processes (GPs) provide a gold standard for performance in online settings, such as sample-efficient control and black box optimization, where we need to update a posterior distribution as we acquire data in a sequential fashion. However, updating a GP posterior to accommodate even a single new observation after having observed $n$ points incurs at least $O(n)$ computations in the exact setting. We show how to use structured kernel interpolation to efficiently recycle computations for constant-time $O(1)$ online updates with respect to the number of points $n$, while retaining exact inference. We demonstrate the promise of our approach in a range of online regression and classification settings, Bayesian optimization, and active sampling to reduce error in malaria incidence forecasting. Code is available at https://github.com/wjmaddox/online_gp.

This paper presents the first general (supervised) statistical learning framework for point processes in general spaces. Our approach is based on the combination of two new concepts, which we define in the paper: i) bivariate innovations, which are measures of discrepancy/prediction-accuracy between two point processes, and ii) point process cross-validation (CV), which we here define through point process thinning. The general idea is to carry out the fitting by predicting CV-generated validation sets using the corresponding training sets; the prediction error, which we minimise, is measured by means of bivariate innovations. Having established various theoretical properties of our bivariate innovations, we study in detail the case where the CV procedure is obtained through independent thinning and we apply our statistical learning methodology to three typical spatial statistical settings, namely parametric intensity estimation, non-parametric intensity estimation and Papangelou conditional intensity fitting. Aside from deriving theoretical properties related to these cases, in each of them we numerically show that our statistical learning approach outperforms the state of the art in terms of mean (integrated) squared error.

Bayesian inference has been long called for Bayesian computation techniques that are scalable to large data sets and applicable in big and complex models with a huge number of unknown parameters to infer. Sampling methods, such as Markov Chain Monte Carlo (MCMC) and Sequential Monte Carlo (SMC), in their current development do not meet this need. Sampling methods have not been successfully used in some modern areas such as deep neural networks. Even in more traditional areas such as graphical modelling and mixture modelling, it is very challenging to use MCMC and SMC. Variational Bayes (VB) is an optimization-based technique for approximate Bayesian inference, and provides a computationally efficient alternative to sampling methods.

Recently, particle-based variational inference (ParVI) methods have gained interest because they directly minimize the Kullback-Leibler divergence and do not suffer from approximation errors from the evidence-based lower bound. However, many ParVI approaches do not allow arbitrary sampling from the posterior, and the few that do allow such sampling suffer from suboptimality. This work proposes a new method for learning to approximately sample from the posterior distribution. We construct a neural sampler that is trained with the functional gradient of the KL-divergence between the empirical sampling distribution and the target distribution, assuming the gradient resides within a reproducing kernel Hilbert space. Our generative ParVI (GPVI) approach maintains the asymptotic performance of ParVI methods while offering the flexibility of a generative sampler. Through carefully constructed experiments, we show that GPVI outperforms previous generative ParVI methods such as amortized SVGD, and is competitive with ParVI as well as gold-standard approaches like Hamiltonian Monte Carlo for fitting both exactly known and intractable target distributions.

This paper discusses the problem of estimation and inference on time-varying treatments. We propose a method for inference on treatment histories, by introducing a \textit{dynamic} covariate balancing method. Our approach allows for (i) treatments to propagate arbitrarily over time; (ii) non-stationarity and heterogeneity of treatment effects; (iii) high-dimensional covariates, and (iv) unknown propensity score functions. We study the asymptotic properties of the estimator, and we showcase the parametric convergence rate of the proposed procedure. We illustrate in simulations and an empirical application the advantage of the method over state-of-the-art competitors.

Differentially private (DP) stochastic convex optimization (SCO) is a fundamental problem, where the goal is to approximately minimize the population risk with respect to a convex loss function, given a dataset of i.i.d. samples from a distribution, while satisfying differential privacy with respect to the dataset. Most of the existing works in the literature of private convex optimization focus on the Euclidean (i.e., $\ell_2$) setting, where the loss is assumed to be Lipschitz (and possibly smooth) w.r.t. the $\ell_2$ norm over a constraint set with bounded $\ell_2$ diameter. Algorithms based on noisy stochastic gradient descent (SGD) are known to attain the optimal excess risk in this setting. In this work, we conduct a systematic study of DP-SCO for $\ell_p$-setups. For $p=1$, under a standard smoothness assumption, we give a new algorithm with nearly optimal excess risk. This result also extends to general polyhedral norms and feasible sets. For $p\in(1, 2)$, we give two new algorithms, whose central building block is a novel privacy mechanism, which generalizes the Gaussian mechanism. Moreover, we establish a lower bound on the excess risk for this range of $p$, showing a necessary dependence on $\sqrt{d}$, where $d$ is the dimension of the space. Our lower bound implies a sudden transition of the excess risk at $p=1$, where the dependence on $d$ changes from logarithmic to polynomial, resolving an open question in prior work [TTZ15] . For $p\in (2, \infty)$, noisy SGD attains optimal excess risk in the low-dimensional regime; in particular, this proves the optimality of noisy SGD for $p=\infty$. Our work draws upon concepts from the geometry of normed spaces, such as the notions of regularity, uniform convexity, and uniform smoothness.

We study the relative power of learning with gradient descent on differentiable models, such as neural networks, versus using the corresponding tangent kernels. We show that under certain conditions, gradient descent achieves small error only if a related tangent kernel method achieves a non-trivial advantage over random guessing (a.k.a. weak learning), though this advantage might be very small even when gradient descent can achieve arbitrarily high accuracy. Complementing this, we show that without these conditions, gradient descent can in fact learn with small error even when no kernel method, in particular using the tangent kernel, can achieve a non-trivial advantage over random guessing.