We study the bidding problem in repeated uniform price multi-unit auctions from the perspective of a value-maximizing buyer. The buyer aims to maximize their cumulative value over $T$ rounds while adhering to per-round return-on-investment (RoI) constraints in a strategic (or adversarial) environment. Using an $m$-uniform bidding format, the buyer submits $m$ bid-quantity pairs $(b_i, q_i)$ to demand $q_i$ units at bid $b_i$, with $m \ll M$ in practice, where $M$ denotes the maximum demand of the buyer. We introduce the notion of safe bidding strategies as those that satisfy the RoI constraints irrespective of competing bids. Despite the stringent requirement, we show that these strategies satisfy a mild no-overbidding condition, depend only on the valuation curve of the bidder, and the bidder can focus on a finite subset without loss of generality. Though the subset size is $O(M^m)$, we design a polynomial-time learning algorithm that achieves sublinear regret, both in full-information and bandit settings, relative to the hindsight-optimal safe strategy. We assess the robustness of safe strategies against the hindsight-optimal strategy from a richer class. We define the richness ratio $α\in (0,1]$ as the minimum ratio of the value of the optimal safe strategy to that of the optimal strategy from richer class and construct hard instances showing the tightness of $α$. Our algorithm achieves $α$-approximate sublinear regret against these stronger benchmarks. Simulations on semi-synthetic auction data show that empirical richness ratios significantly outperform the theoretical worst-case bounds. The proposed safe strategies and learning algorithm extend naturally to more nuanced buyer and competitor models.