Economists have developed different types of models describing the interaction of agents in markets. Early models in general equilibrium theory describe agents taking prices as given and do not consider the incentives of agents to manipulate prices strategically. With appropriate convexity assumptions on the preferences, such models can be cast as convex optimization problems for which efficient algorithms are known to find a competitive equilibrium. Price-taking behavior might be a reasonable approximation of agent behavior in large markets, but it does not adequately capture the incentives and strategies that agents have in smaller markets or in other strategic settings. Modern models in economics, such as those used for modeling auctions, oligopoly competition, or contests, are based on game theory, with the Nash equilibrium as the central solution concept.

The examples are nothing if not relatable: preparing breakfast, or playing a game of chess or tic-tac-toe. Yet the idea of learning from the environment and taking steps that progress toward a goal apparently was under-studied when ACM A.M. Turing Award recipients Andrew G. Barto and Richard S. Sutton took on the topic in the late 1970s. Eventually, their research led to the creation of reinforcement learning algorithms that sought not to recognize patterns but maximize rewards. Barto and Sutton spoke about how it all unfolded, and what's next for the techniques that are so celebrated for their success in AlphaGo and AlphaZero. Let's start with the earliest days of your collaboration.

The early theory of actor-critic methods considered convergence using linear function approximators for the policy and value functions. Recent work has established convergence using neural network approximators with a single hidden layer. In this work we are taking the natural next step and establish convergence using deep neural networks with an arbitrary number of hidden layers, thus closing a gap between theory and practice. We show that actor-critic updates projected on a ball around the initial condition will converge to a neighborhood where the average of the squared gradients is \tilde{O} \left( 1/\sqrt{m} \right) O \left( \epsilon \right), with m being the width of the neural network and \epsilon the approximation quality of the best critic neural network over the projected set.

The curse of dimensionality is severe when modeling high-dimensional discrete data: the number of possible combinations of the variables ex(cid:173) plodes exponentially. In this paper we propose a new architecture for modeling high-dimensional data that requires resources (parameters and computations) that grow only at most as the square of the number of vari(cid:173) ables, using a multi-layer neural network to represent the joint distribu(cid:173) tion of the variables as the product of conditional distributions. The neu(cid:173) ral network can be interpreted as a graphical model without hidden ran(cid:173) dom variables, but in which the conditional distributions are tied through the hidden units. The connectivity of the neural network can be pruned by using dependency tests between the variables. Experiments on modeling the distribution of several discrete data sets show statistically significant improvements over other methods such as naive Bayes and comparable Bayesian networks, and show that significant improvements can be ob(cid:173) tained by pruning the network.

Artificial intelligence (AI) is rapidly advancing in the business world, with an increasing number of companies employing deep learning networks to improve their operations. However, it may come as a surprise that more complex and sophisticated deep learning models may not necessarily be better suited for solving business problems. In fact, in many cases, deploying a simpler network can yield more effective results. In this blog post, we'll explore why complex deep learning networks can be inefficient and even detrimental when applied to business scenarios. In my experience, one of the biggest challenges with deep learning networks is obtaining enough training data to achieve accurate results.

Human Visual System is a marvel of the world. But it is not as simple as it looks like. The human brain has a million neurons and billions of connections between them, which makes this exceptionally complex task of image processing easier. People can effortlessly recognize digits. However, it turns into a challenging task for computers to recognize digits.

Neural Networks typically learn by adjusting weights via nonlinear optimization in a training phase. Often, variants of gradient descent are used. These techniques require some differentiability. Therefore, non-smooth but piecewise linear activation functions like ReLU or the Heaviside function raise the question if techniques of linear and mixed integer linear programming are also suited for network training. Learning to near optimality can be performed with Linear Programs (LP) of exponential size for certain network architectures, see [2].

We develop Banach spaces for ReLU neural networks of finite depth $L$ and infinite width. The spaces contain all finite fully connected $L$-layer networks and their $L^2$-limiting objects under bounds on the natural path-norm. Under this norm, the unit ball in the space for $L$-layer networks has low Rademacher complexity and thus favorable generalization properties. Functions in these spaces can be approximated by multi-layer neural networks with dimension-independent convergence rates. The key to this work is a new way of representing functions in some form of expectations, motivated by multi-layer neural networks. This representation allows us to define a new class of continuous models for machine learning. We show that the gradient flow defined this way is the natural continuous analog of the gradient descent dynamics for the associated multi-layer neural networks. We show that the path-norm increases at most polynomially under this continuous gradient flow dynamics.

In fact, the business plans of the next 10,000 startups are easy to forecast: Take X and add AI. Find something that can be made better by adding online smartness to it Over the past few years, artificial intelligence continues to be one of the hottest topics. The best minds participate in AI research, the largest corporations allocate astronomical sums for the development of competencies in this area, and AI startups collect multibillion-dollar investments annually. If you are engaged in business processes improvement or are looking for new ideas for your business, then you will most likely come across AI. And in order to work effectively with it, you need to understand its constituent parts. Let's find out what artificial intelligence is all about.