We introduce Recurrent All-Pairs Field Transforms (RAFT), a new deep network architecture for optical flow. RAFT extracts per-pixel features, builds multi-scale 4D correlation volumes for all pairs of pixels, and iteratively updates a flow field through a recurrent unit that performs lookups on the correlation volumes. RAFT achieves state-of-the-art performance. On KITTI, RAFT achieves an F1-all error of 5.10%, a 16% error reduction from the best published result (6.10%). On Sintel (final pass), RAFT obtains an end-point-error of 2.855 pixels, a 30% error reduction from the best published result (4.098 pixels). In addition, RAFT has strong cross-dataset generalization as well as high efficiency in inference time, training speed, and parameter count. Code is available at https://github.com/princeton-vl/RAFT.

RAFT: Recurrent All-Pairs Field Transforms for Optical Flow

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Perturbation Analysis Of Optimization Problems

We present a general purpose solver for quadratic programs based on the alternating direction method of multipliers, employing a novel operator splitting technique that requires the solution of a quasi-definite linear system with the same coefficient matrix in each iteration. Our algorithm is very robust, placing no requirements on the problem data such as positive definiteness of the objective function or linear independence of the constraint functions. It is division-free once an initial matrix factorization is carried out, making it suitable for real-time applications in embedded systems. In addition, our technique is the first operator splitting method for quadratic programs able to reliably detect primal and dual infeasible problems from the algorithm iterates. The method also supports factorization caching and warm starting, making it particularly efficient when solving parametrized problems arising in finance, control, and machine learning. Our open-source C implementation OSQP has a small footprint, is library-free, and has been extensively tested on many problem instances from a wide variety of application areas. It is typically ten times faster than competing interior point methods, and sometimes much more when factorization caching or warm start is used.

OSQP: An Operator Splitting Solver for Quadratic Programs

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.

/pdf/differentiable-convex-optimization-layers-19tvs5fa54.pdf

Differentiable Convex Optimization Layers

Efforts to understand the generalization mystery in deep learning have led to the belief that gradient-based optimization induces a form of implicit regularization, a bias towards models of low "complexity." We study the implicit regularization of gradient descent over deep linear neural networks for matrix completion and sensing, a model referred to as deep matrix factorization. Our first finding, supported by theory and experiments, is that adding depth to a matrix factorization enhances an implicit tendency towards low-rank solutions, oftentimes leading to more accurate recovery. Secondly, we present theoretical and empirical arguments questioning a nascent view by which implicit regularization in matrix factorization can be captured using simple mathematical norms. Our results point to the possibility that the language of standard regularizers may not be rich enough to fully encompass the implicit regularization brought forth by gradient-based optimization.

Implicit Regularization in Deep Matrix Factorization

We describe a modular rewriting system for translating optimization problems written in a domain-specific language (DSL) to forms compatible with low-level solver interfaces. Translation is facilit...

A rewriting system for convex optimization problems

We describe a modular rewriting system for translating optimization problems written in a domain-specific language to forms compatible with low-level solver interfaces. Translation is facilitated by reductions, which accept a category of problems and transform instances of that category to equivalent instances of another category. Our system proceeds in two key phases: analysis, in which we attempt to find a suitable solver for a supplied problem, and canonicalization, in which we rewrite the problem in the selected solver's standard form. We implement the described system in version 1.0 of CVXPY, a domain-specific language for mathematical and especially convex optimization. By treating reductions as first-class objects, our method makes it easy to match problems to solvers well-suited for them and to support solvers with a wide variety of standard forms.

A Rewriting System for Convex Optimization Problems

In Massive Open Online Courses (MOOCs), struggling learners often seek help by
posting questions in discussion forums. Unfortunately, given the large volume of discussion in MOOCs, instructors may overlook these learners' posts,
detrimentally impacting the learning process and exacerbating attrition. In this paper, we present YouEDU, an instructional aid that automatically detects and addresses confusion in forum posts. Leveraging our Stanford MOOCPosts corpus, we train a heterogeneous set of classifiers to classify forum posts across multiple dimensions. In particular, classifiers that target sentiment, urgency, and other descriptive variables inform a single classifier that detects confusion. We then employ information retrieval techniques to map confused posts to minute-resolution clips from course videos; the ranking over these clips accounts for textual similarity between posts and closed captions. We measure the performance of our classification model in multiple educational contexts, exploring the nature of confusion within each; we also evaluate the relevancy of materials returned by our ranking algorithm. Experimental results demonstrate that YouEDU achieves both its goals, paving the way for intelligent intervention systems in MOOC discussion forums.

/pdf/youedu-addressing-confusion-in-mooc-discussion-forums-by-51w9ijm2sk.pdf

YouEDU: Addressing Confusion in MOOC Discussion Forums by Recommending Instructional Video Clips.

We consider the problem of efficiently computing the derivative of the solution map of a convex cone program, when it exists. We do this by implicitly differentiating the residual map for its homogeneous self-dual embedding, and solving the linear systems of equations required using an iterative method. This allows us to efficiently compute the derivative operator, and its adjoint, evaluated at a vector. These correspond to computing an approximate new solution, given a perturbation to the cone program coefficients (i.e., perturbation analysis), and to computing the gradient of a function of the solution with respect to the coefficients. Our method scales to large problems, with numbers of coefficients in the millions. We present an open-source Python implementation of our method that solves a cone program and returns the derivative and its adjoint as abstract linear maps; our implementation can be easily integrated into software systems for automatic differentiation.

Akshay Agrawal

Papers

A rewriting system for convex optimization problems

Differentiable Convex Optimization Layers

A Rewriting System for Convex Optimization Problems

YouEDU: Addressing Confusion in MOOC Discussion Forums by Recommending Instructional Video Clips.

Differentiating Through a Conic Program