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Recently, significant progress has been made regarding the statistical understanding of artificial neural networks (ANNs). ANNs are motivated by the functioning of the brain, but differ in several crucial aspects. In particular, it is biologically implausible that the learning of the brain is based on gradient descent. In this work we look at the brain as a statistical method for supervised learning. The main contribution is to relate the local updating rule of the connection parameters in biological neural networks (BNNs) to a zero-order optimization method.

Convolutional neural networks (CNN) have been successful in machine learning applications. Their success relies on their ability to consider space invariant local features. We consider the use of CNN to fit nuisance models in semiparametric estimation of the average causal effect of a treatment. In this setting, nuisance models are functions of pre-treatment covariates that need to be controlled for. In an application where we want to estimate the effect of early retirement on a health outcome, we propose to use CNN to control for time-structured covariates. Thus, CNN is used when fitting nuisance models explaining the treatment and the outcome. These fits are then combined into an augmented inverse probability weighting estimator yielding efficient and uniformly valid inference. Theoretically, we contribute by providing rates of convergence for CNN equipped with the rectified linear unit activation function and compare it to an existing result for feedforward neural networks. We also show when those rates guarantee uniformly valid inference. A Monte Carlo study is provided where the performance of the proposed estimator is evaluated and compared with other strategies. Finally, we give results on a study of the effect of early retirement on hospitalization using data covering the whole Swedish population.

Learning in neural networks is often framed as a problem in which targeted error signals are directly propagated to parameters and used to produce updates that induce more optimal network behaviour. Backpropagation of error (BP) is an example of such an approach and has proven to be a highly successful application of stochastic gradient descent to deep neural networks. We propose constrained parameter inference (COPI) as a new principle for learning. The COPI approach assumes that learning can be set up in a manner where parameters infer their own values based upon observations of their local neuron activities. We find that this estimation of network parameters is possible under the constraints of decorrelated neural inputs and top-down perturbations of neural states for credit assignment. We show that the decorrelation required for COPI allows learning at extremely high learning rates, competitive with that of adaptive optimizers, as used by BP. We further demonstrate that COPI affords a new approach to feature analysis and network compression. Finally, we argue that COPI may shed new light on learning in biological networks given the evidence for decorrelation in the brain.

It is believed that Gradient Descent (GD) induces an implicit bias towards good generalization in training machine learning models. This paper provides a fine-grained analysis of the dynamics of GD for the matrix sensing problem, whose goal is to recover a low-rank ground-truth matrix from near-isotropic linear measurements. It is shown that GD with small initialization behaves similarly to the greedy low-rank learning heuristics (Li et al., 2020) and follows an incremental learning procedure (Gissin et al., 2019): GD sequentially learns solutions with increasing ranks until it recovers the ground truth matrix. Compared to existing works which only analyze the first learning phase for rank-1 solutions, our result provides characterizations for the whole learning process. Moreover, besides the over-parameterized regime that many prior works focused on, our analysis of the incremental learning procedure also applies to the under-parameterized regime. Finally, we conduct numerical experiments to confirm our theoretical findings.

Stochastic versions of proximal methods have gained much attention in statistics and machine learning. These algorithms tend to admit simple, scalable forms, and enjoy numerical stability via implicit updates. In this work, we propose and analyze a stochastic version of the recently proposed proximal distance algorithm, a class of iterative optimization methods that recover a desired constrained estimation problem as a penalty parameter $\rho \rightarrow \infty$. By uncovering connections to related stochastic proximal methods and interpreting the penalty parameter as the learning rate, we justify heuristics used in practical manifestations of the proximal distance method, establishing their convergence guarantees for the first time. Moreover, we extend recent theoretical devices to establish finite error bounds and a complete characterization of convergence rates regimes. We validate our analysis via a thorough empirical study, also showing that unsurprisingly, the proposed method outpaces batch versions on popular learning tasks.

Rule-based reasoning is an essential part of human intelligence prominently formalized in artificial intelligence research via logic programs. Describing complex objects as the composition of elementary ones is a common strategy in computer science and science in general. The author has recently introduced the sequential composition of logic programs in the context of logic-based analogical reasoning and learning in logic programming. Motivated by these applications, in this paper we construct a qualitative and algebraic notion of syntactic logic program similarity from sequential decompositions of programs. We then show how similarity can be used to answer queries across different domains via a one-step reduction. In a broader sense, this paper is a further step towards an algebraic theory of logic programming.

Risk-sensitive reinforcement learning (RL) has become a popular tool to control the risk of uncertain outcomes and ensure reliable performance in various sequential decision-making problems. While policy gradient methods have been developed for risk-sensitive RL, it remains unclear if these methods enjoy the same global convergence guarantees as in the risk-neutral case. In this paper, we consider a class of dynamic time-consistent risk measures, called Expected Conditional Risk Measures (ECRMs), and derive policy gradient updates for ECRM-based objective functions. Under both constrained direct parameterization and unconstrained softmax parameterization, we provide global convergence of the corresponding risk-averse policy gradient algorithms. We further test a risk-averse variant of REINFORCE algorithm on a stochastic Cliffwalk environment to demonstrate the efficacy of our algorithm and the importance of risk control.

We prove linear convergence of gradient descent to a global optimum for the training of deep residual networks with constant layer width and smooth activation function. We show that if the trained weights, as a function of the layer index, admit a scaling limit as the depth increases, then the limit has finite $p-$variation with $p=2$. Proofs are based on non-asymptotic estimates for the loss function and for norms of the network weights along the gradient descent path. We illustrate the relevance of our theoretical results to practical settings using detailed numerical experiments on supervised learning problems.

Maximum Inner Product Search or top-k retrieval on sparse vectors is well-understood in information retrieval, with a number of mature algorithms that solve it exactly. However, all existing algorithms are tailored to text and frequency-based similarity measures. To achieve optimal memory footprint and query latency, they rely on the near stationarity of documents and on laws governing natural languages. We consider, instead, a setup in which collections are streaming -- necessitating dynamic indexing -- and where indexing and retrieval must work with arbitrarily distributed real-valued vectors. As we show, existing algorithms are no longer competitive in this setup, even against naive solutions. We investigate this gap and present a novel approximate solution, called Sinnamon, that can efficiently retrieve the top-k results for sparse real valued vectors drawn from arbitrary distributions. Notably, Sinnamon offers levers to trade-off memory consumption, latency, and accuracy, making the algorithm suitable for constrained applications and systems. We give theoretical results on the error introduced by the approximate nature of the algorithm, and present an empirical evaluation of its performance on two hardware platforms and synthetic and real-valued datasets. We conclude by laying out concrete directions for future research on this general top-k retrieval problem over sparse vectors.

Convergence (virtual) bidding is an important part of two-settlement electric power markets as it can effectively reduce discrepancies between the day-ahead and real-time markets. Consequently, there is extensive research into the bidding strategies of virtual participants aiming to obtain optimal bids to submit to the day-ahead market. In this paper, we introduce a price-based general stochastic optimization framework to obtain optimal convergence bid curves. Within this framework, we develop a computationally tractable linear programming-based optimization model, which produces bid prices and volumes simultaneously. We also show that different approximations and simplifications in the general model lead naturally to state-of-the-art convergence bidding approaches, such as self-scheduling and opportunistic approaches. Our general framework also provides a straightforward way to compare the performance of these models, which is demonstrated by numerical experiments on the California (CAISO) market.