How Well Can Transformers Emulate In-context Newton's Method?

Transformer-based models have demonstrated remarkable in-context learning capabilities, prompting extensive research into its underlying mechanisms. Recent studies have suggested that Transformers can implement first-order optimization algorithms for in-context learning and even second order ones for the case of linear regression. In this work, we study whether Transformers can perform higher order optimization methods, beyond the case of linear regression. We establish that linear attention Transformers with ReLU layers can approximate second order optimization algorithms for the task of logistic regression and achieve $\epsilon$ error with only a logarithmic to the error more layers. As a by-product we demonstrate the ability of even linear attention-only Transformers in implementing a single step of Newton's iteration for matrix inversion with merely two layers. These results suggest the ability of the Transformer architecture to implement complex algorithms, beyond gradient descent.

Stochastic Methods in Variational Inequalities: Ergodicity, Bias and Refinements

For min-max optimization and variational inequalities problems (VIP) encountered in diverse machine learning tasks, Stochastic Extragradient (SEG) and Stochastic Gradient Descent Ascent (SGDA) have emerged as preeminent algorithms. Constant step-size variants of SEG/SGDA have gained popularity, with appealing benefits such as easy tuning and rapid forgiveness of initial conditions, but their convergence behaviors are more complicated even in rudimentary bilinear models. Our work endeavors to elucidate and quantify the probabilistic structures intrinsic to these algorithms. By recasting the constant step-size SEG/SGDA as time-homogeneous Markov Chains, we establish a first-of-its-kind Law of Large Numbers and a Central Limit Theorem, demonstrating that the average iterate is asymptotically normal with a unique invariant distribution for an extensive range of monotone and non-monotone VIPs. Specializing to convex-concave min-max optimization, we characterize the relationship between the step-size and the induced bias with respect to the Von-Neumann's value. Finally, we establish that Richardson-Romberg extrapolation can improve proximity of the average iterate to the global solution for VIPs. Our probabilistic analysis, underpinned by experiments corroborating our theoretical discoveries, harnesses techniques from optimization, Markov chains, and operator theory.

Dissecting Chain-of-Thought: A Study on Compositional In-Context Learning of MLPs

Chain-of-thought (CoT) is a method that enables language models to handle complex reasoning tasks by decomposing them into simpler steps. Despite its success, the underlying mechanics of CoT are not yet fully understood. In an attempt to shed light on this, our study investigates the impact of CoT on the ability of transformers to in-context learn a simple to study, yet general family of compositional functions: multi-layer perceptrons (MLPs). In this setting, we reveal that the success of CoT can be attributed to breaking down in-context learning of a compositional function into two distinct phases: focusing on data related to each step of the composition and in-context learning the single-step composition function. Through both experimental and theoretical evidence, we demonstrate how CoT significantly reduces the sample complexity of in-context learning (ICL) and facilitates the learning of complex functions that non-CoT methods struggle with. Furthermore, we illustrate how transformers can transition from vanilla in-context learning to mastering a compositional function with CoT by simply incorporating an additional layer that performs the necessary filtering for CoT via the attention mechanism. In addition to these test-time benefits, we highlight how CoT accelerates pretraining by learning shortcuts to represent complex functions and how filtering plays an important role in pretraining. These findings collectively provide insights into the mechanics of CoT, inviting further investigation of its role in complex reasoning tasks.

The Expressive Power of Tuning Only the Norm Layers

Feature normalization transforms such as Batch and Layer-Normalization have become indispensable ingredients of state-of-the-art deep neural networks. Recent studies on fine-tuning large pretrained models indicate that just tuning the parameters of these affine transforms can achieve high accuracy for downstream tasks. These findings open the questions about the expressive power of tuning the normalization layers of frozen networks. In this work, we take the first step towards this question and show that for random ReLU networks, fine-tuning only its normalization layers can reconstruct any target network that is $O(\sqrt{\text{width}})$ times smaller. We show that this holds even for randomly sparsified networks, under sufficient overparameterization, in agreement with prior empirical work.

