Implicit Bias of SignGD and Adam on Multiclass Separable Data

In the optimization of overparameterized models, different gradient-based methods can achieve zero training error yet converge to distinctly different solutions inducing different generalization properties. While a decade of research on implicit optimization bias has illuminated this phenomenon in various settings, even the foundational case of linear classification with separable data still has important open questions. We resolve a fundamental gap by characterizing the implicit bias of both Adam and Sign Gradient Descent in multi-class cross-entropy minimization: we prove that their iterates converge to solutions that maximize the margin with respect to the classifier matrix's max-norm and characterize the rate of convergence. We extend our results to general p-norm normalized steepest descent algorithms and to other multi-class losses.

Don't Be So Positive: Negative Step Sizes in Second-Order Methods

The value of second-order methods lies in the use of curvature information. Yet, this information is costly to extract and once obtained, valuable negative curvature information is often discarded so that the method is globally convergent. This limits the effectiveness of second-order methods in modern machine learning. In this paper, we show that second-order and second-order-like methods are promising optimizers for neural networks provided that we add one ingredient: negative step sizes. We show that under very general conditions, methods that produce ascent directions are globally convergent when combined with a Wolfe line search that allows both positive and negative step sizes. We experimentally demonstrate that using negative step sizes is often more effective than common Hessian modification methods.

Why Line Search when you can Plane Search? SO-Friendly Neural Networks allow Per-Iteration Optimization of Learning and Momentum Rates for Every Layer

We introduce the class of SO-friendly neural networks, which include several models used in practice including networks with 2 layers of hidden weights where the number of inputs is larger than the number of outputs. SO-friendly networks have the property that performing a precise line search to set the step size on each iteration has the same asymptotic cost during full-batch training as using a fixed learning. Further, for the same cost a planesearch can be used to set both the learning and momentum rate on each step. Even further, SO-friendly networks also allow us to use subspace optimization to set a learning rate and momentum rate for each layer on each iteration. We explore augmenting gradient descent as well as quasi-Newton methods and Adam with line optimization and subspace optimization, and our experiments indicate that this gives fast and reliable ways to train these networks that are insensitive to hyper-parameters.

BlockLLM: Memory-Efficient Adaptation of LLMs by Selecting and Optimizing the Right Coordinate Blocks

Training large language models (LLMs) for pretraining or adapting to new tasks and domains has become increasingly critical as their applications expand. However, as the model and the data sizes grow, the training process presents significant memory challenges, often requiring a prohibitive amount of GPU memory that may not be readily available. Existing methods such as low-rank adaptation (LoRA) add trainable low-rank matrix factorizations, altering the training dynamics and limiting the model's parameter search to a low-rank subspace. GaLore, a more recent method, employs Gradient Low-Rank Projection to reduce the memory footprint, in the full parameter training setting. However GaLore can only be applied to a subset of the LLM layers that satisfy the "reversibility" property, thus limiting their applicability. In response to these challenges, we introduce BlockLLM, an approach inspired by block coordinate descent. Our method carefully selects and updates a very small subset of the trainable parameters without altering any part of its architecture and training procedure. BlockLLM achieves state-of-the-art performance in both finetuning and pretraining tasks, while reducing the memory footprint of the underlying optimization process. Our experiments demonstrate that fine-tuning with only less than 5% of the parameters, BlockLLM achieves state-of-the-art perplexity scores on the GLUE benchmarks. On Llama model pretrained on C4 dataset, BlockLLM is able to train with significantly less memory than the state-of-the-art, while still maintaining competitive performance.

Enhancing Policy Gradient with the Polyak Step-Size Adaption

Policy gradient is a widely utilized and foundational algorithm in the field of reinforcement learning (RL). Renowned for its convergence guarantees and stability compared to other RL algorithms, its practical application is often hindered by sensitivity to hyper-parameters, particularly the step-size. In this paper, we introduce the integration of the Polyak step-size in RL, which automatically adjusts the step-size without prior knowledge. To adapt this method to RL settings, we address several issues, including unknown f* in the Polyak step-size. Additionally, we showcase the performance of the Polyak step-size in RL through experiments, demonstrating faster convergence and the attainment of more stable policies.

