When are two algorithms the same? How can we be sure a recently proposed algorithm is novel, and not a minor twist on an existing method? In this paper, we present a framework for reasoning about equivalence between a broad class of iterative algorit
hms, with a focus on algorithms designed for convex optimization. We propose several notions of what it means for two algorithms to be equivalent, and provide computationally tractable means to detect equivalence. Our main definition, oracle equivalence, states that two algorithms are equivalent if they result in the same sequence of calls to the function oracles (for suitable initialization). Borrowing from control theory, we use state-space realizations to represent algorithms and characterize algorithm equivalence via transfer functions. Our framework can also identify and characterize some algorithm transformations including permutations of the update equations, repetition of the iteration, and conjugation of some of the function oracles in the algorithm. To support the paper, we have developed a software package named Linnaeus that implements the framework to identify other iterative algorithms that are equivalent to an input algorithm. More broadly, this framework and software advances the goal of making mathematics searchable.
We study a linear-quadratic, optimal control problem on a discrete, finite time horizon with distributional ambiguity, in which the cost is assessed via Conditional Value-at-Risk (CVaR). We take steps toward deriving a scalable dynamic programming ap
proach to upper-bound the optimal value function for this problem. This dynamic program yields a novel, tunable risk-averse control policy, which we compare to existing state-of-the-art methods.
Smooth minimax games often proceed by simultaneous or alternating gradient updates. Although algorithms with alternating updates are commonly used in practice for many applications (e.g., GAN training), the majority of existing theoretical analyses f
ocus on simultaneous algorithms for convenience of analysis. In this paper, we study alternating gradient descent-ascent (Alt-GDA) in minimax games and show that Alt-GDA is superior to its simultaneous counterpart (Sim-GDA) in many settings. In particular, we prove that Alt-GDA achieves a near-optimal local convergence rate for strongly convex-strongly concave (SCSC) problems while Sim-GDA converges at a much slower rate. To our knowledge, this is the emph{first} result of any setting showing that Alt-GDA converges faster than Sim-GDA by more than a constant. We further prove that the acceleration effect of alternating updates remains when the minimax problem has only strong concavity in the dual variables. Lastly, we adapt the theory of integral quadratic constraints and show that Alt-GDA attains the same rate emph{globally} for a class of SCSC minimax problems. Numerical experiments on quadratic minimax games validate our claims. Empirically, we demonstrate that alternating updates speed up GAN training significantly and the use of optimism only helps for simultaneous algorithms.
The theory of integral quadratic constraints (IQCs) allows the certification of exponential convergence of interconnected systems containing nonlinear or uncertain elements. In this work, we adapt the IQC theory to study first-order methods for smoot
h and strongly-monotone games and show how to design tailored quadratic constraints to get tight upper bounds of convergence rates. Using this framework, we recover the existing bound for the gradient method~(GD), derive sharper bounds for the proximal point method~(PPM) and optimistic gradient method~(OG), and provide emph{for the first time} a global convergence rate for the negative momentum method~(NM) with an iteration complexity $mathcal{O}(kappa^{1.5})$, which matches its known lower bound. In addition, for time-varying systems, we prove that the gradient method with optimal step size achieves the fastest provable worst-case convergence rate with quadratic Lyapunov functions. Finally, we further extend our analysis to stochastic games and study the impact of multiplicative noise on different algorithms. We show that it is impossible for an algorithm with one step of memory to achieve acceleration if it only queries the gradient once per batch (in contrast with the stochastic strongly-convex optimization setting, where such acceleration has been demonstrated). However, we exhibit an algorithm which achieves acceleration with two gradient queries per batch.