Looped Transformers as Programmable Computers

We present a framework for using transformer networks as universal computers by programming them with specific weights and placing them in a loop. Our input sequence acts as a punchcard, consisting of instructions and memory for data read/writes. We demonstrate that a constant number of encoder layers can emulate basic computing blocks, including embedding edit operations, non-linear functions, function calls, program counters, and conditional branches. Using these building blocks, we emulate a small instruction-set computer. This allows us to map iterative algorithms to programs that can be executed by a looped, 13-layer transformer. We show how this transformer, instructed by its input, can emulate a basic calculator, a basic linear algebra library, and in-context learning algorithms that employ backpropagation. Our work highlights the versatility of the attention mechanism, and demonstrates that even shallow transformers can execute full-fledged, general-purpose programs.

On the convergence of policy gradient methods to Nash equilibria in general stochastic games

Learning in stochastic games is a notoriously difficult problem because, in addition to each other's strategic decisions, the players must also contend with the fact that the game itself evolves over time, possibly in a very complicated manner. Because of this, the convergence properties of popular learning algorithms - like policy gradient and its variants - are poorly understood, except in specific classes of games (such as potential or two-player, zero-sum games). In view of this, we examine the long-run behavior of policy gradient methods with respect to Nash equilibrium policies that are second-order stationary (SOS) in a sense similar to the type of sufficiency conditions used in optimization. Our first result is that SOS policies are locally attracting with high probability, and we show that policy gradient trajectories with gradient estimates provided by the REINFORCE algorithm achieve an $\mathcal{O}(1/\sqrt{n})$ distance-squared convergence rate if the method's step-size is chosen appropriately. Subsequently, specializing to the class of deterministic Nash policies, we show that this rate can be improved dramatically and, in fact, policy gradient methods converge within a finite number of iterations in that case.

Survival of the strictest: Stable and unstable equilibria under regularized learning with partial information

In this paper, we examine the Nash equilibrium convergence properties of no-regret learning in general N-player games. For concreteness, we focus on the archetypal follow the regularized leader (FTRL) family of algorithms, and we consider the full spectrum of uncertainty that the players may encounter - from noisy, oracle-based feedback, to bandit, payoff-based information. In this general context, we establish a comprehensive equivalence between the stability of a Nash equilibrium and its support: a Nash equilibrium is stable and attracting with arbitrarily high probability if and only if it is strict (i.e., each equilibrium strategy has a unique best response). This equivalence extends existing continuous-time versions of the folk theorem of evolutionary game theory to a bona fide algorithmic learning setting, and it provides a clear refinement criterion for the prediction of the day-to-day behavior of no-regret learning in games

From Learning with Partial Information to Bandits: Only Strict Nash Equilibria are Stable

In this paper, we examine the Nash equilibrium convergence properties of no-regret learning in general $N$-player games. Despite the importance and widespread applications of no-regret algorithms, their long-run behavior in multi-agent environments is still far from understood, and most of the literature has focused by necessity on certain, specific classes of games (typically zero-sum or congestion games). Instead of focusing on a fixed class of games, we instead take a structural approach and examine different classes of equilibria in generic games. For concreteness, we focus on the archetypal "follow the regularized leader" (FTRL) class of algorithms, and we consider the full spectrum of information uncertainty that the players may encounter - from noisy, oracle-based feedback, to bandit, payoff-based information. In this general context, we establish a comprehensive equivalence between the stability of a Nash equilibrium and its support: a Nash equilibrium is stable and attracting with arbitrarily high probability if and only if it is strict (i.e., each equilibrium strategy has a unique best response). This result extends existing continuous-time versions of the "folk theorem" of evolutionary game theory to a bona fide discrete-time learning setting, and provides an important link between the literature on multi-armed bandits and the equilibrium refinement literature.