Faster Convergence of Stochastic Accelerated Gradient Descent under Interpolation

We prove new convergence rates for a generalized version of stochastic Nesterov acceleration under interpolation conditions. Unlike previous analyses, our approach accelerates any stochastic gradient method which makes sufficient progress in expectation. The proof, which proceeds using the estimating sequences framework, applies to both convex and strongly convex functions and is easily specialized to accelerated SGD under the strong growth condition. In this special case, our analysis reduces the dependence on the strong growth constant from $\rho$ to $\sqrt{\rho}$ as compared to prior work. This improvement is comparable to a square-root of the condition number in the worst case and address criticism that guarantees for stochastic acceleration could be worse than those for SGD.

Heavy-Tailed Class Imbalance and Why Adam Outperforms Gradient Descent on Language Models

Adam has been shown to outperform gradient descent in optimizing large language transformers empirically, and by a larger margin than on other tasks, but it is unclear why this happens. We show that the heavy-tailed class imbalance found in language modeling tasks leads to difficulties in the optimization dynamics. When training with gradient descent, the loss associated with infrequent words decreases slower than the loss associated with frequent ones. As most samples come from relatively infrequent words, the average loss decreases slowly with gradient descent. On the other hand, Adam and sign-based methods do not suffer from this problem and improve predictions on all classes. To establish that this behavior is indeed caused by class imbalance, we show empirically that it persist through different architectures and data types, on language transformers, vision CNNs, and linear models. We further study this phenomenon on a linear classification with cross-entropy loss, showing that heavy-tailed class imbalance leads to ill-conditioning, and that the normalization used by Adam can counteract it.

Analyzing and Improving Greedy 2-Coordinate Updates for Equality-Constrained Optimization via Steepest Descent in the 1-Norm

We consider minimizing a smooth function subject to a summation constraint over its variables. By exploiting a connection between the greedy 2-coordinate update for this problem and equality-constrained steepest descent in the 1-norm, we give a convergence rate for greedy selection under a proximal Polyak-Lojasiewicz assumption that is faster than random selection and independent of the problem dimension $n$. We then consider minimizing with both a summation constraint and bound constraints, as arises in the support vector machine dual problem. Existing greedy rules for this setting either guarantee trivial progress only or require $O(n^2)$ time to compute. We show that bound- and summation-constrained steepest descent in the L1-norm guarantees more progress per iteration than previous rules and can be computed in only $O(n \log n)$ time.

Don't be so Monotone: Relaxing Stochastic Line Search in Over-Parameterized Models

Recent works have shown that line search methods can speed up Stochastic Gradient Descent (SGD) and Adam in modern over-parameterized settings. However, existing line searches may take steps that are smaller than necessary since they require a monotone decrease of the (mini-)batch objective function. We explore nonmonotone line search methods to relax this condition and possibly accept larger step sizes. Despite the lack of a monotonic decrease, we prove the same fast rates of convergence as in the monotone case. Our experiments show that nonmonotone methods improve the speed of convergence and generalization properties of SGD/Adam even beyond the previous monotone line searches. We propose a POlyak NOnmonotone Stochastic (PoNoS) method, obtained by combining a nonmonotone line search with a Polyak initial step size. Furthermore, we develop a new resetting technique that in the majority of the iterations reduces the amount of backtracks to zero while still maintaining a large initial step size. To the best of our knowledge, a first runtime comparison shows that the epoch-wise advantage of line-search-based methods gets reflected in the overall computational time.

Searching for Optimal Per-Coordinate Step-sizes with Multidimensional Backtracking

The backtracking line-search is an effective technique to automatically tune the step-size in smooth optimization. It guarantees similar performance to using the theoretically optimal step-size. Many approaches have been developed to instead tune per-coordinate step-sizes, also known as diagonal preconditioners, but none of the existing methods are provably competitive with the optimal per-coordinate stepsizes. We propose multidimensional backtracking, an extension of the backtracking line-search to find good diagonal preconditioners for smooth convex problems. Our key insight is that the gradient with respect to the step-sizes, also known as hypergradients, yields separating hyperplanes that let us search for good preconditioners using cutting-plane methods. As black-box cutting-plane approaches like the ellipsoid method are computationally prohibitive, we develop an efficient algorithm tailored to our setting. Multidimensional backtracking is provably competitive with the best diagonal preconditioner and requires no manual tuning.