Neural Information Processing Systems http://nips.cc/

Recent work has shown how to embed differentiable optimization problems (that is, problems whose solutions can be backpropagated through) as layers within deep learning architectures. This method provides a useful inductive bias for certain problems, but existing software for differentiable optimization layers is rigid and difficult to apply to new settings. In this paper, we propose an approach to differentiating through disciplined convex programs, a subclass of convex optimization problems used by domain-specific languages (DSLs) for convex optimization. We introduce disciplined parametrized programming, a subset of disciplined convex programming, and we show that every disciplined parametrized program can be represented as the composition of an affine map from parameters to problem data, a solver, and an affine map from the solver's solution to a solution of the original problem (a new form we refer to as affine-solver-affine form). We then demonstrate how to efficiently differentiate through each of these components, allowing for end-to-end analytical differentiation through the entire convex program. We implement our methodology in version 1.1 of CVXPY, a popular Python-embedded DSL for convex optimization, and additionally implement differentiable layers for disciplined convex programs in PyTorch and TensorFlow 2.0. Our implementation significantly lowers the barrier to using convex optimization problems in differentiable programs. We present applications in linear machine learning models and in stochastic control, and we show that our layer is competitive (in execution time) compared to specialized differentiable solvers from past work.

The efficient frontier (EF) is a fundamental resource allocation problem where one has to find an optimal portfolio maximizing a reward at a given level of risk. This optimal solution is traditionally found by solving a convex optimization problem. In this paper, we introduce NeuralEF: a fast neural approximation framework that robustly forecasts the result of the EF convex optimizations problems with respect to heterogeneous linear constraints and variable number of optimization inputs. By reformulating an optimization problem as a sequence to sequence problem, we show that NeuralEF is a viable solution to accelerate large-scale simulation while handling discontinuous behavior.

Neurons mainly communicate through spikes, and much effort has been spent to understand how the dynamics of spiking neural networks (SNNs) relates to their connectivity. Meanwhile, most major advances in machine learning have been made with simpler, rate-based networks, with SNNs only recently showing competitive results, largely thanks to transferring insights from rate to spiking networks. However, it is still an open question exactly which computations SNNs perform. Recently, the time-averaged firing rates of several SNNs were shown to yield the solutions to convex optimization problems. Here we turn these findings around and show that virtually all inhibition-dominated SNNs can be understood through the lens of convex optimization, with network connectivity, timescales, and firing thresholds being intricately linked to the parameters of underlying convex optimization problems. This approach yields new, geometric insights into the computations performed by spiking networks. In particular, we establish a class of SNNs whose instantaneous output provides a solution to linear or quadratic programming problems, and we thereby reveal their input-output mapping. Using these insights, we derive local, supervised learning rules that can approximate given convex input-output functions, and we show that the resulting networks are consistent with many features from biological networks, such as low firing rates, irregular firing, E/I balance, and robustness to perturbations and synaptic delays.

In this paper, we propose an accelerated quasi-Newton proximal extragradient method for solving unconstrained smooth convex optimization problems. With access only to the gradients of the objective, we prove that our method can achieve a convergence rate of $\mathcal{O}\bigl(\min\\{\frac{1}{k^2}, \frac{\sqrt{d\log k}}{k^{2.5}}\\}\bigr)$,

Online optimization covers problems such as online resource allocation, online bipartite matching, adwords (a central problem in e-commerce and advertising), and adwords with separable concave returns.

Overall then, restart schemes generically accelerate accelerated methods.