Let a two-player Markov game where both players affect the transition. We will effectively show that the problem of best-responding to a correlated policy σ is526 equivalent to best-responding to the marginal policy of σ for the opponent. The proof follows from527 the equivalence of the two MDPs.528 Before that, given a (possibly correlated) joint policy σ we define a nonlinear program, (PBR), whose539 optimal solutions are best-response policies of each agent k to σ k and the values for each state s540 and timestep h:541 A.2 Proof of Theorem 3.2542 The best-response program. First, we state the following lemma that will prove useful for several543 of our arguments,544 Lemma A.1 (Best-response LP).

In this paper, we prove a conjecture published in 1989 and also partially address an open problem announced at the Conference on Learning Theory (COLT) 2015. With no unrealistic assumption, we first prove the following statements for the squared loss function of deep linear neural networks with any depth and any widths: 1) the function is non-convex and non-concave, 2) every local minimum is a global minimum, 3) every critical point that is not a global minimum is a saddle point, and 4) there exist "bad" saddle points (where the Hessian has no negative eigenvalue) for the deeper networks (with more than three layers), whereas there is no bad saddle point for the shallow networks (with three layers). Moreover, for deep nonlinear neural networks, we prove the same four statements via a reduction to a deep linear model under the independence assumption adopted from recent work. As a result, we present an instance, for which we can answer the following question: how difficult is it to directly train a deep model in theory? It is more difficult than the classical machine learning models (because of the non-convexity), but not too difficult (because of the nonexistence of poor local minima). Furthermore, the mathematically proven existence of bad saddle points for deeper models would suggest a possible open problem. We note that even though we have advanced the theoretical foundations of deep learning and non-convex optimization, there is still a gap between theory and practice.

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In recent years, stochastic gradient descent (SGD) based techniques has become the standard tools for training neural networks. However, formal theoretical understanding of why SGD can train neural networks in practice is largely missing. In this paper, we make progress on understanding this mystery by providing a convergence analysis for SGD on a rich subset of two-layer feedforward networks with ReLU activations. This subset is characterized by a special structure called identity mapping. We prove that, if input follows from Gaussian distribution, with standard $O(1/\sqrt{d})$ initialization of the weights, SGD converges to the global minimum in polynomial number of steps.

In this paper, we prove a conjecture published in 1989 and also partially address an open problem announced at the Conference on Learning Theory (COLT) 2015. For an expected loss function of a deep nonlinear neural network, we prove the following statements under the independence assumption adopted from recent work: 1) the function is non-convex and non-concave, 2) every local minimum is a global minimum, 3) every critical point that is not a global minimum is a saddle point, and 4) the property of saddle points differs for shallow networks (with three layers) and deeper networks (with more than three layers). Moreover, we prove that the same four statements hold for deep linear neural networks with any depth, any widths and no unrealistic assumptions. As a result, we present an instance, for which we can answer to the following question: how difficult to directly train a deep model in theory? It is more difficult than the classical machine learning models (because of the non-convexity), but not too difficult (because of the nonexistence of poor local minima and the property of the saddle points). We note that even though we have advanced the theoretical foundations of deep learning, there is still a gap between theory and practice.

In recent years, transformer-based models have revolutionized deep learning, particularly in sequence modeling.

Let a two-player Markov game where both players affect the transition. As we have seen in Section 2.1, in the case of unilateral deviation from joint policy Let a (possibly correlated) joint policy ˆ σ . By Lemma A.1, we know that Where the equality holds due to the zero-sum property, (1). An approximate NE is an approximate global minimum. An approximate global minimum is an approximate NE.

One such example is two-player zero-sum Markov games, in which efficient ways to compute a Nash equilibrium are known. Inspired by zero-sum polymatrix normal-form games (Cai et al., 2016), we define a class of zero-sum

Several recent works demonstrate that transformers can implement algorithms like gradient descent. By a careful construction of weights, these works show that multiple layers of transformers are expressive enough to simulate iterations of gradient descent.

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