Projection-free first-order methods, e.g., the celebrated Frank-Wolfe (FW) algorithms, have emerged as powerful tools for optimization over simple convex sets such as polyhedra, because of their scalability, fast convergence, and iteration-wise feasibility without costly projections. However, extending these methods effectively to general compact convex sets remains challenging and largely open, as FW methods rely on expensive linear optimization oracles (LOO), while penalty-based methods often struggle with poor feasibility. We tackle this open challenge by presenting **Hom-PGD**, a novel projection-free method without expensive (optimization) oracles. Our method constructs a homeomorphism between the convex constraint set and a unit ball, transforming the original problem into an equivalent ball-constrained formulation, thus enabling efficient gradient-based optimization while preserving the original problem structure. We prove that Hom-PGD attains *optimal* convergence rates matching gradient descent with constant step-size to find an $\epsilon$-approximate (stationary) solution: $\mathcal{O}(\log (1/\epsilon))$ for strongly convex objectives, $\mathcal{O}(\epsilon^{-1})$ for convex objectives, and $\mathcal{O}(\epsilon^{-2})$ for non-convex objectives. Meanwhile, Hom-PGD enjoys a low per-iteration complexity of $\mathcal{O}(n^2)$, without expensive oracles like LOO or projection, where $n$ is the input size. Our framework further extends to certain non-convex sets, broadening its applicability in practical optimization scenarios with complex constraints. Extensive numerical experiments demonstrate that Hom-PGD achieves comparable convergence rates to state-of-the-art projection-free methods, while significantly reducing per-iteration runtime (up to 5 orders of magnitude faster) and thus the total problem-solving time.

We present and analyze an algorithm for optimizing smooth and convex or strongly convex objectives using minibatch stochastic gradient estimates. The algorithm is optimal with respect to its dependence on both the minibatch size and minimum expected loss simultaneously. This improves over the optimal method of Lan [17], which is insensitive to the minimum expected loss; over the optimistic acceleration of Cotter et al. [10], which has suboptimal dependence on the minibatch size; and over the algorithm of Liu and Belkin [19], which is limited to least squares problems and is also similarly suboptimal with respect to the minibatch size.

We cast this problem as stochastic convex optimization with heavy tailed stochastic gradients, and prove that the widely used Clipped-SGD algorithm attains near-optimal sub-Gaussian statistical rates whenever the second moment of the stochastic gradient noise is finite.

We develop a new accelerated stochastic gradient method for efficiently solving the convex regularized empirical risk minimization problem in mini-batch settings. The use of mini-batches has become a golden standard in the machine learning community, because the mini-batch techniques stabilize the gradient estimate and can easily make good use of parallel computing. The core of our proposed method is the incorporation of our new ``double acceleration'' technique and variance reduction technique. We theoretically analyze our proposed method and show that our method much improves the mini-batch efficiencies of previous accelerated stochastic methods, and essentially only needs size $\sqrt{n}$ mini-batches for achieving the optimal iteration complexities for both non-strongly and strongly convex objectives, where $n$ is the training set size. Further, we show that even in non-mini-batch settings, our method achieves the best known convergence rate for non-strongly convex and strongly convex objectives.

Greedy optimization methods such as Matching Pursuit (MP) and Frank-Wolfe (FW) algorithms regained popularity in recent years due to their simplicity, effectiveness and theoretical guarantees. MP and FW address optimization over the linear span and the convex hull of a set of atoms, respectively. In this paper, we consider the intermediate case of optimization over the convex cone, parametrized as the conic hull of a generic atom set, leading to the first principled definitions of non-negative MP algorithms for which we give explicit convergence rates and demonstrate excellent empirical performance. In particular, we derive sublinear (O(1/t)) convergence on general smooth and convex objectives, and linear convergence (O(e^{-t})) on strongly convex objectives, in both cases for general sets of atoms. Furthermore, we establish a clear correspondence of our algorithms to known algorithms from the MP and FW literature. Our novel algorithms and analyses target general atom sets and general objective functions, and hence are directly applicable to a large variety of learning settings.

We propose the algorithms for online convex optimization which lead to cumulative squared constraint violations of the form $\sum\limits_{t=1}^T\big([g(x_t)]_+\big)^2=O(T^{1-\beta})$, where $\beta\in(0,1)$. Previous literature has focused on long-term constraints of the form $\sum\limits_{t=1}^Tg(x_t)$. There, strictly feasible solutions can cancel out the effects of violated constraints. In contrast, the new form heavily penalizes large constraint violations and cancellation effects cannot occur. Furthermore, useful bounds on the single step constraint violation $[g(x_t)]_+$ are derived. For convex objectives, our regret bounds generalize existing bounds, and for strongly convex objectives we give improved regret bounds. In numerical experiments, we show that our algorithm closely follows the constraint boundary leading to low cumulative violation.

Thanks totheir scalability and lowmemory requirements, first-order methods are especially popular inthis setting (Bottouetal., 2018).

We develop a new accelerated stochastic gradient method for efficiently solving the convex regularized empirical risk minimization problem in mini-batch settings. The use of mini-batches has become a golden standard in the machine learning community, because the mini-batch techniques stabilize the gradient estimate and can easily make good use of parallel computing. The core of our proposed method is the incorporation of our new ``double acceleration'' technique and variance reduction technique. We theoretically analyze our proposed method and show that our method much improves the mini-batch efficiencies of previous accelerated stochastic methods, and essentially only needs size $\sqrt{n}$ mini-batches for achieving the optimal iteration complexities for both non-strongly and strongly convex objectives, where $n$ is the training set size. Further, we show that even in non-mini-batch settings, our method achieves the best known convergence rate for non-strongly convex and strongly convex objectives.